<Text-field style="Heading 1" layout="Heading 1">Equation for m-Dyck paths (ascents are multiples of m), Section 4.1.</Text-field>

restart:

libname:="/home/bousquet/Calculs/Gfun/lib", libname;

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with(gfun)

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

gfun[version]

JCIkKVIhIiM=

with(plots):

<Text-field style="Heading 1" layout="Heading 1">Equation for m-Dyck paths (ascents are multiples of m), Section 4.1.</Text-field>

Equation for the series G_m, counting interval of m-Dyck paths

eqG:=proc(m) -G(x,y) +(x*y)^m*t + (x*y)^m*t*G(x,y) + t*y*(x*y)^m/(x-1)/(y-1)*(G(x,y)-G(1,y)) - (x*y)^m*t*(G(x*y,1)-G(1,1))/(y-1)/(x*y-1): end:

eqG(1); eqG(2)

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LCwtSSJHRzYiNiRJInhHRiVJInlHRiUhIiIqKEYnIiIjRihGK0kidEdGJSIiIkYtKipGJ0YrRihGK0YsRi1GI0YtRi0qLkYsRi1GKCIiJEYnRissJkYnRi1GKUYtRiksJkYoRi1GKUYtRiksJkYjRi0tRiQ2JEYtRihGKUYtRi0qLkYnRitGKEYrRixGLSwmLUYkNiQqJkYnRi1GKEYtRi1GLS1GJDYkRi1GLUYpRi1GMkYpLCZGOkYtRilGLUYpRik=

eqG(m)

LCwtSSJHRzYiNiRJInhHRiVJInlHRiUhIiIqJikqJkYnIiIiRihGLUkibUdGJUYtSSJ0R0YlRi1GLSooRitGLUYvRi1GI0YtRi0qLkYvRi1GKEYtRitGLSwmRidGLUYpRi1GKSwmRihGLUYpRi1GKSwmRiNGLS1GJDYkRi1GKEYpRi1GLSosRitGLUYvRi0sJi1GJDYkRixGLUYtLUYkNiRGLUYtRilGLUYzRiksJkYsRi1GKUYtRilGKQ==

Equation for the associated series Q(x,y)

eqQ:=proc(m) map(Z->factor(Z/t/x^m),subs(x=x/y,G(x/y, y)= t*x^m*Q(x,y),G(1, y)= t*y^m*Q(y,y), G(x, 1)= t*x^m*Q(x,1),G(1, 1)= t*Q(1,1),eqG(m))): end:

eqQ(m)

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Generating coefficients from the functional equation

Gser:=proc(m,n) option remember: if n<1 then 0 else convert(map(factor,series(subs(G(x,y)=Gser(m,n-1), G(1,y)=subs(x=1,Gser(m,n-1)), G(1,1)=subs(x=1,y=1,Gser(m,n-1)), G(x*y,1)=subs(y=1,x=x*y,Gser(m,n-1)),eqG(m)+G(x,y)), t,n+1)),polynom): fi: end:

Gser(1,3);

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G(x,y) is a formal power series in (xy) and y

collect(subs(x=x/y,%),t,factor)

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Value at (1,1)

subs(x=1,y=1,Gser(1,10))

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seriestolist(series(%,t,11))

Ny0iIiEiIiIiIiQiIzgiI3AiJDwlIiVmRiImPCY+IidgYDkiKFx3NyIiKCh5ViEq

In the OEIS !

Gser(2,3);

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subs(x=1,y=1,Gser(2,10))

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Not in the OEIS

seriestolist(series(%,t,11))

Ny0iIiEiIiIiIiciI2kiJFspIiYjeTgiJ009RCIoKFE0XSIqdTRGMSIiKyZbI0hyQiIsOXJqNV4m

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<Text-field style="Heading 1" layout="Heading 1">Equation for mirrored m-Dyck paths (descents are multiples of m), Section 4.2. </Text-field>

eqGp:=proc(m) -G(x,y) +x^m*t*y + t*x^m*y*(G(x,y)-G(x,1))/(y-1) +t*y^2*x^m/(x-1)/(y-1)*(G(x,y)-G(1,y)) - t*x^m*y*(G(x,1)-G(1,1))/(x-1)/(y-1) : end:

eqGp(1); eqGp(2)

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LCwtSSJHRzYiNiRJInhHRiVJInlHRiUhIiIqKEYnIiIjSSJ0R0YlIiIiRihGLUYtKixGLEYtRidGK0YoRi0sJkYjRi0tRiQ2JEYnRi1GKUYtLCZGKEYtRilGLUYpRi0qLkYsRi1GKEYrRidGKywmRidGLUYpRi1GKUYyRiksJkYjRi0tRiQ2JEYtRihGKUYtRi0qLkYsRi1GJ0YrRihGLSwmRjBGLS1GJDYkRi1GLUYpRi1GNEYpRjJGKUYp

eqGp(m)

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Symmetric equation, for m=1

collect(eqGp(1),G,factor)

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Ker:=coeff(%,G(x,y)); factor(Ker-subs({x=y,y=x},Ker))

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IiIh

factor(eqGp(1)+G(x,y)-t*x*y)

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eqQp:=proc(m) map(Z->factor(Z/t/y/x^m),subs(G(x,y)=t*x^m*y*Q(x,y), G(x,1)=t*x^m*Q(x,1), G(1,y)=t*y*Q(1,y), G(1,1)=t*Q(1,1), eqGp(m))): end:

eqQp(m)

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Generating coefficients from the functional equation

Gpser:=proc(m,n) option remember: if n<1 then 0 else convert(map(factor,series(subs(G(x,y)=Gpser(m,n-1), G(1,y)=subs(x=1,Gpser(m,n-1)),G(1,1)=subs(x=1,y=1,Gpser(m,n-1)),G(x,1)=subs(y=1,Gpser(m,n-1)),eqGp(m)+G(x,y)), t,n+1)),polynom): fi: end:

Gpser(1,4);

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Value at (1,1)

subs(x=1,y=1,Gpser(1,10))

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Gpser(2,3);

LCgqKEkieEc2IiIiI0kidEdGJSIiIkkieUdGJUYoRigqKkYpRihGJEYmLCwqJEYkRiZGKComRiRGKEYpRihGKEYkRihGKUYoRihGKEYoRidGJkYoKipGKUYoRiRGJiw6KiRGJCIiJUYoKiZGJCIiJEYpRihGJiomRiRGJkYpRiZGKCokRiRGM0YzKiZGJEYmRilGKEYxKiZGJEYoRilGJkYmRiwiIiZGLUY4KiRGKUYmRiZGJEY4RilGOEY4RihGKEYnRjNGKA==

subs(x=1,y=1,Gpser(2,10))

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OEIS

seriestolist(series(%,t,11))

Ny0iIiEiIiIiIiYiI1MiJDIlIiUoeiUiJlhDJyInQEwoKSIpbVcqRyIiKjVFayk+IislPVtrOyQ=

subs(x=1,y=1,Gpser(3,10))

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OEIS

seriestolist(series(%,t,11))

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Ny0iIiEiIiIiIiYiI1MiJDIlIiUoeiUiJlhDJyInQEwoKSIpbVcqRyIiKjVFayk+IislPVtrOyQ=

subs(x=1,y=1,Gpser(4,10))

LDYqJEkidEc2IiIjNSIubllTK19RIiokRiQiIioiLERCLXFMJSokRiQiIikiKyJcU2BTIiokRiQiIigiKVdfYFoqJEYkIiInIig4IipwIiokRiQiIiYiJnZgJyokRiQiIiUiJSp5IyokRiQiIiQiJFIiKiRGJCIiI0YpRiQiIiI=

OEIS

seriestolist(series(%,t,11))

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subs(x=1,y=1,Gpser(5,10))

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OEIS

seriestolist(series(%,t,11))

Ny0iIiEiIiIiIzYiJDYjIiUiRyYiJ2hZOiIoJHBEXSIqJlFZZTwiK2lwKkddJyItJkg0SS9eIyIvJHlbbT9JKyI=

subs(x=1,y=1,Gpser(6,10))

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OEIS

seriestolist(series(%,t,11))

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Walks with finitely many steps

eqR:=proc(m) collect(subs(x=u+1,y=v+1,Q(u + 1, v + 1)=R(u,v),Q(u + 1, 1)=R(u,0),Q(1, v + 1)=R(0,v),Q(1, 1)=R(0,0), eqQp(m)+R(u,v)),R,factor)-R(u,v): end:

eqR(m)

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coeff(coeff(%,R(u,v)),t,1); expand(subs(m=1,%))

KiwpLCZJInVHNiIiIiJGJ0YnSSJtR0YmRicsJkkidkdGJkYnRidGJ0YnLChGJUYnRipGJ0YnRidGJ0YlISIiRipGLA==

LDJJInVHNiIiIiJJInZHRiRGJSomRiNGJUYmISIiRiUiIiRGJSomRiNGKEYmRiVGJSomIiIjRiVGJkYoRiUqJkYsRiVGI0YoRiUqJkYjRihGJkYoRiU=

<Text-field style="Heading 1" layout="Heading 1">The case m=1: Recycling earlier quadrant walks results to determine G'(x,y) (Section 5.1)</Text-field>

We start from the equation for the series G'

eqGp(1)

LCwtSSJHRzYiNiRJInhHRiVJInlHRiUhIiIqKEYnIiIiRihGK0kidEdGJUYrRisqLEYsRitGJ0YrRihGKywmRiNGKy1GJDYkRidGK0YpRissJkYoRitGK0YpRilGKyouRixGKylGKCIiI0YrRidGKywmRidGK0YrRilGKUYxRiksJkYjRistRiQ2JEYrRihGKUYrRisqLkYsRitGJ0YrRihGKywmRi9GKy1GJDYkRitGK0YpRitGNUYpRjFGKUYp

eqQp(1)

LCwtSSJRRzYiNiRJInhHRiVJInlHRiUhIiIiIiJGKioqRidGKkkidEdGJUYqLCYqJkYjRipGKEYqRiktRiQ2JEYnRipGKkYqLCZGKEYqRipGKUYpRikqLClGKCIiI0YqRixGKiwmKiZGI0YqRidGKkYpLUYkNiRGKkYoRipGKkYxRiksJkYnRipGKkYpRilGKSoqRixGKiwmKiZGL0YqRidGKkYpLUYkNiRGKkYqRipGKkYxRilGOUYpRio=

eqR(1)

LC4qKkkidEc2IiIiIkkidkdGJSEiIkkidUdGJUYoLUkiUkdGJTYkIiIhRi1GJkYmKixGJEYmKSwmRidGJkYmRiYiIiNGJkYnRihGKUYoLUYrNiRGLUYnRiZGKCosKSwmRilGJkYmRiZGMUYmRiRGJkYpRihGJ0YoLUYrNiRGKUYtRiZGKEYmRiYqMEY2RiZGJEYmRjBGJiwoRilGJkYnRiZGJkYmRiZGKUYoRidGKC1GKzYkRilGJ0YmRiZGO0Yo

Steps:=expand(coeff(coeff(%,R(u,v)),t))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTdGVwc0dGKCwySSJ1R0YoIiIiSSJ2R0YoRjAqJkYvRjBGMSEiIkYwIiIkRjAqJkYvRjNGMUYwRjAqJiIiI0YwRjFGM0YwKiZGN0YwRi9GM0YwKiZGL0YzRjFGM0YwNyNGLg==

Here we recycle the results of the paper by Bernardi, MBM and Raschel on the quadrant model with the above step polynomial (without the null step)

<Text-field style="Heading 2" layout="Heading 2">The series G'(x,1)</Text-field>

The series Z_old of that paper (Appendix A.7)

tZo := Zo/(Zo^3+4*Zo^2+2*Zo+2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0Wm9HRigqJkkjWm9HRigiIiIsKiokKUYvIiIkRjBGMComIiIlRjApRi8iIiNGMEYwKiZGOEYwRi9GMEYwRjhGMCEiIjcjRi4=

algZo:=numer(t-tZo)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdab0dGKCwsKiYpSSNab0dGKCIiJCIiIkkidEdGKEYzRjMqKCIiJUYzKUYxIiIjRjNGNEYzRjMqKEY4RjNGMUYzRjRGM0YzRjEhIiIqJkY4RjNGNEYzRjM3I0Yu

Zoser:=op(algeqtoseries(algZo,t,Zo,10,true))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZab3NlckdGKCs5SSJ0R0YoIiIjIiIiIiIlRjAiI0MiIiQiJD8iRjIiJE8oIiImIiVXWCIiJyImaydIIiIoIidDeT4iIikiKCFvXjgiIioiKD92USoiIzUtSSJPR0YlNiNGMSIjNjcjRi4=

The series Z in the present paper

tZ:=Z/(1+Z)/(1+2*Z)^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WkdGKCooSSJaR0YoIiIiLCZGMEYwRi9GMCEiIiksJkYwRjAqJiIiI0YwRi9GMEYwRjZGMjcjRi4=

algZ:=numer(t-tZ)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCwqKCIiJSIiIilJIlpHRigiIiRGMUkidEdGKEYxRjEqKCIiKUYxKUYzIiIjRjFGNUYxRjEqKCIiJkYxRjNGMUY1RjFGMUYzISIiRjVGMTcjRi4=

Zser:=op(algeqtoseries(algZ,t,Z,10,true))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVac2VyR0YoKzlJInRHRigiIiJGMCIiJiIiIyIjTCIiJCIkXCMiIiUiJUw/RjEiJiZbPCIiJyInTGc6IiIoIigiR0o5IiIpIikkPjdNIiIiKiIqJjNTeTciIzUtSSJPR0YlNiNGMCIjNjcjRi4=

Connection between Z(t) and Zo evaluated at t/(1-3t)

numer(factor(subs(t=tZ,tZo-t/(1-3*t))))

KiYsJiomIiIjIiIiSSJaRzYiRiZGJkkjWm9HRighIiJGJiwqKihGJUYmKUYnRiVGJkYpRiZGJiomRidGJilGKUYlRiZGJiooIiIlRiZGJ0YmRilGJkYmRiZGKkYm

Let us check: OK

series(subs(t=t/(1-3*t),Zoser/2),t,11)

KzlJInRHNiIiIiJGJSIiJiIiIyIjTCIiJCIkXCMiIiUiJUw/RiYiJiZbPCIiJyInTGc6IiIoIigiR0o5IiIpIikkPjdNIiIiKiIqJjNTeTciIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

Expression of Q_old(0,0) in App. A.7

Q00Zo:=1/8*(Zo^3 - 4*Zo + 4)*Zo/t

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZRMDBab0dGKCwkKiojIiIiIiIpRjEsKCokKUkjWm9HRigiIiRGMUYxKiYiIiVGMUY2RjEhIiJGOUYxRjFGNkYxSSJ0R0YoRjpGMTcjRi4=

series(subs(Zo=Zoser,Q00Zo),t,11)

KzdJInRHNiIiIiIiIiEiIiUiIiMiIiciIiQiI1tGJyIkdyIiIiYiJVM1RikiJSFHJiIiKCImPzIkIiIpIidncDwiIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

series(1/(1-3*t)*subs(t=t/(1-3*t),%),t,11)

KzlJInRHNiIiIiIiIiEiIiRGJSIjOCIiIyIjcEYnIiQ8JSIiJSIlZkYiIiYiJjwmPiIiJyInYGA5IiIoIihcdzciIiIpIigoeVYhKiIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM1

subs(x=1,y=1,Gser(1,10))

LDYqJiIoKHlWISoiIiIpSSJ0RzYiIiM1RiVGJSomIihcdzciRiUpRiciIipGJUYlKiYiJ2BgOUYlKUYnIiIpRiVGJSomIiY8Jj5GJSlGJyIiKEYlRiUqJiIlZkZGJSlGJyIiJ0YlRiUqJiIkPCVGJSlGJyIiJkYlRiUqJiIjcEYlKUYnIiIlRiVGJSomIiM4RiUpRiciIiRGJUYlKiZGRUYlKUYnIiIjRiVGJUYnRiU=

Our series R(0,0)

R00Z:=factor(1/(1-3*t)*subs(t=t/(1-3*t), Zo=2*Z,Q00Zo))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVSMDBaR0YoKigsKComIiIjIiIiKUkiWkdGKCIiJEYyRjIqJkYxRjJGNEYyISIiRjJGMkYyRjRGMkkidEdGKEY3NyNGLg==

G11Z:=factor(t*%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVHMTFaR0YoKiYsKComIiIjIiIiKUkiWkdGKCIiJEYyRjIqJkYxRjJGNEYyISIiRjJGMkYyRjRGMjcjRi4=

Now we want G'(x,1)

The equation for Q_old(x,0) in terms of Zo

eqQx:=subs(Z=Zo, y=x,-256*Z^3*y*(y + 1)^4*Q^2 + 32*(Z^3 + 4*Z^2 + 2*Z + 2)*(Z^6*y^3 - 6*Z^6*y^2 + 8*Z^4*y^4 + Z^6*y - 64*Z^5*y^2 + 20*Z^4*y^3 + 32*Z^3*y^4 - 8*Z^2*y^5 - 176*Z^4*y^2 + 100*Z^3*y^3 + 28*Z^4*y - 184*Z^3*y^2 + 56*Z^2*y^3 + 16*Z*y^4 + 8*Z^4 + 132*Z^3*y - 168*Z^2*y^2 + 48*Z*y^3 + 32*Z^3 + 152*Z^2*y - 80*Z*y^2 + 40*Z^2 + 64*Z*y - 32*y^2 + 16*Z)*Q - (Z^3 + 4*Z^2 + 2*Z + 2)^2*(Z^9*y + 32*Z^8*y + 32*Z^7*y^2 + 248*Z^7*y + 128*Z^6*y^2 - 32*Z^5*y^3 + 32*Z^7 + 584*Z^6*y - 32*Z^5*y^2 + 128*Z^6 - 112*Z^5*y - 320*Z^4*y^2 + 128*Z^3*y^3 + 32*Z^5 - 1472*Z^4*y - 128*Z^3*y^2 - 128*Z^2*y^3 + 256*Z*y^4 - 320*Z^4 - 256*Z^3*y - 896*Z^2*y^2 + 512*Z*y^3 - 128*Z^3 + 1536*Z^2*y - 768*Z*y^2 + 384*Z^2 + 1024*Z*y - 512*y^2 + 256*Z))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVF4R0YoLCgqLCIkYyMiIiIpSSNab0dGKCIiJEYxSSJ4R0YoRjEpLCZGNUYxRjFGMSIiJUYxKUkiUUdGKCIiI0YxISIiKioiI0tGMSwqKiRGMkYxRjEqJkY4RjEpRjNGO0YxRjEqJkY7RjFGM0YxRjFGO0YxRjEsVComKUYzIiInRjEpRjVGNEYxRjEqKEZHRjFGRkYxKUY1RjtGMUY8KigiIilGMSlGM0Y4RjEpRjVGOEYxRjEqJkZGRjFGNUYxRjEqKCIja0YxKUYzIiImRjFGSkYxRjwqKCIjP0YxRk1GMUZIRjFGMSooRj5GMUYyRjFGTkYxRjEqKEZMRjFGQkYxKUY1RlNGMUY8KigiJHciRjFGTUYxRkpGMUY8KigiJCsiRjFGMkYxRkhGMUYxKigiI0dGMUZNRjFGNUYxRjEqKCIkJT1GMUYyRjFGSkYxRjwqKCIjY0YxRkJGMUZIRjFGMSooIiM7RjFGM0YxRk5GMUYxKiZGTEYxRk1GMUYxKigiJEsiRjFGMkYxRjVGMUYxKigiJG8iRjFGQkYxRkpGMUY8KigiI1tGMUYzRjFGSEYxRjEqJkY+RjFGMkYxRjEqKCIkXyJGMUZCRjFGNUYxRjEqKCIjISlGMUYzRjFGSkYxRjwqJiIjU0YxRkJGMUYxKihGUUYxRjNGMUY1RjFGMSomRj5GMUZKRjFGPComRl5vRjFGM0YxRjFGMUY6RjFGMSomKUY/RjtGMSxmbiomKUYzIiIqRjFGNUYxRjEqKEY+RjEpRjNGTEYxRjVGMUYxKihGPkYxKUYzIiIoRjFGSkYxRjEqKCIkWyNGMUZpcEYxRjVGMUYxKigiJEciRjFGRkYxRkpGMUYxKihGPkYxRlJGMUZIRjFGPComRj5GMUZpcEYxRjEqKCIkJWVGMUZGRjFGNUYxRjEqKEY+RjFGUkYxRkpGMUY8KiZGXnFGMUZGRjFGMSooIiQ3IkYxRlJGMUY1RjFGPCooIiQ/JEYxRk1GMUZKRjFGPCooRl5xRjFGMkYxRkhGMUYxKiZGPkYxRlJGMUYxKigiJXM5RjFGTUYxRjVGMUY8KihGXnFGMUYyRjFGSkYxRjwqKEZecUYxRkJGMUZIRjFGPCooRjBGMUYzRjFGTkYxRjEqJkZocUYxRk1GMUY8KihGMEYxRjJGMUY1RjFGPCooIiQnKilGMUZCRjFGSkYxRjwqKCIkNyZGMUYzRjFGSEYxRjEqJkZecUYxRjJGMUY8KigiJU86RjFGQkYxRjVGMUYxKigiJG8oRjFGM0YxRkpGMUY8KiYiJCVRRjFGQkYxRjEqKCIlQzVGMUYzRjFGNUYxRjEqJkZlckYxRkpGMUY8KiZGMEYxRjNGMUYxRjFGPDcjRi4=

We derive from this an equation for R(x,0)

eqR:=collect(op(3,numer(factor(subs({Q=(1-3*t)*R,Zo=2*Z},t=tZ,eqQx)))),R,factor)

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

and then an equation for G'(x,1)

eqGZ:=collect(op(3,numer(map(factor,subs(x=x-1,R=G/tZ/x,eqR)))),G,factor)

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

Let us check

Gpser(1,4)

LCoqKEkieEc2IiIiIkkieUdGJUYmSSJ0R0YlRiZGJioqRiRGJkYnRiYsKEYkRiZGJ0YmRiZGJkYmKUYoIiIjRiZGJioqRiRGJkYnRiYsLiokKUYkRixGJkYmKihGLEYmRiRGJkYnRiZGJiokKUYnRixGJkYmKiYiIiRGJkYkRiZGJiomRjVGJkYnRiZGJkY1RiZGJilGKEY1RiZGJioqRiRGJkYnRiYsNiokKUYkRjVGJkYmKihGNUYmRjBGJkYnRiZGJiooRjVGJkYkRiZGM0YmRiYqJClGJ0Y1RiZGJiomIiInRiZGMEYmRiYqKCIjNUYmRiRGJkYnRiZGJiomRkFGJkYzRiZGJiomIiM4RiZGJEYmRiYqJkZGRiZGJ0YmRiZGRkYmRiYpRigiIiVGJkYm

N:=7: factor(series(subs(Z=Zser,G=subs(y=1,Gpser(1,N)),eqGZ),t,N+1))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

Degree 1 when x=1

factor(subs(x=1,eqGZ))

LCQqKkkiWkc2IiIiIiwmRiZGJkYkRiZGJiksJkYmRiYqJiIiI0YmRiRGJkYmRitGJiwqKiZGK0YmKUYkIiIlRiYhIiIqJkYrRiYpRiRGK0YmRiZJIkdHRiVGJkYkRjBGJkYw

disc:=factor(discrim(eqGZ,G))

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

This is the quantity denoted Delta in the paper

discs:=-op(2,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkaXNjc0dGKCw2KigiIiUiIiIpSSJaR0YoRjBGMUkieEdGKEYxISIiKiZGMEYxRjJGMUYxKigiIzdGMSlGMyIiJEYxRjRGMUY1KiYpRjMiIiNGMSlGNEY9RjFGMSomRjhGMUY5RjFGMSooIiM1RjFGPEYxRjRGMUY1KiYiIzhGMUY8RjFGMSooRj1GMUYzRjFGNEYxRjUqJiIiJ0YxRjNGMUYxRjFGMTcjRi4=

collect(discs,x,factor); latex(%)

LCgqJilJIlpHNiIiIiMiIiIpSSJ4R0YmRidGKEYoKixGJ0YoRiVGKCwmRihGKEYlRihGKCwoKiZGJ0YoRiRGKEYoKiYiIiVGKEYlRihGKEYoRihGKEYqRighIiIqJilGLEYnRigpLCZGKEYoKiZGJ0YoRiVGKEYoRidGKEYo

Z^{2} x^{2}-2 Z \134left(1+Z \134right) \134left(2 Z^{2}+4 Z +1\134right) x +\134left(1+Z \134right)^{2} \134left(1+2 Z \134right)^{2}

Gx1solens:=simplify(factor([solve(eqGZ,G)]),symbolic)

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

The correct solution is the first one

map(factor,map(series,Gx1solens,Z,4))

NyQrJ0kiWkc2IkkieEdGJSIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YnIiIjKy1GJCwkKiYsJkYmRidGJyEiIkYnKUYmIiIkRjNGMyEiIyooLCwqJClGJiIiJUYnRicqJEY0RidGMyomIiIpRicpRiZGLkYnRjMqJiIjPkYnRiZGJ0YnIiM1RjNGJ0Y0RjNGMkYzRjMsJCooLC4qJClGJiIiJkYnRicqJkZIRidGOkYnRjMqJkY1RidGNEYnRicqJiIjS0YnRj9GJ0YnKiYiI3hGJ0YmRidGMyIjVEYnRidGNEYzRjJGM0YzIiIhLCQqKCwuRkZGJyomIiIqRidGOkYnRjMqJiIiKEYnRjRGJ0YnKiYiI3RGJ0Y/RidGJyomIiRvIkYnRiZGJ0YzIiMpKUYnRidGNEYzRjJGM0YzRidGKEYu

From now on we just rewrite the solution

subs(sqrt(Z^2*x^2 + (-4*Z^4 - 12*Z^3 - 10*Z^2 - 2*Z)*x + 4*(Z + 1/2)^2*(1 + Z)^2)=sDelta, Gx1solens[1]*2*Z^2*x^3*(1-x))

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

c1:=factor(-coeff(%,sDelta,1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKComLCwqKCIiIyIiIilJIlpHRihGMUYySSJ4R0YoRjJGMiomRjRGMilGNUYxRjJGMiomIiIlRjJGM0YyISIiKiZGOUYyRjRGMkY6RjJGOkYyLDBGMEYyKiZGMUYyRjNGMkY6KihGOUYyRjRGMkY1RjJGMiokRjdGMkY6KiYiIiRGMkY0RjJGOiomRjFGMkY1RjJGMkYyRjpGMjcjRi4=

c0:=factor(coeff(%%,sDelta,0))

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

collect(c0,x,factor)

LC4qJilJIlpHNiIiIiMiIiIpSSJ4R0YmIiImRihGKCoqRiVGKCwmRihGKEYlRihGKCksJkYoRigqJkYnRihGJUYoRihGJ0YoKUYqIiIlRighIiIqKEYlRigsLComRidGKClGJUYrRihGKComIiInRigpRiUiIiRGKEYzKiYiIihGKEYkRihGMyomRjtGKEYlRihGM0YoRjNGKClGKkY7RihGMyooRi1GKCwuKiYiIz1GKEY3RihGKComIiNZRigpRiVGMkYoRigqJiIjW0YoRjpGKEYoKiYiI0RGKEYkRihGKComRj1GKEYlRihGKEYoRihGKClGKkYnRihGKCoqRi1GKCwqKiYiIilGKEY6RihGKComIiM7RihGJEYoRigqJiIiKkYoRiVGKEYoRidGKEYoRi5GKEYqRihGMyomKUYtRidGKClGL0YyRihGKA==

latex(%)

Z^{2} x^{5}-Z \134left(1+Z \134right) \134left(1+2 Z \134right)^{2} x^{4}-Z \134left(2 Z^{5}-6 Z^{3}-7 Z^{2}-3 Z -1\134right) x^{3}+\134left(1+Z \134right) \134left(18 Z^{5}+46 Z^{4}+48 Z^{3}+25 Z^{2}+7 Z +1\134right) x^{2}-\134left(1+Z \134right) \134left(8 Z^{3}+16 Z^{2}+9 Z +2\134right) \134left(1+2 Z \134right)^{2} x +\134left(1+Z \134right)^{2} \134left(1+2 Z \134right)^{4}

collect(-op(1,c1),x,factor)

LCgqJkkiWkc2IiIiIilJInhHRiUiIiNGJiEiIiooRilGJilGJEYpRiZGKEYmRioqJCksJkYmRiYqJkYpRiZGJEYmRiZGKUYmRiY=

latex(%)

-Z \134,x^{2}-2 Z^{2} x +\134left(1+2 Z \134right)^{2}

collect(-op(2,c1),x,factor)

LCgqJClJInhHNiIiIiMiIiJGKCooRidGKCksJkYoRihJIlpHRiZGKEYnRihGJUYoISIiKiZGK0YoLCZGKEYoKiZGJ0YoRixGKEYoRihGKA==

latex(%)

x^{2}-2 \134left(1+Z \134right)^{2} x +\134left(1+Z \134right) \134left(1+2 Z \134right)

<Text-field style="Heading 2" layout="Heading 2">Asymptotics of the coefficients of G'(1,1)</Text-field>

Now let's study singularities of Z and G11

algZ

LCwqKCIiJSIiIilJIlpHNiIiIiRGJUkidEdGKEYlRiUqKCIiKUYlKUYnIiIjRiVGKkYlRiUqKCIiJkYlRidGJUYqRiVGJUYnISIiRipGJQ==

factor(discrim(algZ,Z))

LCQqKCIjOyIiIkkidEc2IkYlLCgqJClGJiIiI0YlRiUqJiIjNkYlRiZGJUYlRiUhIiJGJUYu

op(3,%)

LCgqJClJInRHNiIiIiMiIiJGKComIiM2RihGJUYoRihGKCEiIg==

solve(%)

NiQsJiMiIzYiIiMhIiIqJiMiIiZGJiIiIilGKiNGK0YmRitGKywmRiRGJ0YoRic=

evalf([%])

NyQkIilVKnAsKiEiKiQhKyUqcCw0NiEiKQ==

A unique dominant singularity

tc:=-11/2 + (5*sqrt(5))/2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0Y0dGKCwmIyIjNiIiIyEiIiomIyIiJkYxIiIiKUY1I0Y2RjFGNkY2NyNGLg==

The growth constant

mu1:=rationalize(1/tc)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRtdTFHRigsJiMiIzYiIiMiIiIqJiMiIiZGMUYyKUY1I0YyRjFGMkYyNyNGLg==

Singular expansion of Z at tc? Theory (Drmota-Lalley-Woods) tells us that Z has a square root singularity

with(plots):

implicitplot(algZ,t=0..0.11,Z=0..1)

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The first series does not have the square root singularity.

map(allvalues,algeqtoseries(subs(t=tc*(1-epsilon),algZ),epsilon, Z, 4, true))

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

Zsing:=op(3,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZac2luZ0dGKCwuKiYjIiIiIiIlRjEpIiImI0YxIiIjRjFGMUYwISIiKigjRjEiIz9GMSksJiIjXUYxKiYiI0lGMUYzRjFGMUY1RjEpSShlcHNpbG9uR0YoRjVGMUY3KiYsJiomI0YxIiM1RjFGM0YxRjEjRjFGNEYxRjFGQUYxRjEqJiwmKiYjIiM2IiQrJUYxRjtGMUY3KigjRjFGTUYxRjtGMUYzRjFGN0YxKUZBIyIiJEY2RjFGMS1JIk9HRiU2IyokKUZBRjZGMUYxNyNGLg==

Back to G11

map(simplify,series(subs(Z=Zsing,G11Z),epsilon,10))

LCgqKCMiIiIiIztGJSwmKiQpIiImI0YlIiIjRiVGJSIiJSEiIkYlLCZGKEYlRiVGLkYlRi4qJiNGJUYtRiVJKGVwc2lsb25HNiJGJUYuLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGMzYjKiQpRjJGLEYlRiU=

One needs to go further in the expansion of Z

Zsing:=map(simplify,op(3,map(allvalues,algeqtoseries(subs(t=tc*(1-epsilon),algZ),epsilon, Z, 6, true))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZac2luZ0dGKCwyKiYjIiIiIiIlRjEpIiImI0YxIiIjRjFGMUYwISIiKigjRjEiIz9GMSksJiIjXUYxKiYiI0lGMUYzRjFGMUY1RjEpSShlcHNpbG9uR0YoRjVGMUY3KigjRjEiIzVGMSwmKiRGM0YxRjFGNkYxRjFGQUYxRjEqKiNGMSIkKyVGMUY7RjEsJiIjNkYxRkZGMUYxKUZBIyIiJEY2RjFGNyooI0YxIiQrJkYxLCYqJiIjUEYxRjNGMUYxIiN2RjFGMSlGQUY2RjFGMSoqI0YxIiYrKyVGMUY7RjEsJiIkUClGMSomIiMoKkYxRjNGMUYxRjEpRkEjRjRGNkYxRjctSSJPR0YlNiMqJClGQUZORjFGMTcjRi4=

map(simplify,series(subs(Z=Zsing,G11Z),epsilon,10))

LCwqKCMiIiIiIztGJSwmKiQpIiImI0YlIiIjRiVGJSIiJSEiIkYlLCZGKEYlRiVGLkYlRi4qJiNGJUYtRiVJKGVwc2lsb25HNiJGJUYuKigjIiIkRiZGJSwmRihGJUYlRiVGJSlGMkYsRiVGJSoqI0YlIiQrIkYlKSwmIiNdRiUqJiIjSUYlRilGJUYlRitGJSwmIiIoRiUqJkY2RiVGKUYlRiVGJSlGMiNGKkYsRiVGLi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRjM2IyokKUYyRjZGJUYl

const:=subs(epsilon=1,op(4,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwkKigjIiIiIiQrIkYxKSwmIiNdRjEqJiIjSUYxKSIiJiNGMSIiI0YxRjFGOkYxLCYiIihGMSomIiIkRjFGOEYxRjFGMSEiIjcjRi4=

unassign('G11')

algG11:=collect(factor(resultant(G11-G11Z,algZ,Z)),G11,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdHMTFHRigsKiooIiRjIyIiIilJJEcxMUdGKCIiJEYxKUkidEdGKCIiJUYxRjEqKiIja0YxKUY2IiIjRjEsKComIiM2RjFGOkYxRjEqJiIjPUYxRjZGMSEiIkYxRkFGMSlGM0Y7RjFGMSomLCwqJiIkVydGMUY1RjFGMSomIiUzPUYxKUY2RjRGMUZBKiYiJF8qRjFGOkYxRjEqJiIkNyJGMUY2RjFGQUY3RjFGMUYzRjFGMSooRjdGMUY2RjEsKiomIiNcRjFGSUYxRjEqJiIkbiJGMUY6RjFGQSomIiNERjFGNkYxRjFGMUZBRjFGMTcjRi4=

implicitplot(algG11,t=0..0.11,G11=0..0.8)

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GAMMA(-5/2)

LCQqJiMiIikiIzoiIiIpSSNQaUclKnByb3RlY3RlZEcjRiciIiNGJyEiIg==

kappa:=simplify(const/%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZrYXBwYUdGKCwkKiojIiIkIiRnIiIiIiksJiIjXUYzKiYiI0lGMykiIiYjRjMiIiNGM0YzRjtGMywmIiIoRjMqJkYxRjNGOUYzRjNGMylJI1BpR0YmRjshIiJGMzcjRi4=

factor(expand(factor(kappa^2))); evalf(kappa)

LCQqKCMiIioiJFMnIiIiLCYiJHYjRicqJiIkQiJGJykiIiYjRiciIiNGJ0YnRidJI1BpRyUqcHJvdGVjdGVkRyEiIkYn

JCIrIjQwInA6ISIq

kappa_paper:=3/8*sqrt((275+123*sqrt(5))/10/Pi)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxrYXBwYV9wYXBlckdGKCwkKigjIiIkIiMhKSIiIikiIzUjRjMiIiNGMykqJiwmIiR2I0YzKiYiJEIiRjMpIiImRjZGM0YzRjNJI1BpR0YmISIiRjZGM0YzNyNGLg==

simplify(kappa-kappa_paper)

IiIh

Let us check numerically

N:=70:ser:=subs(x=1,y=1,Gpser(1,N)):

plot([seq([1/n,coeff(ser,t,n)*tc^n*n^(7/2)],n=1..N)],view=[0..0.2,0.6..1.7],style=point)

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

G11Z

KiYsKComIiIjIiIiKUkiWkc2IiIiJEYmRiYqJkYlRiZGKEYmISIiRiZGJkYmRihGJg==

Connections to walks in a cone (Denisov and Wachtel). The number of excursions in a cone of opening theta has known exponent -Pi/theta -1

subs(theta=2*Pi/5, -Pi/theta -1)

IyEiKCIiIw==

<Text-field style="Heading 2" layout="Heading 2">D-finiteness of G'(1,1) and recurrence relation</Text-field>

algeqtodiffeq(subs(G11=G11(t),algG11),G11(t))

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diffeqtorec(%,G11(t),g(n))

PCUsKComLCYqJiIiIyIiIilJIm5HNiJGJ0YoRihGKkYoRigtSSJnR0YrNiNGKkYoRigqJiwoKiYiI0FGKEYpRihGKComIiMpKUYoRipGKEYoIiMlKUYoRigtRi02IywmRipGKEYoRihGKEYoKiYsKEYmISIiKiYiIzpGKEYqRihGOyIjR0Y7RigtRi02IywmRipGKEYnRihGKEYoLy1GLTYjIiIhRkUvLUYtNiNGKEYo

recg:=map(factor,op(1,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyZWNnR0YoLCgqKEkibkdGKCIiIiwmKiYiIiNGMUYwRjFGMUYxRjFGMS1JImdHRig2I0YwRjFGMSooRjRGMSwoKiYiIzZGMSlGMEY0RjFGMSomIiNXRjFGMEYxRjEiI1VGMUYxLUY2NiMsJkYwRjFGMUYxRjFGMSooLCZGMEYxIiIlRjFGMSwmRjNGMSIiKEYxRjEtRjY2IywmRjBGMUY0RjFGMSEiIjcjRi4=

with(LRETools): MinimalRecurrence(recg,g(n),{g(0) = 0, g(1) = 1})

NicsKCooSSJuRzYiIiIiLCYqJiIiI0YnRiVGJ0YnRidGJ0YnLUkiZ0dGJjYjRiVGJ0YnKihGKkYnLCgqJiIjNkYnKUYlRipGJ0YnKiYiI1dGJ0YlRidGJyIjVUYnRictRiw2IywmRiVGJ0YnRidGJ0YnKigsJkYlRiciIiVGJ0YnLCZGKUYnIiIoRidGJy1GLDYjLCZGJUYnRipGJ0YnISIiUTNSZWxhdGlvbn5ob2xkc35mb3JGJjEiIiFGJVEuSW5pdGlhbH50ZXJtc0YmPCQvLUYsNiNGREZELy1GLDYjRidGJw==

hypergeomsols(recg,g(n),{})

IiIh

<Text-field style="Heading 2" layout="Heading 2">Rational parametrisation for x and G'(x,1)</Text-field>

with(algcurves):

p1:=op(1,factor(parametrization(discs-S^2,x,S,U)))

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p2:=factor(subs(U=U*(1+Z)*(1+2*Z),p1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKCwkKjAjIiIiIiIjRjEsJkYxRjFJIlpHRihGMUYxKSwmRjFGMSomRjJGMUY0RjFGMUYyRjEsJkkiVUdGKEYxRjEhIiJGMSwmRjlGMUYxRjFGMSwsKihGMkYxRjlGMUY0RjFGMSomRjJGMSlGNEYyRjFGOkY5RjEqJiIiJUYxRjRGMUY6RjFGOkY6RjRGOkYxNyNGLg==

collect(op(2,denom(p2)),U,factor)

LCoqJiwmIiIiRiUqJiIiI0YlSSJaRzYiRiVGJUYlSSJVR0YpRiVGJSomRidGJSlGKEYnRiUhIiIqJiIiJUYlRihGJUYtRiVGLQ==

p3:=factor(subs(U=1-U*2,p2))

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If we take U with constant term 0 (resp. non zero), what is the value of G'(x,1) ?

numer(factor(subs(x=p3,eqGZ))): Gcand1:=factor(solve(op(3,%),G)); Gcand2:=factor(solve(op(4,%%),G));

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

Ucand:=map(factor,algeqtoseries(numer(x-p3),x,U,3))

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CombinaisonU1 G2 is correct

factor(series(subs(U=Ucand[1],Gcand1),x,3)); factor(series(subs(U=Ucand[1],Gcand2),x,3))

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

Combinaison U2 G1

factor(series(subs(U=Ucand[2],Gcand1),x,3)); factor(series(subs(U=Ucand[2],Gcand2),x,3))

KytJInhHNiIqKCksJiIiIkYoKiYiIiNGKEkiWkdGJEYoRigiIiVGKCksJkYoRihGK0YoRipGKClGK0YqISIiISIkLCQqKiwqKiZGLEYoKUYrIiIkRihGKComIiIpRihGL0YoRigqJkYsRihGK0YoRihGKEYoRihGL0YwRi5GKClGJ0YqRihGMCEiIyooRi5GKCwsKiZGKkYoKUYrRixGKEYoKiZGKkYoRjZGKEYwKiZGLEYoRi9GKEYwKiZGN0YoRitGKEYwRihGMEYoRitGMEYwLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiE=

p4:=factor(subs(U=1/U,p3))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCouLCYiIiJGMEkiWkdGKEYwRjApLCZGMEYwKiYiIiNGMEYxRjBGMEY1RjBJIlVHRighIiIsJkY2RjBGMEY3RjAsKiomRjZGMClGMUY1RjBGMComRjZGMEYxRjBGMEY0RjBGMEYwRjdGMUY3NyNGLg==

p4*tZ/t

KipJIlVHNiIhIiIsJkYjIiIiRidGJUYnLCoqJkYjRicpSSJaR0YkIiIjRidGJyomRiNGJ0YrRidGJyomRixGJ0YrRidGJ0YnRidGJUkidEdGJEYl

p5:=factor(subs(U=1+U,p4))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCouSSJVR0YoIiIiLCZGMEYwSSJaR0YoRjBGMCksJkYwRjAqJiIiI0YwRjJGMEYwRjZGMCwmRi9GMEYwRjAhIiIsLComRi9GMClGMkY2RjBGMComRi9GMEYyRjBGMCokRjtGMEYwKiYiIiRGMEYyRjBGMEYwRjBGOEYyRjg3I0Yu

Our final choice

xU:=p5*tZ/t

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4VUdGKCoqSSJVR0YoIiIiLCZGL0YwRjBGMCEiIiwsKiZGL0YwKUkiWkdGKCIiI0YwRjAqJkYvRjBGNkYwRjAqJEY1RjBGMComIiIkRjBGNkYwRjBGMEYwRjJJInRHRihGMjcjRi4=

Ucand:=map(factor,algeqtoseries(numer(x-p5),x,U,5))

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

We take for U(x) the only formal power series in x

User:=op(2,%)

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

Computing the value of G'(x,1)

numer(factor(subs(x=p5,eqGZ))): Gcand1:=factor(solve(op(5,%),G)); Gcand2:=factor(solve(op(4,%%),G));

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

The correct value of G'(x,1) is the second one below

factor(series(subs(U=User,Gcand1),x,3)); factor(series(subs(U=User,Gcand2),x,4))

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

KipJIlVHNiIiIiIsJkYjRiVGJUYlISIiLCwqJkYjRiUpSSJaR0YkIiIjRiVGJSomRiNGJUYrRiVGJSokRipGJUYlKiYiIiRGJUYrRiVGJUYlRiVGJ0kidEdGJEYn

When x=1, the value of U

numer(factor(subs(t=tZ,xU-1)))

LCQqJiwoKiZJIlVHNiIiIiJJIlpHRidGKEYoRiZGKEYpISIiRigsKiomRiZGKClGKSIiI0YoRigqJEYtRihGKiomIiIkRihGKUYoRipGKEYqRihGKg==

factor([solve(%,U)])

NyQqJkkiWkc2IiIiIiwmRiZGJkYkRiYhIiIqJiwoKiQpRiQiIiNGJkYmKiYiIiRGJkYkRiZGJkYmRiZGJkYsRig=

U1:=op(1,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNVMUdGKComSSJaR0YoIiIiLCZGMEYwRi9GMCEiIjcjRi4=

factor(subs(U=U1,Gcand2))

KiYsKComIiIjIiIiKUkiWkc2IiIiJEYmRiYqJkYlRiZGKEYmISIiRiZGJkYmRihGJg==

collect(discs,x,factor); discU:=factor(subs(x=p5,%))

LCgqJilJIlpHNiIiIiMiIiIpSSJ4R0YmRidGKEYoKixGJ0YoRiVGKCwmRihGKEYlRihGKCwoKiZGJ0YoRiRGKEYoKiYiIiVGKEYlRihGKEYoRihGKEYqRighIiIqJilGLEYnRigpLCZGKEYoKiZGJ0YoRiVGKEYoRidGKEYo

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkaXNjVUdGKCosKSwmIiIiRjFJIlpHRihGMSIiI0YxKSwmRjFGMSomRjNGMUYyRjFGMUYzRjEpLCwqJilJIlVHRihGM0YxKUYyRjNGMUYxKiZGOkYxRjJGMUYxKiRGPEYxISIiKiYiIiRGMUYyRjFGP0YxRj9GM0YxKSwmRjtGMUYxRjFGM0Y/KSwsKiZGO0YxRjxGMUYxKiZGO0YxRjJGMUYxRj5GMUZARjFGMUYxRjNGPzcjRi4=

discU/xU^2*x^2*tZ^2/t^2

KiwpLCYiIiJGJSomIiIjRiVJIlpHNiJGJUYlRichIiIpLCwqJilJIlVHRilGJ0YlKUYoRidGJUYlKiZGLkYlRihGJUYlKiRGMEYlRioqJiIiJEYlRihGJUYqRiVGKkYnRiVGLkYqKUkieEdGKUYnRiVGMEYl

exprGx:=factor(Gcand2*t/tZ*x/p5)

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

collect(factor(-numer(exprGx)/t/x/(1+U)),U,factor)

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

latex(-denom(exprGx))

-\134left(1+2 Z \134right)^{2} \134left(U \134,Z^{2}-Z^{2}-3 Z -1\134right)

collect(%,Z,factor)

LDoiIiJGIyooKSwmSSJVRzYiRiNGI0YjIiIkRiMsJkYnRiNGIyEiIkYjKUkiWkdGKCIjNkYjRisqKCwoKiYiIiVGIylGJyIiI0YjRiNGJ0YjIiM+RitGIylGJkY0RiMpRi0iIzVGI0YrKihGJkYjLCoqJiIiJ0YjKUYnRilGI0YjKiYiIiZGI0YzRiNGIyomIiMhKUYjRidGI0YrIiMmKkYrRiMpRi0iIipGI0YrKihGJkYjLCoqJkYyRiNGPUYjRiMqJiIiKEYjRjNGI0YrKiYiJG0iRiNGJ0YjRisiJCY+RitGIylGLSIiKUYjRisqKEYmRiMsKiokRj1GI0YjKiYiIztGI0YzRiNGKyomIiQiPUYjRidGI0YrIiRnIkYrRiMpRi1GSUYjRisqJiwqKiZGOEYjRj1GI0YjKiYiJDUiRiNGM0YjRiMqJiIjc0YjRidGI0YjIiM3RitGIylGLUY8RiNGIyomLCoqJkY0RiNGPUYjRiMqJiIjQkYjRjNGI0YjKiYiJEciRiNGJ0YjRisiIyYpRitGIylGLUY/RiNGIyomLCgqJkY0RiNGM0YjRisqJiIkNiJGI0YnRiNGK0Y/RitGIylGLUYyRiNGIyomLCgqJEYzRiNGKyomIiNORiNGJ0YjRisiI2FGI0YjKUYtRilGI0YjKiYsJiomRjJGI0YnRiNGKyIjUEYjRiMpRi1GNEYjRiMqJkY4RiNGLUYjRiM=

factor(series(subs(U=User,Gcand2),x))

Ky9JInhHNiIqKiw0KiQpSSJaR0YkIiIpIiIiRisqJiIjOkYrKUYpIiIoRitGKyomIiNKRispRikiIidGK0YrKiYiIzVGKylGKSIiJkYrRisqJiIjPkYrKUYpIiIlRishIiIqJkYvRispRikiIiRGK0Y8KiYiIipGKylGKSIiI0YrRisqJkYzRitGKUYrRitGK0YrRitGKUYrKSwmRitGKyomRkNGK0YpRitGK0Y7RjwsJkYrRitGKUYrRjxGKyoqLDgqJkY7RispRilGNUYrRisqJiIjb0YrKUYpRkFGK0YrKiYiJFsjRitGKEYrRisqJiIkXCRGK0YuRitGKyomIiRWIkYrRjJGK0YrKiYiJDQiRitGNkYrRjwqJiIkLSJGK0Y6RitGPCokRj5GK0YrKiYiI0RGK0ZCRitGKyomRkFGK0YpRitGK0YrRitGK0ZCRispRkhGQ0Y8KUZGRjNGPEZDKiosPComIiM5RispRikiIzdGK0YrKiYiJHUjRispRikiIzZGK0YrKiYiJW84RitGTEYrRisqJiIlJDQkRitGT0YrRisqJiIleE1GK0YoRitGKyomIiU/OkYrRi5GK0YrKiYiJFkmRitGMkYrRjwqJiIkOClGK0Y2RitGPComIiQvI0YrRjpGK0Y8KiYiIyUpRitGPkYrRisqJiIjZ0YrRkJGK0YrKiYiIzhGK0YpRitGK0YrRitGK0Y+RispRkZGKkY8KUZIRj9GPEY/KiosQComIiNbRispRilGXW9GK0YrKiYiJXM1RispRilGZ3BGK0YrKiYiJU9uRitGXm9GK0YrKiYiJiFlP0YrRmJvRitGKyomIiZXYCRGK0ZMRitGKyomIiZrViRGK0ZPRitGKyomIiZyYSJGK0YoRitGKyomIiUtQkYrRi5GK0Y8KiYiJXFoRitGMkYrRjwqJiIlb0NGK0Y2RitGPComIiN5RitGOkYrRisqJiIkbiRGK0Y+RitGKyomIiRDIkYrRkJGK0YrKiYiIz1GK0YpRitGK0YrRitGK0Y6RispRkhGO0Y8KUZGRjVGPEY7KiosRComIiRsIkYrKUYpIiM7RitGKyomIiViVEYrKUYpRi1GK0YrKiYiJiRSSkYrRl5xRitGKyomIidcLTdGK0ZhcUYrRisqJiIneFJGRitGXm9GK0YrKiYiJzo5UkYrRmJvRitGKyomIid0VE1GK0ZMRitGKyomIidaIWUiRitGT0YrRisqJiIlR0ZGK0YoRitGPComIiYqPlpGK0YuRitGPComIiZiWCNGK0YyRitGPComIiUiMyRGK0Y2RitGPComIiVDP0YrRjpGK0YrKiYiJXE1RitGPkYrRisqJiIkSCNGK0ZCRitGKyomIiNDRitGKUYrRitGK0YrRitGNkYrKUZIRjdGPClGRkZfb0Y8RjctSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGK0Yz

factor(series(subs(Z=Zser,convert(%,polynom)),t,N+1))

Ky1JInRHNiJJInhHRiQiIiIqJkYlRiYsJkYlRiYiIiNGJkYmRikqJkYlRiYsKCokKUYlRilGJkYmKiYiIiZGJkYlRiZGJiIiKEYmRiYiIiQqJkYlRiYsKiokKUYlRjFGJkYmKiYiIipGJkYtRiZGJiomIiNFRiZGJUYmRiYiI0xGJkYmIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiZGLw==

series(subs(y=1,Gpser(1,N)),t,N)

KytJInRHNiJJInhHRiQiIiIqJkYlRiYsJkYlRiYiIiNGJkYmRikqJkYlRiYsKCokKUYlRilGJkYmKiYiIiZGJkYlRiZGJiIiKEYmRiYiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJQ==

factor(series(%-%%,t,N))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiU=

<Text-field style="Heading 2" layout="Heading 2"></Text-field>

<Text-field style="Heading 1" layout="Heading 1">The case m=1: solution of the equation for G(x,y) (Section 5.2 and Appendix A)</Text-field>

eqG(1)

LCwtSSJHRzYiNiRJInhHRiVJInlHRiUhIiIqKEYnIiIiRihGK0kidEdGJUYrRisqKkYnRitGKEYrRixGK0YjRitGKyouRixGKylGKCIiI0YrRidGKywmRidGK0YrRilGKSwmRihGK0YrRilGKSwmRiNGKy1GJDYkRitGKEYpRitGKyouRidGK0YoRitGLEYrLCYtRiQ2JComRidGK0YoRitGK0YrLUYkNiRGK0YrRilGK0YyRiksJkY6RitGK0YpRilGKQ==

eqQ(1)

LCwtSSJRRzYiNiRJInhHRiVJInlHRiUhIiIiIiJGKiooSSJ0R0YlRipGJ0YqRiNGKkYqKiwpRigiIiNGKkYsRiosJiomRiNGKkYnRipGKiomLUYkNiRGKEYoRipGKEYqRilGKiwmRidGKkYoRilGKSwmRihGKkYqRilGKUYqKipGLEYqLCYqJi1GJDYkRidGKkYqRidGKkYpLUYkNiRGKkYqRipGKkY2RiksJkYnRipGKkYpRilGKg==

Expansion of Q(x,y)

Qser:=proc(n) collect(subs(x=x/y,Gser(1,n+1))/t/x,t,factor): end:

Qser(3)

LCoiIiJGIyomLDYqJClJInhHNiIiIiRGI0YjKihGKkYjKUYoIiIjRiNJInlHRilGI0YjKigiIiZGI0YoRiMpRi5GLUYjRiMqJkYwRiMpRi5GKkYjRiMqJkYqRiNGLEYjRiMqKCIiKUYjRihGI0YuRiNGIyomIiM1RiNGMUYjRiMqJkY2RiNGKEYjRiMqJiIjOEYjRi5GI0YjRjtGI0YjKUkidEdGKUYqRiNGIyomLC4qJEYsRiNGIyooRi1GI0YoRiNGLkYjRiMqJkYtRiNGMUYjRiMqJkYtRiNGKEYjRiMqJkYqRiNGLkYjRiNGKkYjRiMpRj1GLUYjRiMqJiwoRihGI0YuRiNGI0YjRiNGPUYjRiM=

Let us check

n:=6: factor(series(subs(Q(x,y)=Qser(n), Q(y,y)=subs(x=y,Qser(n)), Q(x,1)=subs(y=1,Qser(n)), Q(1,1)=subs(x=1,y=1,Qser(n)), eqQ(1)),t,n+1))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIig=

<Text-field style="Heading 2" layout="Heading 2">The kernel, its roots, and the associated group</Text-field>

Ker:=-coeff(eqQ(1),Q(x,y))

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nKer:=numer(Ker)

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RHS:=subs(Q(x,y)=0,eqQ(1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRSSFNHRigsKCIiIkYvKiwpSSJ5R0YoIiIkRi9JInRHRihGLy1JIlFHRig2JEYyRjJGLywmSSJ4R0YoRi9GMiEiIkY6LCZGMkYvRi9GOkY6RjoqKkY0Ri8sJiomLUY2NiRGOUYvRi9GOUYvRjotRjY2JEYvRi9GL0YvRjtGOiwmRjlGL0YvRjpGOkYvNyNGLg==

The roots of the kernel

Ysol:=factor([solve(Ker,y)])

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map(factor,subs(csgn(x - 1)=-1,map(series,%,t,3)))

NyQrK0kidEc2IiIiIiIiISomSSJ4R0YlRiYsJkYpRiZGJiEiIkYrRiYqJilGKSIiJUYmKUYqIiIkRisiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYwKytGJEYpRicsJComKUYpRjBGJkYqRitGK0YmLCRGLEYrRjFGMkYw

Y0sol:=op(2,Ysol): Y1sol:=op(1,Ysol):

Xsol:=factor([solve(Ker,x)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYySSJ5R0YoRjIhIiJGN0YyRjJGOCokKSwwKiYpRjZGM0YyKUY3RjNGMkYyKigiIiVGMkY2RjIpRjciIiRGMkY4KigiIidGMkY2RjJGPkYyRjIqKEYzRjJGNkYyRjdGMkY4KiRGPkYyRjIqJkYzRjJGN0YyRjhGMkYyRjFGMkYyRjJGNkY4LCZGN0YyRjJGOEY4RjIsJCoqRjFGMiwqRjVGMkY5RjJGN0Y4RjJGMkYyRjZGOEZIRjhGODcjRi4=

The associated group has order 10

factor([solve(factor(Ker-subs(x=xp,Ker)),xp)])

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factor([solve(factor(Ker-subs(y=yp,Ker)),yp)])

NyRJInlHNiIqKEkieEdGJCIiIkYjRicsKComRiZGJ0YjRidGJ0YmISIiRiNGJ0Yq

phi:=proc(paire) local x,y: x:=paire[1]: y:=paire[2]: factor([y*(x*y - x + y)/((x - y)*(y - 1)),y]): end:

psi:=proc(paire) local x,y: x:=paire[1]: y:=paire[2]: factor([x,x*y/(x*y - x + y)]): end:

[x,y]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

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NyQqKkkieUc2IiIiIiwoKiZJInhHRiVGJkYkRiZGJkYpISIiRiRGJkYmLCZGKUYmRiRGKkYqLCZGJEYmRiZGKkYqKihGJ0YmRilGKkYsRio=

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NyQqKCwoKiZJInhHNiIiIiJJInlHRidGKEYoRiYhIiJGKUYoRihGKUYqLCZGKUYoRihGKkYqKiYpRikiIiNGKkYkRig=

NyQqKkkieUc2IiEiIkkieEdGJSIiIiwoKiZGJ0YoRiRGKEYoRidGJkYkRihGKCwmRidGKEYkRiZGJiomKUYkIiIjRiZGKUYo

NyQqKkkieUc2IiEiIkkieEdGJSIiIiwoKiZGJ0YoRiRGKEYoRidGJkYkRihGKCwmRidGKEYkRiZGJiomRidGKEYrRiY=

NyQqKkkieUc2IiIiIkkieEdGJUYmLCZGJEYmRiYhIiJGKSwmRidGJkYkRilGKSomRidGJkYqRik=

NyQqKkkieUc2IiIiIkkieEdGJUYmLCZGJEYmRiYhIiJGKSwmRidGJkYkRilGKSooRidGJkYkRiYsKComRidGJkYkRiZGJkYnRilGJEYmRik=

NyRJInhHNiIqKEYjIiIiSSJ5R0YkRiYsKComRiNGJkYnRiZGJkYjISIiRidGJkYq

NyRJInhHNiJJInlHRiQ=

orbite:=[[x, y],[y*(x*y - x + y)/((x - y)*(y - 1)), y],[y*(x*y - x + y)/((x - y)*(y - 1)), (x*y - x + y)/(x*(y - 1))],[(x*y - x + y)/(y*(y - 1)), (x*y - x + y)/(x*(y - 1))],[(x*y - x + y)/(y*(y - 1)), (x*y - x + y)/y^2],[x*(x*y - x + y)/(y*(x - y)), (x*y - x + y)/y^2],[x*(x*y - x + y)/(y*(x - y)), x/(x - y)],[y*x/((y - 1)*(x - y)), x/(x - y)],[y*x/((y - 1)*(x - y)), x*y/(x*y - x + y)],[x, x*y/(x*y - x + y)]]

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdvcmJpdGVHRig3LDckSSJ4R0YoSSJ5R0YoNyQqKkYxIiIiLCgqJkYwRjRGMUY0RjRGMCEiIkYxRjRGNCwmRjBGNEYxRjdGNywmRjFGNEY0RjdGN0YxNyRGMyooRjVGNEYwRjdGOUY3NyQqKEY1RjRGMUY3RjlGN0Y7NyRGPSomKUYxIiIjRjdGNUY0NyQqKkYxRjdGMEY0RjVGNEY4RjdGPzckRkMqJkYwRjRGOEY3NyQqKkYxRjRGMEY0RjlGN0Y4RjdGRTckRkcqKEYwRjRGMUY0RjVGNzckRjBGSTcjRi4=

What follows is a test of algebraicity and can be ignored

Elimination of sections by taking linear combinations of the 10 equations gives a righ-hand side equal to 0

res:=collect(add(a[i]*subs({x=orbite[i][1], y=orbite[i][2]},RHS),i=1..10),Q,factor):

indets(res)

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coeff(res, Q(y*(x*y - x + y)/((x - y)*(y - 1)), 1))

LCQqLCwoKiZJInhHNiIiIiJJInlHRidGKEYoRiYhIiJGKUYoRihJInRHRidGKCwqKihGJkYoKUYpIiIjRigmSSJhR0YnNiMiIiRGKEYoKipGL0YoRiZGKEYpRihGMEYoRioqJkYmRihGMEYoRigqJkYpRigmRjE2I0YvRihGKEYoLCZGKUYoRihGKkYqLCwqJkYmRihGLkYoRigqKEYvRihGJkYoRilGKEYqKiZGL0YoRi5GKEYoRiZGKEYpRipGKkYq

a2sol:=solve(%,a[2])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhMnNvbEdGKCwkKiopLCZJInlHRigiIiJGMyEiIiIiI0YzJkkiYUdGKDYjIiIkRjNJInhHRihGM0YyRjRGNDcjRi4=

factor(subs(a[2]=a2sol,coeff(res,Q(y*x/((y - 1)*(x - y)), 1))))

LCQqLEkidEc2IiIiIkkieEdGJUYmLC4qKEYnRiYpSSJ5R0YlIiIjRiYmSSJhR0YlNiMiIipGJkYmKiYpRidGLEYmJkYuNiMiIilGJkYmKipGLEYmRidGJkYrRiZGM0YmISIiKihGJ0YmRitGJkYtRiZGNyomRipGJkYzRiZGJiomRipGJkYtRiZGJkYmLCgqJEYqRiZGJkYnRiZGK0Y3RjcsJkYnRiZGK0Y3RjdGNw==

a9sol:=factor(solve(%,a[9]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhOXNvbEdGKCwkKiopLCZJInhHRigiIiJJInlHRighIiIiIiNGMyZJImFHRig2IyIiKUYzRjRGNSwoKiZGMkYzRjRGM0YzRjJGNUY0RjNGNUY1NyNGLg==

factor(subs(a[2]=a2sol,a[9]=a9sol,coeff(res,Q(x*(x*y - x + y)/(y*(x - y)), 1))))

LCQqMiwwKigpSSJ4RzYiIiIjIiIiSSJ5R0YoRiomSSJhR0YoNiMiIihGKkYqKipGKUYqRidGKilGK0YpRipGLEYqISIiKiYpRisiIiRGKiZGLTYjIiInRipGKiomRjRGKkYsRipGKiomRiZGKkYsRipGMioqRilGKkYnRipGK0YqRixGKkYqKiZGMUYqRixGKkYyRiosKComRidGKkYrRipGKkYnRjJGK0YqRipGJ0YqSSJ0R0YoRipGK0YyLCgqJkYmRipGK0YqRioqJEYmRipGMiokRjFGKkYqRjIsJkYnRipGK0YyRjIsJkYrRipGKkYyRjJGMg==

a6sol:=factor(solve(%,a[6]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhNnNvbEdGKCwkKiomSSJhR0YoNiMiIigiIiIpLCZJInhHRihGNEkieUdGKCEiIiIiI0Y0KUY4IiIkRjksJkY4RjRGNEY5RjRGOTcjRi4=

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,coeff(res,Q((x*y - x + y)/(y*(y - 1)), 1))))

LCQqMCwwKigpSSJ4RzYiIiIjIiIiKUkieUdGKEYpRiomSSJhR0YoNiMiIiVGKkYqKihGJ0YqKUYsIiIkRipGLUYqISIiKipGKUYqRiZGKkYsRipGLUYqRjQqKkYpRipGJ0YqRitGKkYtRipGKiomRjJGKiZGLjYjIiImRipGKiomRiZGKkYtRipGKiooRidGKkYsRipGLUYqRjRGKkkidEdGKEYqLCgqJkYnRipGLEYqRipGJ0Y0RixGKkYqRixGNCwmRidGKkYsRjRGNCwmRixGKkYqRjRGNCwqRj9GKiokRitGKkY0RidGNComRilGKkYsRipGKkY0RjQ=

a5sol:=factor(solve(%,a[5]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhNXNvbEdGKCwkKiwmSSJhR0YoNiMiIiUiIiJJInhHRihGNCksJkkieUdGKEY0RjQhIiIiIiNGNClGOCIiJEY5LCZGNUY0RjhGOUY0Rjk3I0Yu

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,a[5]=a5sol,coeff(res,Q((x*y - x+ y)/(x*(y - 1)), (x*y - x + y)/(x*(y - 1))))))

LCQqMCwoKiZJInhHNiIiIiJJInlHRidGKEYoRiYhIiJGKUYoRihJInRHRidGKEYmRiosLiooRiZGKClGKSIiI0YoJkkiYUdGJzYjIiIlRihGKComKUYmRi9GKCZGMTYjIiIkRihGKCoqRi9GKEYmRihGKUYoRjZGKEYqKihGJkYoRilGKEYwRihGKiomRi5GKEY2RihGKComRi5GKEYwRihGKEYoLCZGJkYoRilGKkYqRilGKiwmRilGKEYoRipGKkYq

a3sol:=factor(solve(%,a[3]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhM3NvbEdGKCwkKiosKComSSJ4R0YoIiIiSSJ5R0YoRjNGM0YyISIiRjRGM0YzJkkiYUdGKDYjIiIlRjMpLCZGMkYzRjRGNSIiI0Y1RjRGM0Y1NyNGLg==

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,a[5]=a5sol,a[3]=a3sol,coeff(res,Q(x*y/(x*y - x + y), x*y/(x*y - x + y)))))

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

a8sol:=factor(solve(%,a[8]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhOHNvbEdGKCouJkkiYUdGKDYjIiM1IiIiKUkieUdGKCIiJUYzLCgqJkkieEdGKEYzRjVGM0YzRjkhIiJGNUYzRjNGOUY6KSwmRjlGM0Y1RjoiIiRGOiksJkY1RjNGM0Y6IiIjRjo3I0Yu

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,a[5]=a5sol,a[3]=a3sol,a[8]=a8sol,coeff(res,Q((x*y - x + y)/y^2, (x*y - x + y)/y^2))))

KiosNiooKUkieEc2IiIiIyIiIilJInlHRiciIiRGKSZJImFHRic2IyIiJUYpRikqKkYsRilGJUYpKUYrRihGKUYtRikhIiIqKEYmRilGKkYpRi1GKUYpKipGLEYpRiVGKUYrRilGLUYpRikqKkYoRilGJkYpRjJGKUYtRilGMyomRiVGKUYtRilGMyomRiVGKSZGLjYjIiIoRilGKSooRiZGKUYrRilGLUYpRikqKkYoRilGJkYpRitGKUY5RilGMyomRjJGKUY5RilGKUYpSSJ0R0YnRiksKComRiZGKUYrRilGKUYmRjNGK0YpRikpRisiIiZGMw==

a7sol:=factor(solve(%,a[7]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhN3NvbEdGKCwkKixJInhHRigiIiImSSJhR0YoNiMiIiVGMSksJkkieUdGKEYxRjEhIiIiIiNGMSwoKiZGMEYxRjhGMUYxRjBGOUY4RjFGMSksJkYwRjFGOEY5RjpGOUY5NyNGLg==

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,a[5]=a5sol,a[3]=a3sol,a[8]=a8sol,a[7]=a7sol,coeff(res,Q(x/(x - y), x/(x - y)))))

Ki5JInhHNiIiIiJJInRHRiRGJSwwKigpRiMiIiNGJSlJInlHRiRGKkYlJkkiYUdGJDYjIiIlRiVGJSooRiNGJSlGLCIiJEYlRi1GJSEiIioqRipGJUYpRiVGLEYlRi1GJUY0KipGKkYlRiNGJUYrRiVGLUYlRiUqJkYyRiUmRi42IyIjNUYlRjQqJkYpRiVGLUYlRiUqKEYjRiVGLEYlRi1GJUY0RiUsKComRiNGJUYsRiVGJUYjRjRGLEYlRiUsJkYsRiVGJUY0RjQpLCZGI0YlRixGNEYwRjQ=

a10sol:=factor(solve(%,a[10]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhMTBzb2xHRigqLCZJImFHRig2IyIiJSIiIkkieEdGKEYzKSwmSSJ5R0YoRjNGMyEiIiIiI0YzKUY3IiIkRjgsJkY0RjNGN0Y4RjM3I0Yu

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,a[5]=a5sol,a[3]=a3sol,a[8]=a8sol,a[7]=a7sol,a[10]=a10sol,coeff(res,Q(x,1))))

LCQqLiw2KigpSSJ4RzYiIiIjIiIiKUkieUdGKCIiJUYqJkkiYUdGKDYjRi1GKkYqKipGLUYqRiZGKilGLCIiJEYqRi5GKiEiIiooRidGKkYrRipGLkYqRioqKiIiJ0YqRiZGKilGLEYpRipGLkYqRioqKkYzRipGJ0YqRjJGKkYuRipGNCoqRi1GKkYmRipGLEYqRi5GKkY0KipGM0YqRidGKkY4RipGLkYqRioqJkYyRiomRi82I0YqRipGKiomRiZGKkYuRipGKiooRidGKkYsRipGLkYqRjRGKkYnRipJInRHRihGKiwmRidGKkYqRjRGNEYyRjQsJkYsRipGKkY0RjRGNA==

a1sol:=factor(solve(%,a[1]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhMXNvbEdGKCwkKiwpLCZJInlHRigiIiJGMyEiIiIiJEYzLCgqJkkieEdGKEYzRjJGM0YzRjhGNEYyRjNGM0Y4RjMmSSJhR0YoNiMiIiVGMylGMkY1RjRGNDcjRi4=

Nothing is left... this suggests an algebraic solution

factor(subs(a[2]=a2sol,a[9]=a9sol,a[6]=a6sol,a[5]=a5sol,a[3]=a3sol,a[8]=a8sol,a[7]=a7sol,a[10]=a10sol,a[1]=a1sol,res))

IiIh

Ker

LCgiIiJGIyomSSJ4RzYiRiNJInRHRiZGIyEiIiosKUkieUdGJiIiI0YjRidGI0YlRiMsJkYlRiNGK0YoRigsJkYrRiNGI0YoRihGKA==

<Text-field style="Heading 2" layout="Heading 2">Construction of rational invariants</Text-field>

We sum a rational function H(u,v) over all pairs of the orbit. Our first choice, H(u,v)=u, yields a trivial invariant

factor(expand(rationalize(subs(y=Y0sol,factor(convert(map(Z->Z[1],orbite),`+`))))));expand(%)

LCQqKCIiIyIiIiwmSSJ0RzYiRiVGJCEiIkYlRidGKUYp

LCYiIiMhIiIqJiIiJSIiIkkidEc2IkYkRic=

Let us start from H(u,v)=v instead

Ix:=1/2*factor(simplify(factor(expand(rationalize(subs(y=Y0sol,factor(convert(map(Z->Z[2],orbite),`+`)))))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKiopSSJ4R0YoIiIjISIiLDYqJilJInRHRigiIiQiIiIpRjEiIiZGOUY5KiZGNkY5KUYxIiIlRjlGOSooRjJGOSlGN0YyRjlGPUY5RjMqKEY4RjlGQEY5KUYxRjhGOUYzKiZGQEY5RjBGOUYzKiZGN0Y5RkJGOUY5KihGMkY5RjdGOUYwRjlGOSooRjJGOUY3RjlGMUY5RjlGMUY5RjlGM0Y5RjdGMywmKiZGN0Y5RjFGOUY5RjlGM0YzRjM3I0Yu

More attempts

factor(simplify(factor(expand(rationalize(subs(y=Y0sol,factor(convert(map(Z->Z[1]^2,orbite),`+`)))))),symbolic));

LCQqLCIiIyIiIilJInhHNiJGJCEiIiw6KihGJEYlKUkidEdGKCIiJEYlKUYnIiImRiVGJSooRiRGJUYsRiUpRiciIiVGJUYlKiZGLEYlKUYnRi5GJUYlKihGM0YlKUYtRiRGJUYyRiVGKSooRjNGJUY3RiVGNUYlRikqKEYuRiVGN0YlRiZGJUYpKihGM0YlRi1GJUY1RiVGJSooRiRGJUYtRiVGJkYlRiUqKEYzRiVGLUYlRidGJUYlKiZGJEYlRiZGJUYpKiZGJEYlRidGJUYlRiRGKUYlLCYqJkYtRiVGJ0YlRiVGJUYpRilGN0YpRiU=

factor(simplify(factor(expand(rationalize(subs(y=Y0sol,factor(convert(map(Z->Z[1]/Z[2],orbite),`+`)))))),symbolic));

LCQqKCIiIyIiIiwmKiZGJEYlSSJ0RzYiRiVGJUYlRiVGJUYoISIiRiU=

factor(simplify(factor(expand(rationalize(subs(y=Y0sol,factor(convert(map(Z->Z[2]/Z[1],orbite),`+`)))))),symbolic));

LCYqJiIiIyIiIkkidEc2IkYlISIiIiIlRiU=

convert(Ix,parfrac,x)

LDAqJilJInhHNiIiIiMiIiJJInRHRiZGKCEiIiomRiVGKEYpRihGKkYlRihGJ0YoKiQsJiomRilGKEYlRihGKEYoRipGKkYqKigsJkYpRihGKEYoRihGJUYqRilGKkYoKiZGJEYqRilGKkYq

I0x:=factor(Ix-2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMHhHRigsJCoqLDQqJilJInRHRigiIiQiIiIpSSJ4R0YoIiImRjVGNSomRjJGNSlGNyIiJUY1RjUqKCIiI0Y1KUYzRj1GNUY6RjUhIiIqJkY+RjUpRjdGNEY1Rj8qJkY+RjUpRjdGPUY1Rj8qJkYzRjVGQUY1RjUqKEY9RjVGM0Y1RjdGNUY1RjdGNUY1Rj9GNUYzRj9GQ0Y/LCYqJkYzRjVGN0Y1RjVGNUY/Rj9GPzcjRi4=

J0y:=factor(simplify(factor(expand(rationalize(subs(x=Xsol[1],I0x)))),symbolic)); convert(%,parfrac,y)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMHlHRigsJCoqLDYqJkkidEdGKCIiIilJInlHRigiIiZGMyEiIiomKUYyIiIjRjMpRjUiIiRGM0YzKihGOkYzRjJGMylGNSIiJUYzRjMqKEY8RjNGMkYzRjtGM0Y3KihGPEYzRjJGMylGNUY6RjNGMyokRjtGM0Y3KiZGMkYzRjVGM0Y3KiZGPEYzRkJGM0YzKiZGPEYzRjVGM0Y3RjNGM0YzRjJGN0ZCRjcpLCZGNUYzRjNGN0Y6RjdGNzcjRi4=

LCxJInlHNiIiIiIqJkkidEdGJEYlKSwmRiNGJUYlISIiIiIjRipGKiomLCZGJ0YqRiVGJUYlRilGKkYlKigsJkYnRiVGJUYlRiVGI0YqRidGKkYlKiYpRiNGK0YqRidGKkYq

convert(I0x,parfrac,x); convert(J0y,parfrac,y)

LC4qJkkidEc2IiIiIilJInhHRiUiIiNGJiEiIiomRihGJkYkRiZGKkYoRiYqKCwmRiRGJkYmRiZGJkYoRipGJEYqRiYqJkYkRipGJ0YqRioqJCwmKiZGJEYmRihGJkYmRiZGKkYqRio=

LCxJInlHNiIiIiIqJkkidEdGJEYlKSwmRiNGJUYlISIiIiIjRipGKiomLCZGJ0YqRiVGJUYlRilGKkYlKigsJkYnRiVGJUYlRiVGI0YqRidGKkYlKiYpRiNGK0YqRidGKkYq

factor((I0x-J0y)/Ker)

KjIsKCooSSJ0RzYiIiIiSSJ4R0YmRidJInlHRiZGJ0YnRikhIiJGJ0YnRicsLiooRiVGJylGKCIiI0YnKUYpRi5GJ0YnKihGJUYnRi1GJ0YpRidGKkYkRioqJkYoRidGKUYnRipGKEYnRilGJ0YnLCZGKEYnRilGKkYnLCZGKUYnRidGKkYqRi9GKkYlRipGLUYqLCYqJkYlRidGKEYnRidGJ0YqRio=

collect(op(2,numer(%)),t,factor)

LCoqKiwoKiZJInhHNiIiIiJJInlHRidGKEYoRiYhIiJGKEYqRihGJkYoRilGKEkidEdGJ0YoRihGJUYqRiZGKEYpRig=

collect(eqH,H,factor)

LCwqLEkieEc2IiIiIkkidEdGJUYmLCZJInlHRiVGJkYmISIiRiosJkYkRiZGJkYqRiotSSJIR0YlNiRGJkYmRiZGJiosRiRGJkYnRiYtRi02JEYkRiZGJkYoRipGK0YqRioqKiwwKihGJ0YmKUYkIiIjRiZGKUYmRiYqJkYnRiZGNUYmRioqKEYnRiZGJEYmRilGJkYmKiZGJEYmRilGJkYqKiQpRilGNkYmRiZGJEYmRilGKkYmLCZGJEYmRilGKkYqRihGKi1GLTYkRiRGKUYmRiYqLkYnRiZGJEYmRjtGJkYoRipGPEYqLUYtNiRGKUYpRiZGKiomRiRGJkYnRiZGJg==

We multiply out by (y-1) so that there is a term in x only in the RHS

RHSmult:=collect(RHS*(y-1),Q,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShSSFNtdWx0R0YoLCwqKEkidEdGKCIiIiwmSSJ4R0YoRjFGMSEiIkY0LUkiUUdGKDYkRjFGMUYxRjEqKkYzRjFGMEYxRjJGNC1GNjYkRjNGMUYxRjQqKilJInlHRigiIiRGMUYwRjEsJkYzRjFGPUY0RjQtRjY2JEY9Rj1GMUY0Rj1GMUYxRjQ3I0Yu

<Text-field style="Heading 2" layout="Heading 2">A multiplicative decoupling for ty^3/(x-y)</Text-field>

factor(Ker-1+t*x)

LCQqLClJInlHNiIiIiMiIiJJInRHRiZGKEkieEdGJkYoLCZGKkYoRiUhIiJGLCwmRiVGKEYoRixGLEYs

id1:=K-1+t*x=%

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRpZDFHRigvLCgqJkkidEdGKCIiIkkieEdGKEYyRjJJIktHRihGMkYyISIiLCQqLClJInlHRigiIiNGMkYxRjJGM0YyLCZGM0YyRjlGNUY1LCZGOUYyRjJGNUY1RjU3I0Yu

id2:=collect(map(u->factor(u*y/x*(y-1)),-id1),K,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRpZDJHRigvLCYqKkkieUdGKCIiIiwmRjFGMkYyISIiRjJJInhHRihGNEkiS0dGKEYyRjQqKiwmKiZJInRHRihGMkY1RjJGMkYyRjRGMkYxRjJGM0YyRjVGNEY0KigpRjEiIiRGMkY6RjIsJkY1RjJGMUY0RjQ3I0Yu

id2

LywmKipJInlHNiIiIiIsJkYlRidGJyEiIkYnSSJ4R0YmRilJIktHRiZGJ0YpKiosJiomSSJ0R0YmRidGKkYnRidGJ0YpRidGJUYnRihGJ0YqRilGKSooKUYlIiIkRidGL0YnLCZGKkYnRiVGKUYp

New form of the equation

eqQK:=-K*Q(x,y)+subs(Q(x,y)=0,eqQ(1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVFLR0YoLCoqJkkiS0dGKCIiIi1JIlFHRig2JEkieEdGKEkieUdGKEYxISIiRjFGMSosKUY2IiIkRjFJInRHRihGMS1GMzYkRjZGNkYxLCZGNUYxRjZGN0Y3LCZGNkYxRjFGN0Y3RjcqKkY7RjEsJiomLUYzNiRGNUYxRjFGNUYxRjctRjM2JEYxRjFGMUYxRj9GNywmRjVGMUYxRjdGN0YxNyNGLg==

collect(map(u->factor((y-1)*u),eqQK),Q,factor);

LC4qKEkidEc2IiIiIiwmSSJ4R0YlRiZGJiEiIkYpLUkiUUdGJTYkRiZGJkYmRiYqKCwmSSJ5R0YlRiZGJkYpRiZJIktHRiVGJi1GKzYkRihGL0YmRilGL0YmRiZGKSoqKUYvIiIkRiZGJEYmLUYrNiRGL0YvRiYsJkYoRiZGL0YpRilGKSoqRiRGJi1GKzYkRihGJkYmRihGJkYnRilGKQ==

op(5,%)

LCQqKilJInlHNiIiIiQiIiJJInRHRiZGKC1JIlFHRiY2JEYlRiVGKCwmSSJ4R0YmRihGJSEiIkYvRi8=

subsop(5=-op(1,id2)*Q(y,y),%%)

LC4qKEkidEc2IiIiIiwmSSJ4R0YlRiZGJiEiIkYpLUkiUUdGJTYkRiZGJkYmRiYqKCwmSSJ5R0YlRiZGJkYpRiZJIktHRiVGJi1GKzYkRihGL0YmRilGL0YmRiZGKSomLCYqKkYvRiZGLkYmRihGKUYwRiZGKSoqLCYqJkYkRiZGKEYmRiZGJkYpRiZGL0YmRi5GJkYoRilGKUYmLUYrNiRGL0YvRiZGKSoqRiRGJi1GKzYkRihGJkYmRihGJkYnRilGKQ==

The new form of the equation:

eqQ_new:=collect(%/(1-t*x)*x,[K,Q],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShlcVFfbmV3R0YoLCwqJiwmKiosJkkieUdGKCIiIkY0ISIiRjRJInhHRihGNCwmKiZJInRHRihGNEY2RjRGNEY0RjVGNS1JIlFHRig2JEY2RjNGNEY0KipGM0Y0RjJGNEY3RjUtRjs2JEYzRjNGNEY1RjRJIktHRihGNEY0KixGNkY0RjlGNCwmRjZGNEY0RjVGNUY3RjUtRjs2JEY0RjRGNEY1KiwpRjYiIiNGNEY5RjRGQkY1RjdGNS1GOzYkRjZGNEY0RjQqKEYzRjRGMkY0Rj5GNEY1KihGMkY0RjZGNEY3RjVGNTcjRi4=

factor(coeff(%,K))

KigsJkkieUc2IiIiIkYmISIiRiYsJiomLUkiUUdGJTYkSSJ4R0YlRiRGJkYtRiZGJiomLUYrNiRGJEYkRiZGJEYmRidGJiwmKiZJInRHRiVGJkYtRiZGJkYmRidGJw==

<Text-field style="Heading 2" layout="Heading 2">An additive decoupling by guessing and checking</Text-field>

The constant term in the RHS of eqQ_new

op(5,eqQ_new)

LCQqKCwmSSJ5RzYiIiIiRichIiJGJ0kieEdGJkYnLCYqJkkidEdGJkYnRilGJ0YnRidGKEYoRig=

unassign('c1'):

An Ansatz on the fraction B, with a pole at infinity and two finite ones.

Bform:=a0+a1*y +a2*y^2+a3*y^3+b1/(y-c1)+d1/(y-c1)^2+b2/(y-c2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZCZm9ybUdGKCwwSSNhMEdGKCIiIiomSSNhMUdGKEYwSSJ5R0YoRjBGMComSSNhMkdGKEYwKUYzIiIjRjBGMComSSNhM0dGKEYwKUYzIiIkRjBGMComSSNiMUdGKEYwLCZGM0YwSSNjMUdGKCEiIkZARjAqJkkjZDFHRihGMClGPkY3RkBGMComSSNiMkdGKEYwLCZGM0YwSSNjMkdGKEZARkBGMDcjRi4=

The equations for the a_i, b_i, c_i and d_i

collect(numer(factor(x/(1-t*x)-rationalize(subs(y=Ysol[1],Bform)- subs(y=Ysol[2],Bform))/(Ysol[1]-Ysol[2]))),x,factor):

eqs:={coeffs(%,x)}:

The two solutions just differ by the exchange of c1 and c2 (and b1 and b2)

solve(eqs,indets(eqs) minus{t});

NiQ8Ki9JI2ExRzYiLCQqJEkidEdGJiEiIkYqL0kjYTJHRiYiIiEvSSNhM0dGJkYtL0kjYjFHRiYsJCokKUYpIiIjRipGKi9JI2IyR0YmRigvSSNjMUdGJkYtL0kjYzJHRiYiIiIvSSNkMUdGJkYtPCpGJEYrRi4vRjFGKC9GN0YyL0Y5RjwvRjtGLUY9

subs(%,Bform)

LCpJI2EwRzYiIiIiKiZJInRHRiQhIiJJInlHRiRGJUYoKiYpRiciIiNGKEYpRihGKComRidGKCwmRilGJUYlRihGKEYl

By:=subs(a0=0,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNCeUdGKCwoKiZJInRHRighIiJJInlHRigiIiJGMSomKUYwIiIjRjFGMkYxRjEqJkYwRjEsJkYyRjNGM0YxRjFGMzcjRi4=

Ax:=convert(factor(simplify(factor(expand(rationalize(subs(y=Ysol[1],x/(1-t*x)*(y-1)-By)))),symbolic)),parfrac,x)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBeEdGKCwoKiYsJkkieEdGKCIiIiIiI0YyRjJJInRHRighIiJGMiomLCYqJkY0RjJGMUYyRjJGMkY1RjVGNEY1RjIqJkYxRjUpRjRGM0Y1RjI3I0Yu

Ax;By

LCgqJiwmSSJ4RzYiIiIiIiIjRidGJ0kidEdGJiEiIkYnKiYsJiomRilGJ0YlRidGJ0YnRipGKkYpRipGJyomRiVGKilGKUYoRipGJw==

LCgqJkkidEc2IiEiIkkieUdGJSIiIkYmKiYpRiQiIiNGJkYnRiZGJiomRiRGJiwmRidGKEYoRiZGJkYo

Let us check

factor(x/(1-t*x)*(y-1)-Ax-By); factor(%/Ker)

LCQqMCwmKihJInRHNiIiIiJJInhHRidGKEkieUdGJ0YoRihGKCEiIkYoLDAqKEYmRigpRikiIiNGKEYqRihGKComRiZGKEYuRihGK0YlRigqJkYpRihGKkYoRisqJClGKkYvRihGKEYpRihGKkYrRigsJiomRiZGKEYpRihGKEYoRitGKylGJkYvRitGKUYrRipGKywmRipGKEYoRitGK0Yr

Ki4sJiooSSJ0RzYiIiIiSSJ4R0YmRidJInlHRiZGJ0YnRichIiJGJywmRihGJ0YpRipGJ0YpRiosJiomRiVGJ0YoRidGJ0YnRipGKilGJSIiI0YqRihGKg==

<Text-field style="Heading 2" layout="Heading 2">Alternative: an additive decoupling by Thm. 4.11 in [BeBMRa]</Text-field>

orbite

Nyw3JEkieEc2IkkieUdGJTckKipGJiIiIiwoKiZGJEYpRiZGKUYpRiQhIiJGJkYpRiksJkYkRilGJkYsRiwsJkYmRilGKUYsRixGJjckRigqKEYqRilGJEYsRi5GLDckKihGKkYpRiZGLEYuRixGMDckRjIqJilGJiIiI0YsRipGKTckKipGJkYsRiRGKUYqRilGLUYsRjQ3JEY4KiZGJEYpRi1GLDckKipGJkYpRiRGKUYuRixGLUYsRjo3JEY8KihGJEYpRiZGKUYqRiw3JEYkRj4=

The expression that we want to decouple

TD:=x/(1-t*x)*(y-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNUREdGKCooLCZJInlHRigiIiJGMSEiIkYxLCYqJkkidEdGKEYxSSJ4R0YoRjFGMkYxRjFGMkY2RjE3I0Yu

The expression given by the theorem

TD_tau:=factor(-1/5*(subs({x=y*(x*y - x + y)/((x - y)*(y - 1)),y=(x*y - x + y)/(x*(y - 1))}, TD) + 2* subs({x=(x*y - x + y)/(y*(y - 1)),y=(x*y - x + y)/y^2}, TD) +3*subs({x=x*(x*y - x + y)/(y*(x - y)),y=x/(x - y)}, TD)+4*subs({x=y*x/((y - 1)*(x - y)),y=x*y/(x*y - x + y)}, TD))):

Corresponding values of A(x) and B(y)

Acand:=factor(expand(rationalize(subs(y=Ysol[1],TD_tau)+subs(y=Ysol[2],TD_tau))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBY2FuZEdGKCwkKiwjIiIiIiImRjEsOCooIiIqRjEpSSJ0R0YoIiIkRjEpSSJ4R0YoRjJGMUYxKihGNUYxRjZGMSlGOiIiJUYxRjEqKCIjQkYxKUY3IiIjRjFGPEYxISIiKihGQUYxRkBGMSlGOkY4RjFGMSooRjVGMUZARjEpRjpGQUYxRkIqKCIjQEYxRjdGMUZERjFGMSooRkhGMUY3RjFGRkYxRkIqKCIjPUYxRjdGMUY6RjFGMSomIiIoRjFGRkYxRkIqJiIjOUYxRjpGMUYxRjVGQkYxRkBGQkZGRkIsJiomRjdGMUY6RjFGMUYxRkJGQkZCNyNGLg==

Bcand:=factor(simplify(factor(expand(rationalize(subs(x=Xsol[1],x/(1-t*x)*(y-1)-Acand))))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZCY2FuZEdGKCwkKiwjIiIiIiImRjEsOCooIiM5RjFJInRHRihGMSlJInlHRihGMkYxISIiKigiIipGMSlGNiIiI0YxKUY4IiIkRjFGMSooIiNcRjFGNkYxKUY4IiIlRjFGMSooIiNwRjFGNkYxRj5GMUY5KiYiIihGMUZCRjFGMSooIiNWRjFGNkYxKUY4Rj1GMUYxKiYiI0dGMUY+RjFGOSooRjtGMUY2RjFGOEYxRjkqJiIjV0YxRkpGMUYxKiYiI0tGMUY4RjFGOUY7RjFGMUY8RjkpLCZGOEYxRjFGOUY9RjlGSkY5RjE3I0Yu

factor(Acand-Ax)

LCQqLCMiIiIiIiZGJSw4KigiIipGJSlJInRHNiIiIiRGJSlJInhHRixGJkYlRiUqKEYpRiVGKkYlKUYvIiIlRiVGJSooIiM9RiUpRisiIiNGJUYxRiUhIiIqKCIjN0YlRjVGJSlGL0YtRiVGJSooRilGJUY1RiUpRi9GNkYlRjcqKCIjO0YlRitGJUY6RiVGJSooIiNARiVGK0YlRjxGJUY3KihGNEYlRitGJUYvRiVGJSomIiIoRiVGPEYlRjcqJkYpRiVGL0YlRiVGKUY3RiVGNUY3RjxGNywmKiZGK0YlRi9GJUYlRiVGN0Y3Rjc=

convert(%,parfrac,x)

LCoqKCMiIiIiIiZGJSwsKigiIipGJSlJInRHNiIiIiNGJSlJInhHRixGLUYlRiUqKEYpRiVGKkYlRi9GJUYlKihGKUYlRitGJUYvRiUhIiIqJiIjQEYlRitGJUYlIiIoRiVGJUYqRjJGMiooI0YpRiZGJSwmKiZGK0YlRi9GJUYlRiVGMkYyRitGMkYyKipGJEYlLCYqJkYpRiVGK0YlRiVGKUYlRiVGL0YyRipGMkYlKihGN0YlRi5GMkYqRjJGMg==

convert(I0x,parfrac,x)

LC4qJilJInhHNiIiIiMiIiJJInRHRiZGKCEiIiomRiVGKEYpRihGKkYlRigqJCwmKiZGKUYoRiVGKEYoRihGKkYqRioqKCwmRilGKEYoRihGKEYlRipGKUYqRigqJkYkRipGKUYqRio=

factor(%%-9/5*%/t)

LCQqKCMiIigiIiYiIiIsJiomIiIkRidJInRHNiJGJ0YnRidGJ0YnKUYrIiIjISIiRi8=

factor(Acand-(Ax+9/5*I0x/t))

LCQqKCMiIigiIiYiIiIsJiomIiIkRidJInRHNiJGJ0YnRidGJ0YnKUYrIiIjISIiRi8=

factor(Bcand-(By-9/5*J0y/t))

LCQqKCMiIigiIiYiIiIsJiomIiIkRidJInRHNiJGJ0YnRidGJ0YnKUYrIiIjISIiRic=

A new pair of invariants

eqQ_new

LCwqJiwmKiosJkkieUc2IiIiIkYpISIiRilJInhHRihGKSwmKiZJInRHRihGKUYrRilGKUYpRipGKi1JIlFHRig2JEYrRidGKUYpKipGJ0YpRiZGKUYsRiotRjA2JEYnRidGKUYqRilJIktHRihGKUYpKixGK0YpRi5GKSwmRitGKUYpRipGKkYsRiotRjA2JEYpRilGKUYqKiwpRisiIiNGKUYuRilGN0YqRixGKi1GMDYkRitGKUYpRikqKEYnRilGJkYpRjNGKUYqKihGJkYpRitGKUYsRipGKg==

I1x:=Ax-t*x/(1-x*t)/(x-1)*(x*Q(x,1)-Q(1,1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMXhHRigsKiomLCZJInhHRigiIiIiIiNGMkYySSJ0R0YoISIiRjIqJiwmKiZGNEYyRjFGMkYyRjJGNUY1RjRGNUYyKiZGMUY1KUY0RjNGNUYyKixGNEYyRjFGMiwmRjhGNUYyRjJGNSwmRjFGMkYyRjVGNSwmKiYtSSJRR0YoNiRGMUYyRjJGMUYyRjItRkE2JEYyRjJGNUYyRjU3I0Yu

J1y:= -By+ (y-1)*y*Q(y,y)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMXlHRigsKiomSSJ0R0YoISIiSSJ5R0YoIiIiRjMqJilGMCIiI0YxRjJGMUYzKiZGMEYxLCZGMkYzRjNGMUYxRjEqKEYyRjNGOEYzLUkiUUdGKDYkRjJGMkYzRjM3I0Yu

factor(subs(isolate(eqQ(1),Q(x,1)), I1x-J1y))

KjIsMCooSSJ0RzYiIiIiKUkieEdGJiIiI0YnSSJ5R0YmRidGJyomRiVGJ0YoRichIiIqKEYlRidGKUYnRitGJ0YnKiZGKUYnRitGJ0YtKiQpRitGKkYnRidGKUYnRitGLUYnLDIqKi1JIlFHRiY2JEYpRitGJylGJUYqRidGKEYnRjFGJ0YnKiotRjU2JEYrRitGJ0Y3RidGKUYnKUYrIiIkRidGLSoqRjRGJ0Y3RidGKEYnRitGJ0YtKipGOUYnRjdGJ0YpRidGMUYnRicqKEYoRidGJUYnRitGJ0YnKihGMUYnRiVGJ0YpRidGLUYpRi1GK0YnRidGN0YtLCYqJkYlRidGKUYnRidGJ0YtRi1GKUYtLCZGKUYnRitGLUYtLCZGK0YnRidGLUYtRitGLQ==

ratio1:=collect(factor(%/Ker/(x-y)/(y-1)*(1-t*x)),Q,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdyYXRpbzFHRigsKCooSSJ4R0YoIiIiLCZGMEYxSSJ5R0YoISIiRjQtSSJRR0YoNiRGMEYzRjFGMSooRjNGMUYyRjQtRjY2JEYzRjNGMUY0KiwsJiooSSJ0R0YoRjFGMEYxRjNGMUYxRjFGNEYxLCZGM0YxRjFGNEY0KUY+IiIjRjRGMEY0RjNGNEYxNyNGLg==

<Text-field style="Heading 2" layout="Heading 2">Construction of trivial invariants, and equations in one catalytic variable</Text-field>

factor(series(J1y,y=1,1)); factor(series(J0y,y=1,2))

KyksJkkieUc2IiIiIkYmISIiLCQqJEkidEdGJUYnRidGJyomLCZGKkYmRiZGJkYmKUYqIiIjRiciIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

KyksJkkieUc2IiIiIkYmISIiLCRJInRHRiVGJyEiIywmRilGJ0YmRiZGJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIh

A series with no pole at y=1

factor(series(J0y+t^3*J1y^2-(3*t+1)*t*J1y,y=1,1))

KyUsJkkieUc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIiE=

Jy:=J0y+t^3*J1y^2-(3*t+1)*t*J1y: Ix:=I0x+t^3*I1x^2-(3*t+1)*t*I1x:

factor(subs(isolate(eqQ(1),Q(x,1)), Ix-Jy)):

ratio:=collect(factor(%/Ker/(x-y)/(y-1)),Q,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyYXRpb0dGKCwsKi4pSSJ0R0YoIiIkIiIiKUkieEdGKCIiI0YzLDAqKEYxRjNGNEYzSSJ5R0YoRjNGMyomRjFGM0Y0RjMhIiIqKEYxRjNGNUYzRjlGM0YzKiZGNUYzRjlGM0Y7KiQpRjlGNkYzRjNGNUYzRjlGO0YzKSwmRjVGM0Y5RjtGNkY7KSwmKiZGMUYzRjVGM0YzRjNGO0Y2RjspLUkiUUdGKDYkRjVGOUY2RjNGOyomLCYqMEY2RjMpRjEiIiVGM0Y0RjMpRjlGMkYzRkBGO0ZCRjstRkc2JEY5RjlGM0YzKiosMCooRjZGMylGMUY2RjMpRjVGMkYzRjMqKkY2RjNGVEYzRjRGM0Y5RjNGMyomRlRGM0Y0RjNGOyooRjJGM0YxRjNGNEYzRjsqKEYyRjNGMUYzRjVGM0YzRjVGM0Y2RjtGM0YxRjNGQkY7RkFGO0Y7RjNGRkYzRjMqLkYwRjNGP0YzLDJGOEYzKipGNkYzRjFGM0Y1RjNGP0YzRjtGOkY7RjxGM0Y9RjtGPkYzRjVGM0Y5RjtGM0ZARjtGQkY7KUZPRjZGM0YzKiwsMioqRk1GM0ZURjNGNUYzRj9GM0YzKihGVEYzRjVGM0Y5RjNGO0Y8RjsqKEY2RjNGMUYzRj9GM0Y7KihGNkYzRjFGM0Y1RjNGMyomRjFGM0Y5RjNGM0Y5RjNGNkY7RjNGMUYzRkJGO0ZBRjtGT0YzRjMqKCwmRl1vRjNGM0Y7RjNGMUYzRkJGO0Y7NyNGLg==

This ratio has no pole at y=1, clearly.

Now at y=x:

factor(series(ratio,y=x,2))

KyUsJkkieUc2IiIiIkkieEdGJSEiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIh

So I and J are trivial

Let us check

n:=10: factor(series(subs(Q(x,1)=subs(y=1,Qser(n)), Q(1,1)=subs(x=1,y=1,Qser(n)),Ix),t,n))

KzlJInRHNiIiIiMiIiEhIiUiIiIhIiNGJSEiJyIiJCEjRSIiJSEkUSIiIiYhJE0pIiInISU9YiIiKCEmTSFSIiIpIScxMkgiIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjNQ==

Strange! obvious factorisations are not performed here.

n:=10: factor(series(subs(Q(y,y)=subs(x=y,Qser(n)),Jy),t,n))

KzlJInRHNiIiIiMiIiEhIiUiIiIqJiwmKiZGJUYoSSJ5R0YkRighIiJGJUYoRigsJkYsRihGKEYtRi1GJSomLCYqJiIiJ0YoRixGKEYtRjJGKEYoRi5GLSIiJComLCYqJiIjRUYoRixGKEYtRjdGKEYoRi5GLSIiJSomLCYqJiIkUSJGKEYsRihGLUY8RihGKEYuRi0iIiYqJiwmKiYiJE0pRihGLEYoRi1GQUYoRihGLkYtRjIqJiwmKiYiJT1iRihGLEYoRi1GRUYoRihGLkYtIiIoKiYsJiomIiZNIVJGKEYsRihGLUZKRihGKEYuRi0iIikqJiwmKiYiJzEySEYoRixGKEYtRk9GKEYoRi5GLSIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiM1

map(factor,%)

KzlJInRHNiIiIiMiIiEhIiUiIiIhIiNGJSEiJyIiJCEjRSIiJSEkUSIiIiYhJE0pIiInISU9YiIiKCEmTSFSIiIpIScxMkgiIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjNQ==

Value of the constant

factor(series(Jy,y=1,3))

KycsJkkieUc2IiIiIkYmISIiLCgqKCIiI0YmLUkiUUdGJTYkRiZGJkYmKUkidEdGJUYqRiZGJyomIiIlRiZGL0YmRidGKkYmIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiZGJg==

C:=convert(%,polynom)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJDR0YoLCgqKCIiIyIiIi1JIlFHRig2JEYxRjFGMSlJInRHRihGMEYxISIiKiYiIiVGMUY2RjFGN0YwRjE3I0Yu

factor(series(Ix,x=1,2))

KycsJkkieEc2IiIiIkYmISIiKiYsOComKS1JIlFHRiU2JEYmRiYiIiNGJilJInRHRiUiIiZGJkYmKipGL0YmRixGJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiY2I0YtRi5GJkYwRiZGJiomKUY0Ri9GJkYwRiZGJiooIiIkRiZGLEYmKUYxIiIlRiZGJiooRkBGJkY0RiZGQUYmRiYqJkYsRiYpRjFGL0YmRicqJkY0RiZGRUYmRicqJkYvRiYpRjFGQEYmRicqJiIiKkYmRkVGJkYmKiYiIilGJkYxRiZGJ0YvRiZGJiksJkYxRiZGJkYnRi9GJyIiIS1JIk9HRjhGO0Ym

factor(series(Ix,x=0,3))

KydJInhHNiIsKCooIiIjIiIiLUkiUUdGJDYkRihGKEYoKUkidEdGJEYnRighIiIqJiIiJUYoRi1GKEYuRidGKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

<Text-field style="Heading 2" layout="Heading 2">Equation for Q(y,y)</Text-field>

eqcaty:=collect(numer(factor(subs(Q(y,y)=Qyy,Q(1,1)=Q11,(Jy-C)/t))),[Qyy,Q11,t],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdHlHRigsKioqKUkidEdGKCIiIyIiIiksJkkieUdGKEYzRjMhIiJGMkYzKUY2RjJGMylJJFF5eUdGKEYyRjNGMyomLCYqKEY2RjMsKComRjJGM0Y4RjNGMyomIiImRjNGNkYzRjdGM0YzRjNGMUYzRjMqJkY1RjMsJkY2RjNGMkY3RjNGN0YzRjpGM0YzKihGMkYzSSRRMTFHRihGM0YxRjNGM0ZCRjM3I0Yu

factor(%)

LDwqKClJJFF5eUc2IiIiIyIiIilJInRHRiZGJ0YoKUkieUdGJiIiJUYoRigqKkYnRihGJEYoRilGKClGLCIiJEYoISIiKihGJEYoRilGKClGLEYnRihGKCoqRidGKEYlRihGKkYoRi9GKEYoKioiIiZGKEYlRihGKkYoRjNGKEYxKihGJUYoRipGKEYsRihGKComRiVGKEYzRihGMSooRidGKEkkUTExR0YmRihGKkYoRigqKEYwRihGJUYoRixGKEYoKiRGM0YoRigqJkYnRihGJUYoRjEqJkYwRihGLEYoRjFGJ0Yo

The good old quadratic method [Brown, in the sixties]. Let us examine the series Y(t) that solve the following equation:

deq:=collect(map(factor,diff(eqcaty,Qyy)),[Qyy,t],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkZXFHRigsKCosIiIjIiIiKUkieUdGKEYwRjEpLCZGM0YxRjEhIiJGMEYxSSRReXlHRihGMSlJInRHRihGMEYxRjEqKEYzRjEsKComRjBGMUYyRjFGMSomIiImRjFGM0YxRjZGMUYxRjFGOUYxRjEqJkY1RjEsJkYzRjFGMEY2RjFGNjcjRi4=

We see that there are two solutions. We can compute their first coefficients.

unassign(coY)

Yser:=proc(N) local n: add(coY[n]*t^n,n=0..N): end:

n:=0:

factor(series(subs(Qyy=subs(x=y,Qser(n)),y=Yser(n),deq),t,n+1))

KydJInRHNiIsJComLCYmSSRjb1lHRiQ2IyIiISIiIkYsISIiRiwsJkYoRiwiIiNGLUYsRi1GKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRiw=

coY[0]:=1:

Now we iterate the next two lines

n:=n+1: factor(series(subs(Qyy=subs(x=y,Qser(n)),y=Yser(n),deq),t,n+1))

KydJInRHNiIsJiIoaUQnRyEiIiZJJGNvWUdGJDYjIiIpIiIiRistSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLCIiKg==

coY[n]:=solve(coeff(%,t,n))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkkY29ZR0YoNiMiIikiKGlEJ0c3I0Yx

The other solution

unassign(coY)

n:=0:

factor(series(subs(Qyy=subs(x=y,Qser(n)),y=Yser(n),deq),t,n+1))

KydJInRHNiIsJComLCYmSSRjb1lHRiQ2IyIiISIiIkYsISIiRiwsJkYoRiwiIiNGLUYsRi1GKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRiw=

coY[0]:=2:

n:=n+1: factor(series(subs(Qyy=subs(x=y,Qser(n)),y=Yser(n),deq),t,n+1))

KydJInRHNiIsJiIpbStLNiEiIiZJJGNvWUdGJDYjIiIqRidGKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjNQ==

coY[n]:=solve(coeff(%,t,n))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkkY29ZR0YoNiMiIiohKW0rSzY3I0Yx

By Brown's result, each of these two series is a double root of the following discriminant

disc:=collect(numer(factor(discrim(eqcaty, Qyy))),[y],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoLC4iIiUiIiIqKEYvRjBJInRHRihGMClJInlHRigiIiZGMCEiIiomLCoqKCIiKUYwSSRRMTFHRihGMClGMiIiJEYwRjYqJiIiKEYwKUYyIiIjRjBGNiomIiNBRjBGMkYwRjBGMEYwRjApRjRGL0YwRjAqJiwqKigiIztGMEY7RjBGPEYwRjAqJiIjPUYwRkBGMEYwKiYiI1NGMEYyRjBGNiIiJ0Y2RjApRjRGPUYwRjAqJiwqRjlGNkY+RjYqJiIjRUYwRjJGMEYwIiM4RjBGMClGNEZBRjBGMComLCYqJkYvRjBGMkYwRjYiIzdGNkYwRjRGMEYwNyNGLg==

latex(%)

4-4 t \134,y^{5}+\134left(-8 \134mathit{Q11} \134,t^{3}-7 t^{2}+22 t +1\134right) y^{4}+\134left(16 \134mathit{Q11} \134,t^{3}+18 t^{2}-40 t -6\134right) y^{3}+\134left(-8 \134mathit{Q11} \134,t^{3}-7 t^{2}+26 t +13\134right) y^{2}+\134left(-4 t -12\134right) y

factor(discrim(disc,y))

LCQqKCImJVE7IiIiKUkidEc2IiIiKUYlKSw8KigiI2tGJSlJJFExMUdGKCIiJEYlKUYnIiInRiVGJSooIiR3IkYlKUYvIiIjRiUpRiciIiZGJUYlKigiJClHRiVGNUYlKUYnIiIlRiUhIiIqKCIjO0YlRjVGJSlGJ0YwRiVGPSooIiRoIkYlRi9GJUY7RiVGJSooIiRfJUYlRi9GJUZARiVGPSooIiRRI0YlRi9GJSlGJ0Y2RiVGJSomIiNcRiVGQEYlRiUqKCIjR0YlRi9GJUYnRiVGPSomIiRuIkYlRkdGJUY9Ri9GJSomIiNERiVGJ0YlRiVGJUY9RjZGJUYl

algQ11:=collect(op(1,op(3,%)),Q11,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdRMTFHRigsMCooIiNrIiIiKUkkUTExR0YoIiIkRjEpSSJ0R0YoIiInRjFGMSoqIiM7RjEpRjZGNEYxLCgqJiIjNkYxKUY2IiIjRjFGMSomIiM9RjFGNkYxISIiRjFGQkYxKUYzRj9GMUYxKiYsLComIiRoIkYxKUY2IiIlRjFGMSomIiRfJUYxRjpGMUZCKiYiJFEjRjFGPkYxRjEqJiIjR0YxRjZGMUZCRjFGMUYxRjNGMUYxKiYiI1xGMUY6RjFGMSomIiRuIkYxRj5GMUZCKiYiI0RGMUY2RjFGMUYxRkI3I0Yu

latex(%)

64 \134mathit{Q11}^{3} t^{6}+16 t^{3} \134left(11 t^{2}-18 t -1\134right) \134mathit{Q11}^{2}+\134left(161 t^{4}-452 t^{3}+238 t^{2}-28 t +1\134right) \134mathit{Q11} +49 t^{3}-167 t^{2}+25 t -1

Let us use the parametrization of t

numer(factor(subs(t=tZ,algQ11)))

KiYsMiomIiIpIiIiKUkiWkc2IiIiJ0YmISIiKiYiIztGJilGKCIiJkYmRisqJiIiI0YmKUYoIiIlRiZGKyomIiM1RiYpRigiIiRGJkYmKiZGMUYmKUYoRjFGJkYmSSRRMTFHRilGJiomRjdGJkYoRiZGK0YmRitGJixSKigiJDcmRiZGOkYmKUYoIiM3RiZGKyooIiU/XkYmRjpGJilGKCIjNkYmRisqJkY+RiZGP0YmRiYqKCImMzEjRiZGOkYmKUYoRjVGJkYrKiZGQkYmRkNGJkYmKigiJiNSVkYmRjpGJilGKCIiKkYmRisqJiImIyo0I0YmRkhGJkYmKigiJjNFJkYmRjpGJilGKEYlRiZGKyomIiZrayVGJkZMRiZGJiooIiNrRiYpRjpGMUYmRidGJkYmKigiJk8kUUYmRjpGJilGKCIiKEYmRisqJiImYz0nRiZGUkYmRiYqKCImJypvIkYmRjpGJkYnRiZGKyomIiZ3PCZGJkZaRiZGJiooIiUpRyVGJkY6RiZGLkYmRisqJiImK3MjRiZGJ0YmRiYqKCIkRyZGJkY6RiZGMkYmRisqJiIlZyQpRiZGLkYmRiYqKEYtRiZGOkYmRjZGJkYrKiYiJSc0IkYmRjJGJkYmKiYiJC8iRiZGNkYmRisqJiIjUkYmRjlGJkYrKiZGMUYmRihGJkYmRiZGJkYm

Q11Z:=factor(solve(op(1,%),Q11))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVRMTFaR0YoKigsJiIiIkYwSSJaR0YoRjBGMCwoKiYiIiNGMClGMSIiJEYwRjAqJkY0RjBGMUYwISIiRjBGMEYwKSwmRjBGMEY3RjBGNEYwNyNGLg==

G11:=factor(tZ*Q11Z)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRHMTFHRigqJiwoKiYiIiMiIiIpSSJaR0YoIiIkRjJGMiomRjFGMkY0RjIhIiJGMkYyRjJGNEYyNyNGLg==

Let us go back to the bivariate series Q(y,y)

algQyyZ:=subs(Q(y,y)=Qyy,collect(numer(factor(subs(t=tZ,Q11=Q11Z,eqcaty))),Qyy,factor))

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

genus(algQyyZ,y,Qyy)

IiIh

genus(algQyyZ,Z,Qyy)

IiIh

n:=10: factor(series(subs(Qyy=subs(x=y,Qser(n)),Z=Zser,algQyyZ),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

degree 1 when y=1

factor(subs(y=1,algQyyZ))

LCQqLCIiIyIiIiksJkYlRiUqJkYkRiVJIlpHNiJGJUYlRiRGJSwmRiVGJUYpRiVGJUYpRiUsMiomIiIpRiUpRikiIidGJSEiIiomIiM7RiUpRikiIiZGJUYxKiZGJEYlKUYpIiIlRiVGMSomIiM1RiUpRikiIiRGJUYlKiZGJEYlKUYpRiRGJUYlSSRReXlHRipGJSomRjxGJUYpRiVGMUYlRjFGJUYx

Back to G(1,y)=tyQ(y,y)

algG1yZ:=collect(numer(factor(subs(Qyy=1/tZ/y*G1y,algQyyZ))),G1y,factor)

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

Let us check this equation

Gser(1,3); Zser

LCgqKEkieEc2IiIiIkkieUdGJUYmSSJ0R0YlRiZGJioqRiRGJkYnRiYsKComRiRGJkYnRiZGJkYnRiZGJkYmRiYpRigiIiNGJkYmKiosLiomKUYkRi1GJilGJ0YtRiZGJiooRi1GJkYkRiZGMkYmRiYqKEYtRiZGJEYmRidGJkYmKiZGLUYmRjJGJkYmKiYiIiRGJkYnRiZGJkY3RiZGJkYkRiZGJ0YmKUYoRjdGJkYm

KzlJInRHNiIiIiJGJSIiJiIiIyIjTCIiJCIkXCMiIiUiJUw/RiYiJiZbPCIiJyInTGc6IiIoIigiR0o5IiIpIikkPjdNIiIiKiIqJjNTeTciIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

n:=8: factor(series(subs(G1y=subs(x=1,Gser(1,n)),Z=Zser,algG1yZ),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

genus(algG1yZ,y,G1y)

IiIh

factor(discrim(algG1yZ,G1y))

LCQqLCwsKigiIiUiIiIpSSJaRzYiIiIjRidJInlHRipGJ0YnKiZGJkYnRihGJyEiIiooRiZGJ0YpRidGLEYnRicqJkYmRidGKUYnRi5GJ0YuRicpLCoqKEYrRidGKUYnRixGJ0YnKiZGK0YnRilGJ0YuRixGJ0YrRi5GK0YnKSwoRixGLkYnRidGNEYnRitGJyksJkYnRidGKUYnRiZGJyksJkYnRidGNEYnIiIpRidGLg==

The two roots

collect(subs(sqrt(-4*Z^2*y + 4*Z^2 - 4*Z*y + 4*Z + 1)=sqrt(Delta),factor(simplify(factor([solve(algG1yZ,G1y)]),symbolic))),Delta,factor)

NyQsJiowIyIiIiIiI0YmLCoqKEYnRiZJIlpHNiJGJkkieUdGK0YmRiYqJkYnRiZGKkYmISIiRixGJkYnRi5GJiwoRixGLkYmRiZGLUYmRiZGLEYuRipGLiksJkYsRiZGJkYuRidGLilJJkRlbHRhR0YrRiVGJkYmKixGJUYmLDwqKCIiJUYmKUYqIiIkRiYpRixGJ0YmRiYqKCIjN0YmRjhGJkYsRiZGLiooIiIpRiYpRipGJ0YmRjpGJkYmKihGJ0YmRipGJilGLEY5RiZGLiomRj5GJkY4RiZGJiooIiNDRiZGP0YmRixGJkYuKigiIzVGJkYqRiZGOkYmRiYqJiIjO0YmRj9GJkYmKihGSEYmRipGJkYsRiZGLiokRjpGJkYmKiZGRkYmRipGJkYmKiZGOUYmRixGJkYuRidGJkYmRixGLkYqRi5GMEYuRiYsJkYkRi5GNEYm

map(factor,map(series,subs (Delta=-4*Z^2*y + 4*Z^2 - 4*Z*y + 4*Z + 1,%),Z,3))

NyQrJ0kiWkc2IkkieUdGJSIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YnIiIjKytGJCooLCZGJkYnRi4hIiJGJ0YmRjIsJkYmRidGJ0YyRjJGMiwkKipGLkYnLCoqJClGJiIiJEYnRicqJiIiJkYnKUYmRi5GJ0YyKiYiIilGJ0YmRidGJ0Y7RjJGJ0YmRjIpRjNGLkYyRjIiIiEsJCooLCpGN0YnKiRGPEYnRjJGPUYyIiM7RidGJ0YmRjJGM0YyRjJGJ0YoRi4=

G1yZ:=subs(Delta=-4*Z^2*y + 4*Z^2 - 4*Z*y + 4*Z + 1,op(1,%%))

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

Check

factor(series(subs(Z=Zser,G1yZ),t,12))

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

map(factor,subs(x=1,Gser(1,9)))

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

factor(series(%-%%,t,10))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

collect(subs(sqrt(-4*Z^2*y + 4*Z^2 - 4*Z*y + 4*Z + 1)=sDelta,numer(G1yZ)),sDelta,factor)

LD4qKCwqKigiIiMiIiJJIlpHNiJGJ0kieUdGKUYnRicqJkYmRidGKEYnISIiRipGJ0YmRixGJywoRipGLEYnRidGK0YnRidJJ3NEZWx0YUdGKUYnRicqKCIiJUYnKUYoIiIkRicpRipGJkYnRicqKCIjN0YnRjFGJ0YqRidGLCooIiIpRicpRihGJkYnRjNGJ0YnKihGJkYnRihGJylGKkYyRidGLComRjdGJ0YxRidGJyooIiNDRidGOEYnRipGJ0YsKigiIzVGJ0YoRidGM0YnRicqJiIjO0YnRjhGJ0YnKihGQUYnRipGJ0YoRidGLCokRjNGJ0YnKiZGP0YnRihGJ0YnKiZGMkYnRipGJ0YsRiZGJw==

D1:=-coeff(%,sDelta); D0:=collect(subs(sDelta=0,%%),y,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNEMUdGKCwkKiYsKiooIiIjIiIiSSJaR0YoRjNJInlHRihGM0YzKiZGMkYzRjRGMyEiIkY1RjNGMkY3RjMsKEY1RjdGM0YzRjZGM0YzRjc3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNEMEdGKCwqKigiIiMiIiJJIlpHRihGMSlJInlHRigiIiRGMSEiIiomLCoqJiIiJUYxKUYyRjVGMUYxKiYiIilGMSlGMkYwRjFGMSomIiM1RjFGMkYxRjFGMUYxRjEpRjRGMEYxRjEqJiwqKiYiIzdGMUY7RjFGNiomIiNDRjFGPkYxRjYqJiIjO0YxRjJGMUY2RjVGNkYxRjRGMUYxKihGMEYxLCZGMUYxRjJGMUYxKSwmRjFGMSomRjBGMUYyRjFGMUYwRjFGMTcjRi4=

latex(D0)

-2 Z \134,y^{3}+\134left(4 Z^{3}+8 Z^{2}+10 Z +1\134right) y^{2}+\134left(-12 Z^{3}-24 Z^{2}-16 Z -3\134right) y +2 \134left(1+Z \134right) \134left(1+2 Z \134right)^{2}

collect(-4*Z^2*y + 4*Z^2 - 4*Z*y + 4*Z + 1,y,factor)

LCYqKiIiJSIiIiwmRiVGJUkiWkc2IkYlRiVGJ0YlSSJ5R0YoRiUhIiIqJCksJkYlRiUqJiIiI0YlRidGJUYlRi9GJUYl

Parametrization in y

yV:=V/(1+V)^2/Z/(1+Z)*(1+2*Z)^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5VkdGKCosSSJWR0YoIiIiKSwmRjBGMEYvRjAiIiMhIiJJIlpHRihGNCwmRjBGMEY1RjBGNCksJkYwRjAqJkYzRjBGNUYwRjBGM0YwNyNGLg==

numer(factor(subs(y=yV, algG1yZ)))

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

factor([solve(%,G1y)])

NyQqKCw4KigiIiMiIiIpSSJWRzYiRiZGJylJIlpHRioiIiVGJ0YnKihGLUYnRilGJ0YrRidGJyooRiZGJ0YoRicpRixGJkYnISIiKihGLUYnRilGJylGLCIiJEYnRicqJkYmRidGK0YnRicqJkYoRidGLEYnRicqKEYtRidGKUYnRjBGJ0YxKiZGLUYnRjNGJ0YnKiRGKEYnRicqKEY0RidGKUYnRixGJ0YxKiZGJkYnRjBGJ0YnRidGKUYxKSwoKiZGKUYnRixGJ0YnRilGJ0YsRjFGJkYxKihGKUYnLDhGJUYnKihGLUYnRihGJ0YzRidGJ0YuRidGL0YnRjJGJ0Y1RidGN0YxRjpGMUY7RjFGLEYnRidGJ0YnKSwoRj5GJ0YsRjFGJ0YxRiZGMQ==

map(factor,map(series,%,Z,2))

NyQrKUkiWkc2IiokSSJWR0YlISIiIiIhLCQqJilGJyIiI0YoLCYiIiJGL0YnRi9GL0YoRi8tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGL0YtKylGJEYnRiksJComRidGL0YuRi9GKEYvRjBGLQ==

G1yZV:=op(2,%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZHMXlaVkdGKCooSSJWR0YoIiIiLDgqKCIiI0YwKUYvRjNGMClJIlpHRigiIiVGMEYwKihGN0YwRjRGMClGNiIiJEYwRjAqKEY3RjBGL0YwRjVGMEYwKihGM0YwRjRGMClGNkYzRjBGMCooRjdGMEYvRjBGOUYwRjAqJkYzRjBGNUYwRjAqKEY3RjBGL0YwRj1GMCEiIiooRjpGMEYvRjBGNkYwRkEqJkYzRjBGPUYwRkFGNkYwRjBGMEYwKSwoKiZGL0YwRjZGMEYwRjZGQUYwRkFGM0ZBNyNGLg==

collect(op(2,numer(G1yZV)),V,factor);

LCgqKiIiIyIiIilJIlpHNiJGJEYlKSwmRiVGJUYnRiVGJEYlKUkiVkdGKEYkRiVGJSooRidGJSwqKiYiIiVGJSlGJyIiJEYlRiUqJkYwRiVGJkYlRiUqJkYwRiVGJ0YlISIiRjJGNUYlRixGJUYlKiZGKkYlLCgqJkYkRiVGMUYlRiUqJkYkRiVGJkYlRjVGJUYlRiVGJQ==

latex(%)

2 Z^{2} \134left(1+Z \134right)^{2} V^{2}+Z \134left(4 Z^{3}+4 Z^{2}-4 Z -3\134right) V +\134left(1+Z \134right) \134left(2 Z^{3}-2 Z^{2}+1\134right)

Value of V when y=1

solve(factor(yV-1),V)

NiQqJiwmIiIiRiVJIlpHNiJGJUYlRiYhIiIqJkYmRiVGJEYo

V1:=Z/(1+Z)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNWMUdGKComSSJaR0YoIiIiLCZGMEYwRi9GMCEiIjcjRi4=

Check

factor(subs(V=V1,G1yZV))

KiYsKComIiIjIiIiKUkiWkc2IiIiJEYmRiYqJkYlRiZGKEYmISIiRiZGJkYmRihGJg==

<Text-field style="Heading 2" layout="Heading 2">Equation for Q(x,1)</Text-field>

eqcatx:=collect(op(3,numer(factor(subs(Q(x,1)=Qx1,Q(1,1)=Q11,Ix-C)))),[Qx1,Q11],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdHhHRigsLCooKUkkUXgxR0YoIiIjIiIiKUkidEdGKCIiJEYzKUkieEdGKEY2RjNGMyomLCYqKkYyRjNJJFExMUdGKEYzRjRGMylGOEYyRjMhIiIqJiwmRjhGM0YzRj5GMywuKihGMkYzKUY1RjJGM0Y3RjNGMyomRkNGM0Y9RjNGMyooRjZGM0Y1RjNGPUYzRj4qKEY2RjNGNUYzRjhGM0YzRjhGM0YyRj5GM0YzRjNGMUYzRjMqKClGPEYyRjNGNEYzRjhGM0YzKihGQEYzLCoqKEY2RjNGQ0YzRjhGM0YzKiZGNUYzRjhGM0YzRjVGPkYzRj5GM0Y8RjNGPiomKUZARjJGMywmKihGMkYzRjVGM0Y4RjNGM0YzRj5GM0YzNyNGLg==

factor(%)

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

Q11Z; tZ

KigsJiIiIkYkSSJaRzYiRiRGJCwoKiYiIiNGJClGJSIiJEYkRiQqJkYpRiRGJUYkISIiRiRGJEYkKSwmRiRGJEYsRiRGKUYk

KihJIlpHNiIiIiIsJkYlRiVGI0YlISIiKSwmRiVGJSomIiIjRiVGI0YlRiVGK0Yn

algQx1Z:=collect(numer(factor(subs(t=tZ,Q11=Q11Z,eqcatx))),[Qx1],factor)

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

with(algcurves): genus(algQx1Z,x,Qx1)

IiIh

genus(algQx1Z,Z,Qx1); degree(algQx1Z,x)

IiIi

IiIl

Back to G(x,1)= txQ(x,1)

algGx1Z:=collect(op(3,numer(factor(subs(Qx1=Gx1/tZ/x,algQx1Z)))),Gx1,factor)

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

Check

n:=10:factor(series(subs(Z=Zser,Gx1=subs(y=1,Gser(1,n)),algGx1Z),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Quadratic with a double root when x=1

factor(subs(x=1,algGx1Z));factor({solve(%,Gx1)})

KiYpSSJaRzYiIiIjIiIiKSwqKiZGJkYnKUYkIiIlRichIiIqJkYmRidGI0YnRidJJEd4MUdGJUYnRiRGLUYmRic=

PCMqJiwoKiYiIiMiIiIpSSJaRzYiIiIkIiIiIiIiKiYiIiMiIiJJIlpHNiIiIiIhIiIiIiIiIiIiIiJJIlpHNiIiIiI=

factor(discrim(algGx1Z,Gx1))

LCQqLCksJkkieEc2IiIiIkYoISIiIiIjRigsKComSSJaR0YnRihGJkYoRihGLUYpRihGKUYoLCwqJiIiJUYoKUYtIiIkRihGKComIiIpRigpRi1GKkYoRihGLEYpKiYiIiZGKEYtRihGKEYoRihGKCksKiomRjBGKEY1RihGKCooRipGKEYtRihGJkYoRikqJkYwRihGLUYoRihGKEYoRipGKCksKkY6RigqJiIiJ0YoRi1GKEYoRiZGKUYqRihGKkYoRik=

collect(4*Z^3 + 8*Z^2 - Z*x + 5*Z + 1,x,factor)

LCYqJkkiWkc2IiIiIkkieEdGJUYmISIiKiYsJkYmRiZGJEYmRiYpLCZGJkYmKiYiIiNGJkYkRiZGJkYuRiZGJg==

simplify(factor([solve(algGx1Z,Gx1)]),symbolic) assuming(Z>0,Z<1/2,x>0,x<1/2)

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

map(factor,map(series,%,Z,3))

NyQrJUkiWkc2Ii1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyIiIkYsKytGJCwkKigsJkkieEdGJUYsRiwhIiJGLCwmRjFGLCIiI0YyRiwpRjFGNEYyRjIhIiMqKEYwRiwsKComIiIkRixGNUYsRiwqJiIjOEYsRjFGLEYyIiM/RixGLEY1RjJGMiwkKipGNEYsRjBGLCwqKiQpRjFGOkYsRiwqJiIiKEYsRjVGLEYyKiYiI0dGLEYxRixGLCIjVEYyRixGNUYyRjIiIiFGJkYs

Gx1Z:=op(1,%%)

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

factor(subs(x=1,Gx1Z))

KiYsKComIiIjIiIiKUkiWkc2IiIiJEYmRiYqJkYlRiZGKEYmISIiRiZGJkYmRihGJg==

num:=collect(subs(sqrt(-Z*x + Z + 1)=sDelta/ sqrt(1 + 4*Z^3 + 8*Z^2 + (-x + 5)*Z), Gx1Z*2*x^2*Z^2),sDelta,factor)

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

C1:=-coeff(%,sDelta)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNDMUdGKCooLCoqJiIiJSIiIilJIlpHRigiIiNGMkYyKihGNUYyRjRGMkkieEdGKEYyISIiKiZGMUYyRjRGMkYyRjJGMkYyLCpGMEYyKiYiIidGMkY0RjJGMkY3RjhGNUYyRjIsJkY3RjJGMkY4RjI3I0Yu

collect(op(2,%),x,factor)

LCZJInhHNiIhIiIqKCIiIyIiIiwmRihGKEkiWkdGJEYoRigsJkYoRigqJkYnRihGKkYoRihGKEYo

C0:=collect(coeff(num,sDelta,0),x,factor)

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

latex(C0)

-2 Z^{2} x^{4}+Z \134left(12 Z^{3}+24 Z^{2}+16 Z +3\134right) x^{3}+\134left(-12 Z^{6}-64 Z^{5}-132 Z^{4}-134 Z^{3}-70 Z^{2}-16 Z -1\134right) x^{2}+3 \134left(1+Z \134right) \134left(4 Z^{3}+8 Z^{2}+6 Z +1\134right) \134left(1+2 Z \134right)^{2} x -2 \134left(1+Z \134right)^{2} \134left(1+2 Z \134right)^{4}

A parametrisation

xU:=U*(1+Z)/Z/(1+U)*(1+2*Z-U*Z^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4VUdGKCosSSJVR0YoIiIiLCZGMEYwSSJaR0YoRjBGMEYyISIiLCZGL0YwRjBGMEYzLCgqJkYvRjApRjIiIiNGMEYzKiZGOEYwRjJGMEYwRjBGMEYwNyNGLg==

Not so nice at x=1 (does not factor)

factor(xU-1)

LCQqKCwuKiYpSSJVRzYiIiIjIiIiKUkiWkdGKCIiJEYqRioqJkYmRiopRixGKUYqRioqKEYpRipGJ0YqRi9GKiEiIiooRilGKkYnRipGLEYqRjFGJ0YxRixGKkYqRixGMSwmRidGKkYqRipGMUYx

User_set:=algeqtoseries(numer(xU-x), Z,U,2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlVc2VyX3NldEdGKDckKylJIlpHRigiIiIhIiMsJiIiI0YxSSJ4R0YoISIiRjYtSSJPR0YlNiNGMSIiISspRjBGNUYxKiZGNUYxLCZGNUYxIiIkRjZGMUY0RjdGPjcjRi4=

User:=op(2,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVVc2VyR0YoKylJIlpHRihJInhHRigiIiIqJkYwRjEsJkYwRjEiIiQhIiJGMSIiIy1JIk9HRiU2I0YxRjQ3I0Yu

<Text-field style="Heading 3" layout="Heading 3">Can one find a parametrisation that does not have this problem? Probably not</Text-field>

Another one, sent by Mark van Hoeij (May 2024)

xU1:= subs(U=-U,(Z+1)*(U+1+1/Z)*(Z+Z/U+1)); factor(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4VUdGKCooLCYiIiJGMEkiWkdGKEYwRjAsKEkiVUdGKCEiIkYwRjAqJEYxRjRGMEYwLChGMUYwKiZGMUYwRjNGNEY0RjBGMEYwNyNGLg==

LCQqLCwmIiIiRiVJIlpHNiJGJUYlLCgqJkkiVUdGJ0YlRiZGJUYlRiYhIiJGJUYrRiUsKEYpRiVGKkYlRiZGK0YlRiZGK0YqRitGKw==

factor(xU1-1)

LCQqKCw2KiYpSSJVRzYiIiIjIiIiKUkiWkdGKCIiJEYqRioqKEYpRipGJkYqKUYsRilGKkYqKihGKUYqRidGKkYrRiohIiIqJkYmRipGLEYqRioqKCIiJUYqRidGKkYvRipGMSokRitGKkYqKihGKUYqRidGKkYsRipGMSomRilGKkYvRipGKkYnRjFGLEYqRipGLEYxRidGMUYx

factor((z+1)*(U+1+1/z)*(z+z/U+1)-1)

KigsNiomKUkiVUc2IiIiIyIiIilJInpHRiciIiRGKUYpKihGKEYpRiVGKSlGK0YoRilGKSooRihGKUYmRilGKkYpRikqJkYlRilGK0YpRikqKCIiJUYpRiZGKUYuRilGKSokRipGKUYpKihGKEYpRiZGKUYrRilGKSomRihGKUYuRilGKUYmRilGK0YpRilGJiEiIkYrRjY=

numer(factor(subs(x=xU, algGx1Z)))

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

Gx1U1sol:=factor([solve(%,Gx1)])

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

The second expression is the right one

series(subs(U=User, Gx1U1sol[1]),Z); series(subs(U=User, Gx1U1sol[2]),Z)

KylJIlpHNiIsJComLCgqJClJInhHRiQiIiMiIiJGLComIiIkRixGKkYsISIiRitGLEYsRilGL0YvISIjLCgqKEYrRixGJ0YsRipGL0YsKipGK0YsRidGLEYpRi8sJkYqRixGLkYvRixGLComLC4qJClGKkYuRixGLyomIiIoRixGKUYsRiwqJiIjO0YsRipGLEYvIiIpRiwqKEYrRixGKUYsRjRGLEYsKihGLkYsRipGLEY0RixGL0YsRilGL0YvRi8tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLCIiIQ==

KylJIlpHNiJJInhHRiQiIiIqJkYlRiYsJkYlRiYiIiQhIiJGJiIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmRik=

Gx1U1sol[2]

LCQqKkkiVUc2IiIiIiwmRiZGJkkiWkdGJUYmRiYsRiooIiIjRiYpRiQiIiRGJilGKCIiKEYmRiYqKEYrRiZGLEYmKUYoIiInRiZGJiooIiIpRiYpRiRGK0YmRi5GJkYmKigiIiVGJkY1RiZGMUYmRiYqKEYyRiZGJEYmRi5GJkYmKihGNEYmRjVGJilGKCIiJkYmISIiKihGMkYmRiRGJkYxRiZGPCooRjtGJkY1RiYpRihGN0YmRjwqKCIjP0YmRiRGJkY6RiZGPComRjRGJkYxRiZGPCooRjtGJkYkRiZGP0YmRjwqJiIjN0YmRjpGJkY8KigiIipGJkYkRiYpRihGLUYmRiYqKEY3RiZGJEYmKUYoRitGJkYmKiZGNEYmRkhGJkYmKiRGSkYmRiYqJkYtRiZGKEYmRjxGJkY8RiYpLCgqJkYkRiZGSkYmRiYqJkYrRiZGKEYmRjxGJkY8RitGPEY8

collect(op(4,numer(Gx1U1sol[2])),U,factor)

LCoqKiIiIyIiIilJIlpHNiIiIidGJSwmRiVGJUYnRiVGJSlJIlVHRigiIiRGJUYlKigpRiciIiVGJSwqKiYiIilGJSlGJ0YtRiVGJSomRjBGJSlGJ0YkRiVGJSomRjNGJUYnRiUhIiIiIiZGOEYlKUYsRiRGJUYlKihGNkYlLC4qJkYpRiUpRidGOUYlRiUqJkYpRiVGL0YlRjgqJiIjP0YlRjRGJUY4KiZGOUYlRjZGJUY4KiYiIipGJUYnRiVGJUYwRiVGJUYsRiVGJSomLCZGJUYlKiZGJEYlRidGJUYlRiUsLiomRjBGJUY+RiVGJSomRjBGJUYvRiVGJSomRiRGJUY0RiVGOComRi1GJUY2RiVGOEYnRiVGJUYlRiVGOA==

latex(%)

2 Z^{6} \134left(1+Z \134right) U^{3}+Z^{4} \134left(8 Z^{3}+4 Z^{2}-8 Z -5\134right) U^{2}+Z^{2} \134left(6 Z^{5}-6 Z^{4}-20 Z^{3}-5 Z^{2}+9 Z +4\134right) U -\134left(1+2 Z \134right) \134left(4 Z^{5}+4 Z^{4}-2 Z^{3}-3 Z^{2}+Z +1\134right)

<Text-field style="Heading 1" layout="Heading 1">Asymptotics for m-Dyck paths (Section 6.1)</Text-field>

eqG(m)

LCwtSSJHRzYiNiRJInhHRiVJInlHRiUhIiIqJikqJkYnIiIiRihGLUkibUdGJUYtSSJ0R0YlRi1GLSooRitGLUYvRi1GI0YtRi0qLkYvRi1GKEYtRitGLSwmRidGLUYtRilGKSwmRihGLUYtRilGKSwmRiNGLS1GJDYkRi1GKEYpRi1GLSosRitGLUYvRi0sJi1GJDYkRixGLUYtLUYkNiRGLUYtRilGLUYzRiksJkYsRi1GLUYpRilGKQ==

eqQ(m)

LCwtSSJRRzYiNiRJInhHRiVJInlHRiUhIiIiIiJGKiooSSJ0R0YlRiopRidJIm1HRiVGKkYjRipGKiosKUYoIiIjRipGLEYqLCYqJkYtRipGI0YqRioqJilGKEYuRiotRiQ2JEYoRihGKkYpRiosJkYnRipGKEYpRiksJkYoRipGKkYpRilGKioqRixGKiwmKiZGLUYqLUYkNiRGJ0YqRipGKS1GJDYkRipGKkYqRipGOUYpLCZGJ0YqRipGKUYpRio=

Steps:=(x)^m+ (x)^m*y^2/((x - y)*(y - 1)) ; factor(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTdGVwc0dGKCwmKUkieEdGKEkibUdGKCIiIioqRi9GMilJInlHRigiIiNGMiwmRjBGMkY1ISIiRjgsJkY1RjJGMkY4RjhGMjcjRi4=

KiopSSJ4RzYiSSJtR0YlIiIiLCgqJkYkRidJInlHRiVGJ0YnRiQhIiJGKkYnRicsJkYkRidGKkYrRissJkYqRidGJ0YrRis=

The positive critical point of S(x,y)

op(solve({diff(Steps,x),diff(Steps,y)},[x,y]))

NyQvSSJ4RzYiKiYsJkkibUdGJSIiIiomIiIjRiktSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywoKiQpSSNfWkdGLkYrRilGKSomLCZGKEYpRishIiJGKUY1RilGKUYoRjhGKUYpRilGKEY4L0kieUdGJUYs

sol_ens:=map(Z->factor(expand(Z)),[allvalues(%)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzb2xfZW5zR0YoNyQ3JC9JInhHRigqJiwmIiIjIiIiKiQpLCYqJClJIm1HRihGNEY1RjUiIiVGNSNGNUY0RjVGNUY1RjshIiIvSSJ5R0YoLCgqJkY9RjVGO0Y1Rj5GNUY1KiZGPUY1RjdGNUY1NyQvRjEsJComLCZGNEY+RjZGNUY1RjtGPkY+L0ZALChGQkY+RjVGNUZDRj43I0Yu

plot(-m/2 + 1 + sqrt(m^2 + 4)/2,m=1..4); plot(-m/2 + 1 - sqrt(m^2 + 4)/2,m=1..4)

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""-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"Ig==

op(1,sol_ens)

NyQvSSJ4RzYiKiYsJiIiIyIiIiokKSwmKiQpSSJtR0YlRihGKUYpIiIlRikjRilGKEYpRilGKUYvISIiL0kieUdGJSwoKiZGMUYpRi9GKUYyRilGKSomRjFGKUYrRilGKQ==

xc:=op(2,op(1,%)); yc:=op(2,op(2,%%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4Y0dGKComLCYiIiMiIiIqJCksJiokKUkibUdGKEYwRjFGMSIiJUYxI0YxRjBGMUYxRjFGNyEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5Y0dGKCwoKiYjIiIiIiIjRjFJIm1HRihGMSEiIkYxRjEqJkYwRjEpLCYqJClGM0YyRjFGMSIiJUYxRjBGMUYxNyNGLg==

m*xc-m-2*yc

IiIh

yxc:=m*(x-1)/2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR5eGNHRigsJCooIyIiIiIiI0YxLCZJInhHRihGMUYxISIiRjFJIm1HRihGMUYxNyNGLg==

x_c tends to 1 and y_c to 1 as well

asympt(xc,m,2);asympt(yc,m,2)

LCgiIiJGIyomIiIjRiNJIm1HNiIhIiJGIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2IyokKUYmRiVGKEYj

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plot([xc,yc,xc-yc],m=1..5)

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The growth constant

mup:=factor(expand(rationalize((subs(y=yxc,x=xc,Steps)))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRtdXBHRigsJCooIyIiIiIiI0YxKSomLCZGMkYxKiQpLCYqJClJIm1HRihGMkYxRjEiIiVGMUYwRjFGMUYxRjshIiJGO0YxLCgqJkY7RjFGN0YxRjFGOUYxRjJGMUYxRjE3I0Yu

expand(subs(m=1,mup)); expand(subs(m=2,mup)); evalf(%); expand(subs(m=3,mup)); evalf(%); expand(subs(m=4,mup)); evalf(%);

LCYjIiM2IiIjIiIiKiYjIiImRiVGJilGKSNGJkYlRiZGJg==

LCYqJiIiJyIiIikiIikjRiUiIiNGJUYlIiM8RiU=

JCIrdmkwKFIkISIp

LCYqJiMiJEwmIiNhIiIiKSIjOCNGJyIiI0YnRicjIiVAPkYmRic=

JCIrYStBO3IhIik=

LCYqJiMiI2IiIiUiIiIpIiM/I0YnIiIjRidGJyMiJEIiRitGJw==

JCIrJXA9Kkg3ISIo

The exponent

cxy:=diff(diff(Steps,x),y)/sqrt(diff(Steps,x$2)*diff(Steps,y$2))

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

yxc

LCQqKCMiIiIiIiNGJSwmSSJ4RzYiRiVGJSEiIkYlSSJtR0YpRiVGJQ==

cx:=simplify(factor(subs(y=yxc,x=x,cxy))):

simplify(factor(expand(rationalize(subs(x=xc,cx^2)))))

KiYsKCokKUkibUc2IiIiIyIiIkYpKiQpLCZGJEYpIiIlRikjRilGKEYpISIiRihGKUYpLCYqJkYoRilGJUYpRikiIidGKUYv

Note that c is negative

plot(subs(x=xc,y=yc,cxy),m=1..5)

6*-%'CURVESG6$7dw7$$"#5!""$!2cZ\P%*p,4$!#<7$$"2D]+,UI5-"!#;$!2cICb**4'RJ!#<7$$"2X!4=w)G$R5!#;$!20v2<^U?=$!#<7$$"2a4>QS2*f5!#;$!2%yUIwa6HK!#<7$$"2C['HRBi!3"!#;$!0K3l)H"eF$!#:7$$"2v],.#)Q75"!#;$!2()*Q:L$3;K$!#<7$$"1KkGPHN?6!#:$!1aFE!fkMO$!#;7$$"2tX"H)\W,9"!#;$!2`)o0W')>1M!#<7$$"1v\**eHhg6!#:$!2`5efE^(\M!#<7$$"1Fa3xd,"="!#:$!1.)*o*=>D\$!#;7$$"2Ge;Le-??"!#;$!1d#*z;c%e`$!#;7$$"2<MoOx([?7!#;$!2P\F`r`Md$!#<7$$"1&**)z>xHT7!#:$!2$Gf')**p<:O!#<7$$"2$Ryc8J>i7!#;$!1&>BvZBkl$!#;7$$"2AV'G(pHBG"!#;$!2C!GHnVc&p$!#<7$$"2/6AWk:1I"!#;$!2&pcZ8pfIP!#<7$$"2b5@USfBK"!#;$!0pQ4/G;x$!#:7$$"2E_/4Az2M"!#;$!21?ok$3'e!Q!#<7$$"1mJjYe?i8!#:$!1;G&3E#3XQ!#;7$$"1D]+T?<"Q"!#:$!1m?Q-&o#zQ!#;7$$"2`06A'4)>S"!#;$!1'>R_d4i"R!#;7$$"1-/3wfz@9!#:$!1<'>r?U3&R!#;7$$"2#3;Kk3ZU9!#;$!/pxN'>k)R!#97$$"2LkGd'oXh9!#;$!2$3)H;`&f=S!#<7$$"2#He;Le$>["!#;$!2b;!33tx_S!#<7$$"207C[s2K]"!#;$!2v*oI0Yr(3%!#<7$$"2hAX!p\s@:!#;$!2X%)G(oJm<T!#<7$$"2C]+,5C<a"!#;$!12$3!Q)H&\T!#;7$$"2\)pRf^Qi:!#;$!1#)Qa'zO>=%!#;7$$"1x`2vzf#e"!#:$!1xW.yB98U!#;7$$"2#Ryc8\:-;!#;$!2uzwPOtGC%!#<7$$"1&)pRf&pQi"!#:$!2VB7aZg`F%!#<7$$"1iBZM6QV;!#:$!12H??&)3/V!#;7$$"2%Qw_0R@k;!#;$!2OXV#fdGMV!#<7$$"2:JiC\"4$o"!#;$!1j_M)*3BhV!#;7$$"2..17oHPq"!#;$!2EuNR\T-R%!#<7$$"16AW3'[Js"!#:$!1#HX(4!=rT%!#;7$$"1oNr#)pWV<!#:$!0%>2jKyWW!#:7$$"2V%)oPN#Hj<!#;$!2cZkH`99Z%!#<7$$"2@X!4='oSy"!#;$!2t'e#\,h))\%!#<7$$"2*[(\*H'yS!=!#;$!2Wy&oJD)[_%!#<7$$"2(Hf=d<aC=!#;$!20!e-\:3^X!#<7$$"1&)pRRa$[%=!#:$!1k8Sf$fmd%!#;7$$"2LkGdA$[j=!#;$!16&)ReO")*f%!#;7$$"2=LmKpb[)=!#;$!2E'zq5p%fi%!#<7$$"2`2:I[rR!>!#;$!0F()f:h*[Y!#:7$$"2ya4>!HNC>!#;$!1#4^x]JJn%!#;7$$"0,-/_gQ%>!#9$!2b5r+@;fp%!#<7$$"1$f=PA@b'>!#:$!1QP%3IA3s%!#;7$$"0.17!yF%)>!#9$!20(RZV/1UZ!#<7$$"206AWomb+#!#;$!29.BA#G!ew%!#<7$$"1f<N!>p\-#!#:$!2`:--"46(y%!#<7$$"2b18E[#>Y?!#;$!2`5x;6k+"[!#<7$$"1kGd%f/X1#!#:$!2:G[uDw&H[!#<7$$"19Gc_*f`3#!#:$!2$*=y^(RZ^[!#<7$$"2(pRz)>0b5#!#;$!1X6)Hy.B([!#;7$$"2b6BYGPc7#!#;$!1Ezx)>4G*[!#;7$$"2NqS"3`pX@!#;$!0Q+HkNH"\!#:7$$"2OtY$*yk\;#!#;$!1F<^k/*>$\!#;7$$"1@U%)3cz&=#!#:$!2bc$*)*f&G_\!#<7$$"2jD^-*)[c?#!#;$!2'R`!**))R8(\!#<7$$"1_.2MNaEA!#:$!1^"=I1&4"*\!#;7$$"1d9H=#eaC#!#:$!1bz;<)=(3]!#;7$$"2/8E_[ijE#!#;$!1FYSwY"z-&!#;7$$"1c7DI^Q'G#!#:$!1^%Ri^Gg/&!#;7$$"2.*zfR!fjI#!#;$!1Y400u$Q1&!#;7$$"1LmKl=AFB!#:$!1Gc)3jl@3&!#;7$$"2jBZ%*GPkM#!#;$!1;5PpR!))4&!#;7$$"2V!3;7l6mB!#;$!1dS6()pg:^!#;7$$"14=O7Z%yQ#!#:$!1Rx3")f)Q8&!#;7$$"1Z%*)y"y^2C!#:$!1y%f(>H>]^!#;7$$"2<LmKDPwU#!#;$!1tB,"3Nm;&!#;7$$"2/-/3?)4[C!#;$!1Kw$4=<J=&!#;7$$"2'He;$z0pY#!#;$!0`%o6%e!)>&!#:7$$"2&RzexS&p[#!#;$!1j&H$H'oP@&!#;7$$"1He;`(\o]#!#:$!1'\#RI?9H_!#;7$$"1V&3<%>8GD!#:$!1GWZA:NX_!#;7$$"2'yd:^D"pa#!#;$!1%z&QMsXf_!#;7$$"2X*)ydZ"eoD!#;$!1sg0b_]v_!#;7$$"1T#['4g8)e#!#:$!1h&)>*=#y*G&!#;7$$"0(Rz)>%[2E!#9$!1dr`j-s.`!#;7$$"2W%*)y<(*GGE!#;$!1w)Gf*G]=`!#;7$$"1&)pR*Rz"\E!#:$!1RmS)3NJL&!#;7$$"2Z"He;f=oE!#;$!19a*QRoiM&!#;7$$"2$Ryc$**p$)o#!#;$!12mXW5.g`!#;7$$"*on!3F!")$!1[!\I3#Gt`!#;7$$"2t\**)z,UHF!#;$!1'4YM3\uQ&!#;7$$"1CZ%*Gg*yu#!#:$!1Wqf'pW&*R&!#;7$$"2;LmK@;"pF!#;$!1B7*['QD8a!#;7$$"1`06-V1*y#!#:$!1A$)y`d'fU&!#;7$$"1a3<9(>)3G!#:$!1T5y'>!RQa!#;7$$"2`/4=%3fGG!#;$!0Ym)RKm]a!#:7$$"2E^-09c([G!#;$!1%*\+yv,ja!#;7$$"2C['H>58qG!#;$!0J))>&f$fZ&!#:7$$"1^-0!pH(*)G!#:$!1L4P2Ni([&!#;7$$"2nOtY4w!4H!#;$!1Q-d?b,*\&!#;7$$"029G3d(HH!#9$!1&*)HHkO5^&!#;7$$"2lD^-0$R]H!#;$!1*fEK*G(G_&!#;7$$"2DT#[c]"*oH!#;$!1plr*)QOLb!#;7$$"2PrU&3d&3*H!#;$!1..b@:jXb!#;7$$"2&=PuGY@4I!#;$!1Wf(yeldb&!#;7$$"1,-/[uoII!#:$!1U#zawpuc&!#;7$$"2mMpQ<"=^I!#;$!1X*[->$\yb!#;7$$"2([(\*H'z%pI!#;$!1CVAEk@)e&!#;7$$"0%zed"e+4$!#9$!1_#))f))=!*f&!#;7$$"2mKlI4t26$!#;$!1<Z\uPv4c!#;7$$"2<NqSd*QJJ!#;$!0*)>.N,.i&!#:7$$"1w_0"p.0:$!#:$!1K&=P%3'*Hc!#;7$$"1-.1__HqJ!#:$!1N%f*GN%)Rc!#;7$$"1>Qw7Pw!>$!#:$!1Mh@Z(Q*\c!#;7$$"1rU&3`m6@$!#:$!1z$)*3rw)fc!#;7$$"1Fa3PL:KK!#:$!13;W=7(*pc!#;7$$"1'=Pu_Q1D$!#:$!17N,cnvyc!#;7$$"1Ryct%[9F$!#:$!1'R[ojI&)o&!#;7$$"2OoOt'QM#H$!#;$!1[9*4WA#)p&!#;7$$"2mFb5X![7L!#;$!1YZ"GL[uq&!#;7$$"2X&4>)Rm2L$!#;$!1BEf`4t:d!#;7$$"2(\**)z:5DN$!#;$!1&f?a_jas&!#;7$$"2nOtY(*H4P$!#;$!0;Ts*4hLd!#:7$$"2.,-/gcBR$!#;$!1H="f)z(Hu&!#;7$$"2'pQx%zA8T$!#;$!1\L:w<<^d!#;7$$"2%**)zfrJ@V$!#;$!1nP7Gz0gd!#;7$$"2jC\)Hn%>X$!#;$!1x.=H.Uod!#;7$$"2BX!4=;isM!#;$!0yu7)R/xd!#:7$$"2x[(\>wg"\$!#;$!0r._<t[y&!#:7$$"2LnMpe'37N!#;$!1)y-L"HA$z&!#;7$$"1lHfy%eL`$!#:$!1AzEsIz,e!#;7$$"2229Gsv=b$!#;$!1PX*3Sp"4e!#;7$$"2mMpQ&[(=d$!#;$!1@GLV&\q"e!#;7$$"1He;8f`#f$!#:$!1Hm5(f(4De!#;7$$"28AW)G([Fh$!#;$!19iq(4")G$e!#;7$$"2MoOtm0Bj$!#;$!1"4iRlG.%e!#;7$$"2$He;8.-aO!#;$!0Z!p8L][e!#:7$$"117C))=`tO!#:$!1^Q0=]wbe!#;7$$"2D['HfYO%p$!#;$!1N<)yXKM'e!#;7$$"2c:JiCUKr$!#;$!1`/A]ZIqe!#;7$$"1u[(\V!)Qt$!#:$!1bg%3.Px(e!#;7$$"2b06AO*H`P!#;$!1tT[\Tl%)e!#;7$$"17C[OxftP!#:$!1*p5M'o!=*e!#;7$$"2%)oPv5VMz$!#;$!0tu#HVs)*e!#:7$$"2nHf=P>U"Q!#;$!1;T"*ot)e!f!#;7$$"1$f=PQHU$Q!#:$!1Z+7g5r7f!#;7$$"2Oxa4^#paQ!#;$!1\9fMTh>f!#;7$$"2%He;$>')\(Q!#;$!0GN`7'QEf!#:7$$"1([(\zRj$*Q!#:$!1`+q<\aKf!#;7$$"2b<NqW1]"R!#;$!1c\m\$H&Rf!#;7$$"0#RyOA7MR!#9$!1Jx-#45d%f!#;7$$"1#Rycl.X&R!#:$!1%4(G*=KA&f!#;7$$"2W&3<u7,uR!#;$!14$*e***4%ef!#;7$$"1Pu[x>n&*R!#:$!0T>ak'>lf!#:7$$"1u[(\bGW,%!#:$!1'GyP>75(f!#;7$$"1b4>QurNS!#:$!1k">;4Xv(f!#;7$$"1.17W*>^0%!#:$!1'30B@PM)f!#;7$$"14>QOKMwS!#:$!1LHEZc")*)f!#;7$$"14<M[`l%4%!#:$!1MfO&*RE&*f!#;7$$"2&e;L12^:T!#;$!1***=+n29+'!#;7$$"19Gc_flNT!#:$!1&3eO(4G2g!#;7$$"0'>RQ!)ybT!#9$!1,yAJ748g!#;7$$"1[&4>1Ye<%!#:$!1ohau?#)=g!#;7$$"1zb6Vb6&>%!#:$!17/Y5QFCg!#;7$$"2b18EOYf@%!#;$!1\FQ!z3,.'!#;7$$"205?Sk*zNU!#;$!1'R+2<9c.'!#;7$$"1'>RyG%pcU!#:$!1r")pc.NTg!#;7$$"1-.1s*3cF%!#:$!1vSd=C\Yg!#;7$$"1v\**QK^'H%!#:$!1F!)3J-7_g!#;7$$"1,,-%)e`;V!#:$!1lQEerXdg!#;7$$"1NoO$z4lL%!#:$!1"R:6iHF1'!#;7$$"1xa4>EPdV!#:$!1&>2oe#=og!#;7$$"03;K/)ewV!#9$!1^a4'ocJ2'!#;7$$"1['HfEniR%!#:$!1XlH*\.#yg!#;7$$"2NlIhY&*zT%!#;$!1"\i5)3s$3'!#;7$$"1#He;dowV%!#:$!1U#*z#Rn')3'!#;7$$"1w^.2!)ydW!#:$!1A=O%fyO4'!#;7$$"1kGda*[#yW!#:$!1rzY:is)4'!#;7$$"1uY$pacq\%!#:$!1EdugKK.h!#;7$$"1%yc8$[5<X!#:$!1Nv\3"z"3h!#;7$$"1uY$p]+q`%!#:$!1-IsCK&H6'!#;7$$"1(Qxap#GeX!#:$!1=dUN8,=h!#;7$$"1BY#\Ijqd%!#:$!0k)*HiLC7'!#:7$$"1RxaHAt)f%!#:$!1#p=Ie)[Fh!#;7$$"1&3<Mw'G=Y!#:$!1%38tA2?8'!#;7$$"1:Gc_\jPY!#:$!1$[E!*\Qk8'!#;7$$"1*yd:ZS%eY!#:$!1PbB]+;Th!#;7$$"1He;`,LzY!#:$!1>S)*\f&e9'!#;7$$"1f<NqmL)p%!#:$!1)G)Rm+4]h!#;7$$"1$oOtu?&=Z!#:$!1^La.paah!#;7$$"1X)oPV=#QZ!#:$!1&Q=o\d)eh!#;7$$"1U$oO$4dfZ!#:$!1bR3Vu[jh!#;7$$"2vc8FyY!yZ!#;$!1w*Rf;eu;'!#;7$$"1w^.npE*z%!#:$!0PNJgy><'!#:7$$"1(Rze0:#>[!#:$!0yZ2Y*=wh!#:7$$"1*pRzYq*Q[!#:$!1GjLaKK!='!#;7$$"1*)yd&fT(e[!#:$!1BjyCXU%='!#;7$$"1c8F%*o!*y[!#:$!1%ek:6r&)='!#;7$$"1F`1t<G+\!#:$!1'H&pVn#H>'!#;7$$"1'4>QW!))>\!#:$!1dC5&*\)o>'!#;7$$"16AW[oAR\!#:$!1$yS(e%f2?'!#;7$$"19HeOy!*f\!#:$!1seIbb'[?'!#;7$$"/,-/Qa!)\!#8$!1(yWgXE*3i!#;7$$"#]!""$!1#R+'HCs7i!#;-%&COLORG6'%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-%+_ATTRIBUTEG6#/%'sourceG%,mathdefaultG-%%VIEWG6$;$"#5!""$"#]!""%(DEFAULTG-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"m6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"Ig==

cval:=-sqrt(%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjdmFsR0YoLCQqJCkqJiwoKiQpSSJtR0YoIiIjIiIiRjcqJCksJkYzRjciIiVGNyNGN0Y2RjchIiJGNkY3RjcsJiomRjZGN0Y0RjdGNyIiJ0Y3Rj1GPEY3Rj03I0Yu

c1:=simplify(normal(subs(m=1,x=subs(m=1,xc),y=subs(m=1,yc),cxy))); evalf(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCwmKiYjIiIiIiIlRjEpIiImI0YxIiIjRjEhIiJGMEYxNyNGLg==

JCErVSpwLDQkISM1

arccos(-c1)

LCQqJiMiIiMiIiYiIiJJI1BpRyUqcHJvdGVjdGVkR0YnRic=

asympt(cval,m,3)

LCoqJiMiIiIiIiNGJSlGJkYkRiUhIiIqKCNGJSIiJUYlRidGJUkibUc2IkYoRiUqKCMiIiYiIztGJUYnRiUpRixGJkYoRiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtNiMqJClGLCIiJEYoRiU=

minimal (or: annihilating) polynomial of c

map(Z->Z^2,isolate(cval^2-c^2,sqrt(m^2 + 4)))

LywmKiQpSSJtRzYiIiIjIiIiRikiIiVGKSokKSwoKiYpSSJjR0YnRihGKSwmKiZGKEYpRiVGKUYpIiInRilGKSEiIkYkRilGKEYpRihGKQ==

polmin:=collect(op(2,factor(op(2,%)-op(1,%))),c,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdwb2xtaW5HRigsKComLCYqJiIiJSIiIilJIm1HRigiIiNGM0YzIiM3RjNGMylJImNHRihGMkYzRjMqJiwmRjEhIiIiIilGPEYzKUY5RjZGM0YzKiRGNEYzRjM3I0Yu

factor(subs(c=cval,polmin))

IiIh

latex(polmin)

\134left(4 m^{2}+12\134right) c^{4}+\134left(-4 m^{2}-8\134right) c^{2}+m^{2}

polz:=collect(numer(factor(subs(c=(z+1/z)/2,polmin))),z,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVwb2x6R0YoLC4qJiwmKiQpSSJtR0YoIiIjIiIiRjUiIiRGNUY1KUkiekdGKCIiKUY1RjUqJiIiJUY1KUY4IiInRjVGNSomLCYqJkY0RjVGMkY1RjVGNEY1RjUpRjhGO0Y1RjUqJkY7RjUpRjhGNEY1RjVGMUY1RjZGNTcjRi4=

latex(%)

\134left(m^{2}+3\134right) z^{8}+4 z^{6}+\134left(2 m^{2}+2\134right) z^{4}+4 z^{2}+m^{2}+3

with(NumberTheory): CyclotomicPolynomial(10,z)

LCwqJClJInpHNiIiIiUiIiJGKCokKUYlIiIkRighIiIqJClGJSIiI0YoRihGJUYsRihGKA==

seq(degree(CyclotomicPolynomial(n,z)),n=1..40)

NkoiIiJGIyIiI0YkIiIlRiQiIidGJUYmRiUiIzVGJSIjN0YmIiIpRikiIztGJiIjPUYpRihGJyIjQUYpIiM/RihGK0YoIiNHRikiI0lGKkYtRioiI0NGKCIjT0YrRjBGKg==

Looking for cyclotmopic polynomials of degree at most 8

ens:={}: for n from 1 to 200 do if degree(CyclotomicPolynomial(n,z)) <9 then ens:=ens union{[n,CyclotomicPolynomial(n,z)]} fi: od:

nops(ens)

IiM9

ens[1]; ens[18];

NyQiIiIsJkkiekc2IkYjRiMhIiI=

NyQiI0ksMCokKUkiekc2IiIiKSIiIkYqKiQpRiciIihGKkYqKiQpRiciIiZGKiEiIiokKUYnIiIlRipGMSokKUYnIiIkRipGMUYnRipGKkYq

for n from 1 to 18 do print(n,ens[n][1],lcoeff(factor(rem(polz,ens[n][2],z)),z)) od

NiUiIiJGIywmKiYiIiVGIylJIm1HNiIiIiNGI0YjIiM7RiM=

NiUiIiNGIywmKiYiIiUiIiIpSSJtRzYiRiNGJ0YnIiM7Ric=

NiUiIiRGIywmKiQpSSJtRzYiIiIjIiIiRioiIiYhIiI=

NiUiIiVGIywkKiZGIyIiIilJIm1HNiIiIiNGJkYm

NiUiIiZGIywkKiYsJkkibUc2IiIiIkYpISIiRiksJkYnRilGKUYpRilGKg==

NiUiIidGIywmKiQpSSJtRzYiIiIjIiIiISIiIiImRio=

NiUiIihGIyEiJQ==

NiUiIilGIyIiJQ==

NiUiIipGIywmKiQpSSJtRzYiIiIjIiIiISIiIiIkRis=

NiUiIzVGIyomLCZJIm1HNiIiIiJGKCEiIkYoLCZGJkYoRihGKEYo

NiUiIzYiIzcsJiokKUkibUc2IiIiIyIiIkYrIiIkRis=

NiUiIzciIzkiIiU=

NiUiIzgiIzosJiokKUkibUc2IiIiIyIiIkYrIiIkRis=

NiUiIzkiIzsiIiU=

NiUiIzoiIz0sJiokKUkibUc2IiIiIyIiIkYrIiIkRis=

NiUiIzsiIz8sJiokKUkibUc2IiIiIyIiIkYrIiIoRis=

NiUiIzwiI0MiIiU=

NiUiIz0iI0ksJiokKUkibUc2IiIiIyIiIiEiIiIiJEYs

ens[5];ens[10];factor( subs(m=1,polz))

NyQiIiYsLCokKUkiekc2IiIiJSIiIkYqKiQpRiciIiRGKkYqKiQpRiciIiNGKkYqRidGKkYqRio=

NyQiIzUsLCokKUkiekc2IiIiJSIiIkYqKiQpRiciIiRGKiEiIiokKUYnIiIjRipGKkYnRi5GKkYq

LCQqKCIiJSIiIiwsKiQpSSJ6RzYiRiRGJUYlKiQpRikiIiRGJUYlKiQpRikiIiNGJUYlRilGJUYlRiVGJSwsRidGJUYrISIiRi5GJUYpRjJGJUYlRiVGJQ==

polz; ens[18]

LC4qJiwmKiQpSSJtRzYiIiIjIiIiRioiIiRGKkYqKUkiekdGKCIiKUYqRioqJiIiJUYqKUYtIiInRipGKiomLCYqJkYpRipGJkYqRipGKUYqRiopRi1GMEYqRioqJkYwRiopRi1GKUYqRipGJUYqRitGKg==

NyQiI0ksMCokKUkiekc2IiIiKSIiIkYqKiQpRiciIihGKkYqKiQpRiciIiZGKiEiIiokKUYnIiIlRipGMSokKUYnIiIkRipGMUYnRipGKkYq

rem(polz, ens[18][2],z)

LDAqJiwmKiQpSSJtRzYiIiIjIiIiISIiIiIkRitGKilJInpHRigiIihGKkYqKiYiIiVGKilGLiIiJ0YqRioqJiwmRiVGKkYsRipGKilGLiIiJkYqRioqJiwmKiZGLEYqRiZGKkYqRjdGKkYqKUYuRjFGKkYqKiZGNUYqKUYuRixGKkYqKiZGMUYqKUYuRilGKkYqKiZGJEYqRi5GKkYq

latex(%)

\134left(-m^{2}-3\134right) z^{7}+4 z^{6}+\134left(m^{2}+3\134right) z^{5}+\134left(3 m^{2}+5\134right) z^{4}+\134left(m^{2}+3\134right) z^{3}+4 z^{2}+\134left(-m^{2}-3\134right) z

<Text-field style="Heading 1" layout="Heading 1">Example: m=1</Text-field>

N:=100: ser:=series(subs(x=1,y=1,Gser(1,N)),t,N+1):

Let us check asymptotics numerically

Estimate of the radius

plot([seq([1/n,coeff(ser,t,n)/coeff(ser,t,n-1)],n=2..N)],view=[0..0.2,5..15],style=point)

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

evalf(subs(m=1,mup))

JCIrJSpwLDQ2ISIp

Estimates of the exponent

plot([seq([1/n,n^2*(1-coeff(ser,t,n-1)*coeff(ser,t,n+1)/coeff(ser,t,n)^2)],n=2..N-1)],view=[0..0.2,-3.6..-2.7],style=point)

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NiI=

evalf(-1-Pi/arccos(-subs(m=1,cval)))

JCErLCsrK04hIio=

<Text-field style="Heading 1" layout="Heading 1">Example: m=2</Text-field>

N:=100: ser:=series(subs(x=1,y=1,Gser(2,N)),t,N+1):

Let us check asymptotics numerically

Estimate of the radius

plot([seq([1/n,coeff(ser,t,n)/coeff(ser,t,n-1)],n=2..N)],view=[0..0.2,20..40],style=point)

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NiI=

evalf(subs(m=2,mup))

JCIrdWkwKFIkISIp

Estimates of the exponent

plot([seq([1/n,n^2*(1-coeff(ser,t,n-1)*coeff(ser,t,n+1)/coeff(ser,t,n)^2)],n=2..N-1)],view=[0..0.2,-4.2..-3],style=point)

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JSlDSElMRFJFTkc2Ig==Ig==

evalf(-1-Pi/arccos(-subs(m=2,cval)))

JCErXWM3QlIhIio=

<Text-field style="Heading 1" layout="Heading 1">Asymptotics for mirrored m-Dyck paths (Section 6.2)</Text-field>

eqGp(m)

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eqQp(m)

LCwtSSJRRzYiNiRJInhHRiVJInlHRiUhIiIiIiJGKioqKUYnSSJtR0YlRipJInRHRiVGKiwmKiZGI0YqRihGKkYpLUYkNiRGJ0YqRipGKiwmRihGKkYqRilGKUYpKiwpRigiIiNGKkYuRiosJiomRixGKkYjRipGKi1GJDYkRipGKEYpRipGM0YpLCZGJ0YqRipGKUYpRioqKkYuRiosJiomRixGKkYxRipGKS1GJDYkRipGKkYqRipGM0YpRjtGKUYq

Steps:=x^m*y*(x+y-1)/(x-1)/(y-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTdGVwc0dGKCosKUkieEdGKEkibUdGKCIiIkkieUdGKEYyLChGMEYyRjNGMkYyISIiRjIsJkYwRjJGMkY1RjUsJkYzRjJGMkY1RjU3I0Yu

The positive critical point of S(x,y) (in the domain of convergence)

op(solve({diff(Steps,x),diff(Steps,y)},[x,y]))

NiU3JC9JInhHNiIiIiEvSSJ5R0YmIiIiNyQvRiUsKCokKS1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLChGKkYqKiYpSSNfWkdGMiIiI0YqSSJtR0YmRipGKiomLCYqJkY6RipGO0YqISIiRipGP0YqRjlGKkYqRjpGKkYqKiZGOkYqRjBGKkY/RipGKi9GKUYwNyQvRiVGKi9GKUYn

op(2,[%])

NyQvSSJ4RzYiLCgqJCktSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywoIiIiRjAqJilJI19aR0YrIiIjRjBJIm1HRiVGMEYwKiYsJiomRjRGMEY1RjAhIiJGMEY5RjBGM0YwRjBGNEYwRjAqJkY0RjBGKUYwRjlGMEYwL0kieUdGJUYp

sol_ens:=map(Z->factor(expand(Z)),[allvalues(%)])

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plot([-(-2*m^2 + sqrt(4*m^2 + 1) - 1)/(2*m^2), -(-2*m - 1 + sqrt(4*m^2 + 1))/(2*m)],m=1..4);

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op(1,sol_ens)

NyQvSSJ4RzYiLCQqKCMiIiIiIiNGKSwoKiZGKkYpKUkibUdGJUYqRilGKSokKSwmKiYiIiVGKUYtRilGKUYpRilGKEYpRilGKUYpRilGLSEiIkYpL0kieUdGJSwkKihGKEYpRi5GNCwoKiZGKkYpRi5GKUYpRilGKUYvRilGKUYp

xc:=op(2,op(1,%)); yc:=op(2,op(2,%%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4Y0dGKCwkKigjIiIiIiIjRjEsKComRjJGMSlJIm1HRihGMkYxRjEqJCksJiomIiIlRjFGNUYxRjFGMUYxRjBGMUYxRjFGMUYxRjUhIiJGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5Y0dGKCwkKigjIiIiIiIjRjFJIm1HRighIiIsKComRjJGMUYzRjFGMUYxRjEqJCksJiomIiIlRjEpRjNGMkYxRjFGMUYxRjBGMUYxRjFGMTcjRi4=

factor(m*xc-m+1-yc)

IiIh

yxc:=m*(x-1)+1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR5eGNHRigsJiomLCZJInhHRigiIiJGMiEiIkYySSJtR0YoRjJGMkYyRjI3I0Yu

x_c tends to 1 and y_c to 1 as well

asympt(xc,m,3);asympt(yc,m,2)

LCoiIiJGIyokSSJtRzYiISIiRiMqJiNGIyIiI0YjKUYlRipGJ0YjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjKiQpRiUiIiRGJ0Yj

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plot([xc,yc],m=1..5)

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The growth constant

mup0:=factor(expand(rationalize((subs(y=yxc,x=xc,Steps)))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVtdXAwR0YoKigpIyIiIiIiI0kibUdGKEYxKSomLCgqJkYyRjEpRjNGMkYxRjEqJCksJiomIiIlRjFGOEYxRjFGMUYxRjBGMUYxRjFGMUYxRjghIiJGM0YxLCZGOUYxKiZGMkYxRjNGMUYxRjE3I0Yu

The expression of the paper

mup:=(2*m+sqrt(1+4*m^2))*(1+sqrt(1+4*m^2))^(2*m)/(2*m)^(2*m)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRtdXBHRigqKCwmKiQpLCYqJiIiJSIiIilJIm1HRigiIiNGNUY1RjVGNSNGNUY4RjVGNSomRjhGNUY3RjVGNUY1KSwmRjBGNUY1RjUsJEY6RjVGNSlGPUY9ISIiNyNGLg==

seq(simplify(subs(m=i,mup0-mup)),i=1..6)

NigiIiFGI0YjRiNGI0Yj

The exponent

cxy:=diff(diff(Steps,x),y)/sqrt(diff(Steps,x$2)*diff(Steps,y$2))

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

yxc

LCYqJiwmSSJ4RzYiIiIiRichIiJGJ0kibUdGJkYnRidGJ0Yn

cx:=simplify(factor(subs(y=yxc,x=x,cxy))):

simplify(factor(expand(rationalize(simplify(subs(x=xc,cx^2))))))

KiYsKComSSJtRzYiIiIiKSwmKiYiIiVGJylGJSIiI0YnRidGJ0YnI0YnRi1GJyEiIiomRi1GJ0YsRidGJ0YnRidGJywmKiYiIidGJ0YsRidGJ0YtRidGLw==

Note that c is negative

plot(subs(x=xc,y=yc,cxy),m=1..6)

6*-%'CURVESG6$7dw7$$"#5!""$!0Z\P%*p,4$!#:7$$"2"Gc7D!)GE5!#;$!2FxC:gw*GI!#<7$$"238E_4h"\5!#;$!1d0R./TxH!#;7$$"2$pQxaU)[2"!#;$!2tlZMl77#H!#<7$$"1.17Czx+6!#:$!2&4uFQo[mG!#<7$$"2V)oPD&[l7"!#;$!1)>(*=&\y8G!#;7$$"2./3;<T/:"!#;$!1,%\=zLkw#!#;7$$"2;KkGi!=v6!#;$!1XObSj()=F!#;7$$"2'=Pu)>m2?"!#;$!2N[2%fm?rE!#<7$$"1%yc8spiA"!#:$!1)eezkh^i#!#;7$$"2&Gd9HK]_7!#;$!2uHdP5q#zD!#<7$$"2tU&3<(4cF"!#;$!2'[)>XmO+a#!#<7$$"2Pu[(\@i,8!#;$!1>(\LM[r\#!#;7$$"1*zf>RTxK"!#:$!2_5K'4URbC!#<7$$"2-/3;77HN"!#;$!2E7!)RF\jT#!#<7$$"2"Qw_b&pdP"!#;$!1.@DY&o=Q#!#;7$$"2=Qw_D\HS"!#;$!2/XvW:I?M#!#<7$$"2LlIh-ufU"!#;$!1saV$GL#4B!#;7$$"2tX"H3tv_9!#;$!2Gs57;Q@F#!#<7$$"29Gc70lkZ"!#;$!2()*)RX$[@SA!#<7$$"2#>Qw-iZ-:!#;$!1ZCy2y81A!#;7$$"2E]+,(\CF:!#;$!2t0vkNwX<#!#<7$$"2.,-/e)3`:!#;$!1sG1cR`U@!#;7$$"2U!3;#3@od"!#;$!1)Q!*HRzQ6#!#;7$$"2kGd9z>Cg"!#;$!2.VQ3JoP3#!#<7$$"23:Igl4!H;!#;$!29td+&\L`?!#<7$$"2F`18@c@l"!#;$!1%Qp(4t^F?!#;7$$"2$Gc7D^:x;!#;$!1**4enRJ+?!#;7$$"27BY#\9)Hq"!#;$!14XMSl#H(>!#;7$$"1@U%)ouCG<!#:$!2;:d'\F"o%>!#<7$$"1*zf>k$p_<!#:$!1.K!)RO;A>!#;7$$"26BY#\p$)z<!#;$!2redSG"[&*=!#<7$$"2CX!4=kA/=!#;$!2od&4025s=!#<7$$"2za4>Qn-$=!#;$!2#eE@>mtZ=!#<7$$"2&*)ydlV'Q&=!#;$!2n%)RaGwh#=!#<7$$"2x`2:5i'z=!#;$!1%=]E%*[J!=!#;7$$"2Qw_0wNR!>!#;$!12L&[)Q)>y"!#;7$$"2*f>RG(3$H>!#;$!2`KipMj.w"!#<7$$"2_06AW:T&>!#;$!2)f=j:kqR<!#<7$$"2a18Ex&3!)>!#;$!1Ap.ZNd=<!#;7$$"2j=PuG)40?!#;$!2Nc?'*["o)p"!#<7$$"2CT#['>x1.#!#;$!2#Q7O8@zy;!#<7$$"28BY#*HWg0#!#;$!22%[&=f0&f;!#<7$$"1/3;KSNz?!#:$!2&)))=$Gh:U;!#<7$$"2Z"He;'pg5#!#;$!2_!GBVspA;!#<7$$"1W)oPNk*H@!#:$!2bmIgXmcg"!#<7$$"2X$pQF6Wb@!#;$!2o=<0f%)ye"!#<7$$"2C^-0lD)z@!#;$!2))[qLD=7d"!#<7$$"1T#['H:!p?#!#:$!1E$[8z/Jb"!#;7$$"2v`2:DZ.B#!#;$!2#Q$)=tNuP:!#<7$$"2&Qw_b$epD#!#;$!22N_L^h1_"!#<7$$"2$)pRz[67G#!#;$!2=)\%oh5a]"!#<7$$"1KjE.1u2B!#:$!2N!z&[+l!*["!#<7$$"2/3;KuI1L#!#;$!2AAD@4O_Z"!#<7$$"2w^.2%**pcB!#;$!2G>#=Svyf9!#<7$$"2BY#\)\")=Q#!#;$!2E&QFh3;X9!#<7$$"1&*)ydgYqS#!#:$!2m:f$Hf#3V"!#<7$$"2&ze<N">@V#!#;$!2N$e"yY;oT"!#<7$$"2uT$o')f?cC!#;$!2#y^2qrg.9!#<7$$"2kFb5^WA[#!#;$!2Qv(ov\f*Q"!#<7$$"2.29G6hq]#!#;$!2e*olA>\w8!#<7$$"0%ze<%zJ`#!#9$!1Ry:w%fHO"!#;7$$"29KkGxAob#!#;$!2uCS?SJ4N"!#<7$$"2MmKl5`He#!#;$!211Tx.xyL"!#<7$$"2129GT")zg#!#;$!1NvD*G-cK"!#;7$$"2w[(\*z[Hj#!#;$!2Q^#*Q$[d88!#<7$$"1#He;LF!fE!#:$!2)H465zB,8!#<7$$"2_/4=hYIo#!#;$!1M3$G.u+H"!#;7$$"2a+,-kXwq#!#;$!2$o**fgL$)y7!#<7$$"29E_/*e![t#!#;$!2'>#Hx-VmE"!#<7$$"14=OssRfF!#:$!2lhI_R+eD"!#<7$$"2Y"Hemla%y#!#;$!28R)3)H)*[C"!#<7$$"1D]+^F75G!#:$!2`,&p4)**RB"!#<7$$"2oGd9CKO$G!#;$!2W7b'Gd9C7!#<7$$"1D\)pf#peG!#:$!2"4[3r,"Q@"!#<7$$"2kGd9>iN)G!#;$!2EbCe?@P?"!#<7$$"2%yc8F\;5H!#;$!1E:R*z4J>"!#;7$$"2NsW*)oSO$H!#;$!2ouC:6(*Q="!#<7$$"2%=OsWosgH!#;$!2")4:%)pRM<"!#<7$$"28Ig?,q^)H!#;$!2C8^*ys:k6!#<7$$"2CY#\[_N4I!#;$!2i-![rP6b6!#<7$$"22=Os9i`.$!#;$!1(Qoy&>aX6!#;7$$"27BY#\UZhI!#;$!2ZU9[3(3O6!#<7$$"1V'GdRK_3$!#:$!2s>x&\phF6!#<7$$"1*zf>\i/6$!#:$!2j#*\X.d(=6!#<7$$"2.+++g%3NJ!#;$!2^-e]tT-6"!#<7$$"1sV([Av<;$!#:$!1HF6FS:,6!#;7$$"2Y!4=O+([=$!#;$!2$*[HE-3M4"!#<7$$"1:Hem_R6K!#:$!2/<m.$Gk%3"!#<7$$"1">QwPIjB$!#:$!2&H*\<RGl2"!#<7$$"1oNrUY-hK!#:$!2<!3.u$4'o5!#<7$$"2nIhAbQdG$!#;$!2@98Q=)zg5!#<7$$"229Gc<X4J$!#;$!2AZ*=sj%H0"!#<7$$"2Gg?TxjwL$!#;$!2o0bEuZZ/"!#<7$$"2V"Gc7@;iL!#;$!2%HH&zFSt."!#<7$$"2%3<M=^M'Q$!#;$!2'>k*)y!H,."!#<7$$"1)e<NN'>7M!#:$!2&)3^<,HD-"!#<7$$"2/29GJ"*zV$!#;$!1kF\"*\0:5!#;7$$"1:Ig?Q9hM!#:$!2X@#)R&yV35!#<7$$"1#Rycjp&)[$!#:$!2*fmDP(32+"!#<7$$"1['HfG=:^$!#:$!12"eN'pIV**!#<7$$"1^-05$f$QN!#:$!1Q;.A)>(p)*!#<7$$"2MoOtYwRc$!#;$!1iI)RQ%[+)*!#<7$$"1'=Pu`\oe$!#:$!1cf@"Gr%R(*!#<7$$"1D\)ppsDh$!#:$!1lq*e=Y<n*!#<7$$"2&e;LmjYQO!#;$!1&*>@Bd]/'*!#<7$$"0%zenpBkO!#9$!1rNe"3'\Q&*!#<7$$"2b4>QhH")o$!#;$!19`-EH4y%*!#<7$$"1x`2l!pGr$!#:$!1cbSQ3M;%*!#<7$$"2Oxa4ka%QP!#;$!1G<eM"3LN*!#<7$$"2%Rycj"eRw$!#;$!1>v*zn-8H*!#<7$$"1%yc8n">!z$!#:$!1dWzY^OG#*!#<7$$"2B['Hf")H8Q!#;$!1r'*4'HCO<*!#<7$$"1*zf>f5$RQ!#:$!1cY@NGw7"*!#<7$$"2W&3<M)Ha'Q!#;$!1H)Q:B]C0*!#<7$$"1'4>Qc+1*Q!#:$!1#Q6dvq]**)!#<7$$"2OpQx*zX8R!#;$!18[6D\eV*)!#<7$$"2tV([(pP1%R!#;$!1:$end<J)))!#<7$$"2$3<MoCmjR!#;$!18wvjN_K))!#<7$$"18D]]dW!*R!#:$!16n,-SQu()!#<7$$"1PtY$\`T,%!#:$!1ZH$*=%\Ns)!#<7$$"1u[(\kk,/%!#:$!1WubrFWo')!#<7$$"1d:J7M$\1%!#:$!1J"\fT0mh)!#<7$$"1lIhAqx!4%!#:$!02SLGpJc)!#;7$$"0'=PC&4X6%!#9$!1F!)*QOtY^)!#<7$$"1U$oOB3,9%!#:$!1fvJ_/(HY)!#<7$$"117C)4)pmT!#:$!1W%Q$4/#*4%)!#<7$$"1)e<NlW)*=%!#:$!1l[y^RFk$)!#<7$$"2NoOtcV[@%!#;$!1wZ%R,BbJ)!#<7$$"1'Gd9*)p1C%!#:$!1@VDImul#)!#<7$$"1w_06f$fE%!#:$!1x$e`b=w@)!#<7$$"1a3<%3#Q!H%!#:$!1R+)*p"y:<)!#<7$$"1(Gd9RDvJ%!#:$!0*QTKQ0@")!#;7$$"12:Ig[">M%!#:$!1`Wwf5=w!)!#<7$$"1.17Ce&zO%!#:$!1TZSQ&4)G!)!#<7$$"1X*)y2Gb"R%!#:$!1b5zXuN')z!#<7$$"1$f=Pa]tT%!#:$!1HAS_FXSz!#<7$$"1>Qw-UiTW!#:$!1Ba*e]Nx*y!#<7$$"1:Igqr*pY%!#:$!0)o0@%oN&y!#;7$$"16@U%)Q!=\%!#:$!0eDPMe3"y!#;7$$"2/7C[@ux^%!#;$!1)e&e%*fjmx!#<7$$"1U#['HnyUX!#:$!1=*["y=^Cx!#<7$$"1nMpQcOoX!#:$!1X4&)[.!>o(!#<7$$"1(Gd9uKPf%!#:$!1,!*=K+5Sw!#<7$$"1f=PuC/<Y!#:$!1/qyIg3-w!#<7$$"0(Rze!ePk%!#9$!1ds&o%p(*ev!#<7$$"/*zfz_wm%!#8$!1'Qk%Q)G3_(!#<7$$"0*zfp&HJp%!#9$!1s9`vQd![(!#<7$$"1oNr#49vr%!#:$!07GB[VCW(!#;7$$"2lHf=(**eWZ!#;$!12=.GVb+u!#<7$$"1$f=PpN!oZ!#:$!1YQ7**ylkt!#<7$$"1$pQxzYYz%!#:$!12Dq/>LCt!#<7$$"1a2:I***)=[!#:$!0LFnJfzG(!#;7$$"2lQxa/Ha%[!#;$!1@gr]@e[s!#<7$$"1OrU&=>$o[!#:$!1oj5dW%\@(!#<7$$"1tX"HQ)Q%*[!#:$!1,HYQ3,xr!#<7$$"1=NqS*p&>\!#:$!1bzwPKuSr!#<7$$"2&\**)z/NZ%\!#;$!1(f`9Nh[5(!#<7$$"1MpQxv!)p\!#:$!1nwo_gYpq!#<7$$"1tW*)GW*Q*\!#:$!18OqZ2zNq!#<7$$"1KjE`H$*>]!#:$!0t#G_Ou**p!#;7$$"1E^-b&\Z/&!#:$!10([Z;Gd'p!#<7$$"1&**)zfy'32&!#:$!1uu7h:GIp!#<7$$"1x`2:7^%4&!#:$!1tT.h;])*o!#<7$$"1>Pu[:k?^!#:$!1"=$)3F9P'o!#<7$$"1E^-b)pc9&!#:$!0[=)p%>2$o!#;7$$"1V&3<CP1<&!#:$!1'\13q<")z'!#<7$$"1Z$pQx:n>&!#:$!1:x#yv$Rkn!#<7$$"1,,-a]t?_!#:$!10')3]diLn!#<7$$"1h?T#3M`C&!#:$!1G&**RB-Cq'!#<7$$"2lJjEL%\s_!#;$!1:5(*H0Eom!#<7$$"1lGd9d3(H&!#:$!13BD<]kPm!#<7$$"0(Rz3]BA`!#9$!1bOkMFi1m!#<7$$"1"3;K>6yM&!#:$!01LUDo`d'!#;7$$"1U$oOo?8P&!#:$!1BD#3h(*oa'!#<7$$"21)f>R5Q'R&!#;$!1`x!eh<o^'!#<7$$"1U$oOj]7U&!#:$!1xEEorB(['!#<7$$"1MnMpL&yW&!#:$!1*)R>a&*)eX'!#<7$$"1zd:J"H8Z&!#:$!12Cd-fZGk!#<7$$"1uY$pG:%)\&!#:$!1)=#p=E8(R'!#<7$$"1d8Fa%eG_&!#:$!1<6](f2"pj!#<7$$"1=Nq!pVqa&!#:$!0%4'or<;M'!#;7$$"1PsW*e]Id&!#:$!1Az@,&>BJ'!#<7$$"1(Gd9pi"*f&!#:$!1\<WAD<$G'!#<7$$"1*pRz$3#Hi&!#:$!1Dz^F[)oD'!#<7$$"1b3<M4:[c!#:$!1@p"o02#Hi!#<7$$"1b5@UIxsc!#:$!2`J+J>IC?'!#=7$$"2vU&3nOY*p&!#;$!1kM0q8mth!#<7$$"0'>Ry%eDs&!#9$!2cQcb#=)*[h!#=7$$"0(Rz3P3\d!#9$!0%)\2nx37'!#;7$$"1Z#\)>)=Sx&!#:$!1Uhad/p%4'!#<7$$"1CY#\38()z&!#:$!1(RAD2v*og!#<7$$"1iBZ%*pUBe!#:$!2%fKy$eaM/'!#=7$$"1'>RyhL'[e!#:$!2%=OHXWk<g!#=7$$"1f;L;ANve!#:$!14m#RlC0*f!#<7$$"1pQxa0&)**e!#:$!1J&QdVre'f!#<7$$"1kFbgN.Cf!#:$!2c)z6rGtTf!#=7$$"1W'Gdz%))\f!#:$!1E`twK9;f!#<7$$"2b7D]vzc(f!#;$!1"H`.>F3*e!#<7$$"#g!""$!1Z(QTAbr'e!#<-%&COLORG6'%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-%+_ATTRIBUTEG6#/%'sourceG%,mathdefaultG-%%VIEWG6$;$"#5!""$"#g!""%(DEFAULTG-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"m6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"G6"

cval:=-sqrt(%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjdmFsR0YoLCQqJCkqJiwoKiZJIm1HRigiIiIpLCYqJiIiJUY1KUY0IiIjRjVGNUY1RjUjRjVGO0Y1ISIiKiZGO0Y1RjpGNUY1RjVGNUY1LCYqJiIiJ0Y1RjpGNUY1RjtGNUY9RjxGNUY9NyNGLg==

c1:=expand(rationalize(simplify(normal(subs(m=1,x=subs(m=1,xc),y=subs(m=1,yc),cxy))))); evalf(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCwmKiYjIiIiIiIlRjEpIiImI0YxIiIjRjEhIiJGMEYxNyNGLg==

JCErVSpwLDQkISM1

arccos(-c1)

LCQqJiMiIiMiIiYiIiJJI1BpRyUqcHJvdGVjdGVkR0YnRic=

asympt(cval,m,3)

LCYqKCMiIiIiIiVGJSkiIiMjRiVGKEYlSSJtRzYiISIiRiwtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YrNiMqJClGKiIiJEYsRiU=

minimal (or: annihilating) polynomial of c

map(Z->Z^2,isolate(cval^2-c^2,sqrt(1+4*m^2 )))

LywmKiYiIiUiIiIpSSJtRzYiIiIjRiZGJkYmRiYqJiksKComKUkiY0dGKUYqRiYsJiomIiInRiZGJ0YmRiZGKkYmRiZGJiomRipGJkYnRiYhIiJGJkY1RipGJkYnRjU=

polmin:=collect(op(2,factor(op(2,%)-op(1,%))),c,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdwb2xtaW5HRigsKCIiIkYvKiYsJiomIiM3Ri8pSSJtR0YoIiIjRi9GLyIiJUYvRi8pSSJjR0YoRjdGL0YvKiYsJiomIiIpRi9GNEYvISIiRjdGPkYvKUY5RjZGL0YvNyNGLg==

factor(subs(c=cval,polmin))

IiIh

polz:=collect(numer(factor(subs(c=(z+1/z)/2,polmin))),z,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVwb2x6R0YoLC4qJiwmKiYiIiQiIiIpSSJtR0YoIiIjRjNGM0YzRjNGMylJInpHRigiIilGM0YzKigiIiVGM0Y0RjMpRjgiIidGM0YzKiYsJiomRjZGM0Y0RjNGM0Y2RjNGMylGOEY7RjNGMyooRjtGM0Y0RjMpRjhGNkYzRjNGMUYzRjNGMzcjRi4=

latex(%)

\134left(m^{2}+3\134right) z^{8}+4 z^{6}+\134left(2 m^{2}+2\134right) z^{4}+4 z^{2}+m^{2}+3

with(NumberTheory): CyclotomicPolynomial(10,z)

LCwqJClJInpHNiIiIiUiIiJGKCokKUYlIiIkRighIiIqJClGJSIiI0YoRihGJUYsRihGKA==

seq(degree(CyclotomicPolynomial(n,z)),n=1..40)

NkoiIiJGIyIiI0YkIiIlRiQiIidGJUYmRiUiIzVGJSIjN0YmIiIpRikiIztGJiIjPUYpRihGJyIjQUYpIiM/RihGK0YoIiNHRikiI0lGKkYtRioiI0NGKCIjT0YrRjBGKg==

Looking for cyclotmopic polynomials of degree at most 8

ens:={}: for n from 1 to 200 do if degree(CyclotomicPolynomial(n,z)) <9 then ens:=ens union{[n,CyclotomicPolynomial(n,z)]} fi: od:

nops(ens)

IiM9

ens[1]; ens[18];

NyQiIiIsJkkiekc2IkYjRiMhIiI=

NyQiI0ksMCokKUkiekc2IiIiKSIiIkYqKiQpRiciIihGKkYqKiQpRiciIiZGKiEiIiokKUYnIiIlRipGMSokKUYnIiIkRipGMUYnRipGKkYq

for n from 1 to 18 do print(n,ens[n][1],lcoeff(factor(rem(polz,ens[n][2],z)),z)) od

NiUiIiJGIywmKiYiIztGIylJIm1HNiIiIiNGI0YjIiIlRiM=

NiUiIiNGIywmKiYiIzsiIiIpSSJtRzYiRiNGJ0YnIiIlRic=

NiUiIiRGIywmKiYiIiYiIiIpSSJtRzYiIiIjRichIiJGJ0Yn

NiUiIiVGI0Yj

NiUiIiZGIyomLCZJIm1HNiIiIiJGKCEiIkYoLCZGJkYoRihGKEYo

NiUiIidGIywmKiYiIiYiIiIpSSJtRzYiIiIjRidGJ0YnISIi

NiUiIihGIywkKiYiIiUiIiIpSSJtRzYiIiIjRichIiI=

NiUiIilGIywkKiYiIiUiIiIpSSJtRzYiIiIjRidGJw==

NiUiIipGIywmKiYiIiQiIiIpSSJtRzYiIiIjRichIiJGJ0Ys

NiUiIzVGIywkKiYsJkkibUc2IiIiIkYpISIiRiksJkYnRilGKUYpRilGKg==

NiUiIzYiIzcsJiomIiIkIiIiKUkibUc2IiIiI0YoRihGKEYo

NiUiIzciIzksJComIiIlIiIiKUkibUc2IiIiI0YoRig=

NiUiIzgiIzosJiomIiIkIiIiKUkibUc2IiIiI0YoRihGKEYo

NiUiIzkiIzssJComIiIlIiIiKUkibUc2IiIiI0YoRig=

NiUiIzoiIz0sJiomIiIkIiIiKUkibUc2IiIiI0YoRihGKEYo

NiUiIzsiIz8sJiomIiIoIiIiKUkibUc2IiIiI0YoRihGKEYo

NiUiIzwiI0MsJComIiIlIiIiKUkibUc2IiIiI0YoRig=

NiUiIz0iI0ksJiomIiIkIiIiKUkibUc2IiIiI0YoISIiRihGLQ==

<Text-field style="Heading 1" layout="Heading 1">Example: m=1 (same as before)</Text-field>

N:=100: ser:=series(subs(x=1,y=1,Gser(1,N)),t,N+1):

Let us check asymptotics numerically

Estimate of the radius

plot([seq([1/n,coeff(ser,t,n)/coeff(ser,t,n-1)],n=2..N)],view=[0..0.2,5..15],style=point)

evalf(subs(m=1,mup))

JCIrJSpwLDQ2ISIp

Estimates of the exponent

plot([seq([1/n,n^2*(1-coeff(ser,t,n-1)*coeff(ser,t,n+1)/coeff(ser,t,n)^2)],n=2..N-1)],view=[0..0.2,-3.6..-2.7],style=point)

evalf(-1-Pi/arccos(-subs(m=1,cval)))

JCErLCsrK04hIio=

<Text-field style="Heading 1" layout="Heading 1">Example: m=2</Text-field>

N:=100: ser:=series(subs(x=1,y=1,Gpser(2,N)),t,N+1):

Let us check asymptotics numerically

Estimate of the radius

plot([seq([1/n,coeff(ser,t,n)/coeff(ser,t,n-1)],n=2..N)],view=[0..0.2,10..24],style=point)

NistJSdDVVJWRVNHNiQ3X3E3JCQiIiYhIiIkIiNdISIiNyQkIjFMTExMTExMTCEjOyQiIyEpISIiNyQkIiNEISIjJCImdiwiISIkNyQkIiIjISIiJCIyJ3lTaXlTaXk2ISM6NyQkIjJtbW1tbW1tbSIhIzwkIjI4LU1XNF48SSIhIzo3JCQiMiZHOWRHOWRHOSEjPCQiMjFzKyE+VmEpUiIhIzo3JCQiJEQiISIkJCIyXEJqT1UnW3c5ISM6NyQkIjE2NjY2NjY2NiEjOyQiMjpPUnYxRTBhIiEjOjckJCIiIiEiIiQiMk44XTF5VVNmIiEjOjckJCIxIjQ0NDQ0NDQqISM8JCIyblEsPiFSVFI7ISM6NyQkIjFMTExMTExMJCkhIzwkIjEiZUM+VGMkeTshIzk3JCQiMSRwMkJwMkJwKCEjPCQiMiVvM2dQJVJAciIhIzo3JCQiMVVyJkc5ZEc5KCEjPCQiMXp6ViFRPzx1IiEjOTckJCIxbm1tbW1tbW0hIzwkIjJtZ2sielgkeXciISM6NyQkIiREJyEiJSQiMkUmSEV6ZTAieiIhIzo3JCQiMjBadzYlSE4jKWUhIz0kIjI6QiplJFxRPSI9ISM6NyQkIjFiYmJiYmJiYiEjPCQiMSVlVlJgWTAkPSEjOTckJCIxVW90JSp5OmpfISM8JCIxQilwXEJ2dSU9ISM5NyQkIiImISIjJCIxZnVsbGsnRyc9ISM5NyQkIjI7dy8+dy8+dyUhIz0kIjFfQicqUi0jcCg9ISM5NyQkIjJjYWFhYWFhYSUhIz0kIjFKbiJ5UC4pKik9ISM5NyQkIjI7X2NwM0V5TSUhIz0kIjIuKjM3MGhsLD4hIzo3JCQiMmttbW1tbW07JSEjPSQiMTZCJj45KGY3PiEjOTckJCIiJSEiIyQiMSUzRHVtRkYjPiEjOTckJCIyayVROllROllRISM9JCIyazE6KVJXOEs+ISM6NyQkIjJOcS5QcS5QcSQhIz0kIjF2NVV1QSozJT4hIzk3JCQiMXImRzlkRzlkJCEjPCQiMmlIKDNHZzFcPiEjOjckJCIyYicqbz8nZUZbTSEjPSQiMl8/azY/N24mPiEjOjckJCIxTExMTExMTEwhIzwkIjIkKSlbPFMtKVEnPiEjOjckJCIxLkhoXmshZUEkISM8JCIybFQpUS9PaHE+ISM6NyQkIiVESiEiJiQiMiZwNnJXMSZwKD4hIzo3JCQiMi8uLi4uLi4uJCEjPSQiMSJwJkhLYCNIKT4hIzk3JCQiMmBCKWVxazxUSCEjPSQiMjFWIypmJ3ljKSk+ISM6NyQkIjFkRzlkRzlkRyEjPCQiMnlwKFwiPTBSKj4hIzo3JCQiMnd4eHh4eHh4IyEjPSQiMWJ2RCFRaCopKj4hIzk3JCQiMS5GcS1GcS1GISM8JCIxb15cIzRlUCsjISM5NyQkIjFAJW90JSp5OmojISM8JCIxRFcnZXk5JDM/ISM5NyQkIjFrRDVrRDVrRCEjPCQiMkxUJSoqRyFcRSwjISM6NyQkIiNEISIkJCIyTDxIdXJ3biwjISM6NyQkIjJEIVJDIVJDIVJDISM9JCIyWkN1IW9Bcj8/ISM6NyQkIjIzUV80UV80USMhIz0kIjJRcV9yeW9XLSMhIzo3JCQiMnMkKVtgUiJlREIhIz0kIjJkeSZ5OiNlIUc/ISM6NyQkIjJHRkZGRkZGRiMhIz0kIjJQeilmYzlcSj8hIzo3JCQiMkJBQUFBQUFBIyEjPSQiMDd2VFt5Wi4jISM4NyQkIjIzRXlNLzhSPCMhIz0kIjFWOXJUJUd6LiMhIzk3JCQiMSYzb1dkZnc3IyEjPCQiMVJOJWVzXDQvIyEjOTckJCIyS0xMTExMTDMjISM9JCIxPHVIaCsmUS8jISM5NyQkIjE3MWBFaiIzLyMhIzwkIjImZTd3bWxqWT8hIzo3JCQiIiMhIiMkIjIjZiJmSiJlSlw/ISM6NyQkIjBcRFBKJXlnPiEjOyQiMjhnSGAoUSo9MCMhIzo3JCQiMksjcDJCcDJCPiEjPSQiMUJKJ3pQd1YwIyEjOTckJCIyJyk9SUdYI3onKT0hIz0kIjElW0hsYG9uMCMhIzk3JCQiMjwmPSY9Jj0mPSY9ISM9JCIwTz9RPnYhZj8hIzg3JCQiMT09PT09PT09ISM8JCIyWCJla18zSWg/ISM6NyQkIjJjRzlkRzlkeSIhIz0kIjF6JltecVxNMSMhIzk3JCQiMjFHN1wnZlFhPCEjPSQiMllKUCJmY19sPyEjOjckJCIyRltNNSR6OEM8ISM9JCIyO2EneWhCYG4/ISM6NyQkIjEpR1BVRDpccCIhIzwkIjJPUDA1QXQlcD8hIzo3JCQiMm1tbW1tbW1tIiEjPSQiMSg9UVpVXjgyIyEjOTckJCIxIzMmSGlVTVI7ISM8JCIxMXFVZSpwSjIjISM5NyQkIjI7WDFlQS5IaCIhIz0kIjJ1aE8zaUpcMiMhIzo3JCQiMnNlLHRlLHRlIiEjPSQiMUsoPSVRIVJtMiMhIzk3JCQiJkRjIiEiJyQiMls5cSR5WUh5PyEjOjckJCIyJlE6WVE6WVE6ISM9JCIyMkpAMCczISp6PyEjOjckJCIyX15eXl5eXl4iISM9JCIyJWZkPG8oZjkzIyEjOjckJCIyZSRHVjh0YCNcIiEjPSQiMk94MSFlTShIMyMhIzo3JCQiMnc2JUhOIyllcTkhIz0kIjIlM2M1cFFXJTMjISM6NyQkIjIxJSk9Qk92I1w5ISM9JCIyQyJcRUpHKGUzIyEjOjckJCIyJkc5ZEc5ZEc5ISM9JCIyRnFyRDJpczMjISM6NyQkIjJATkRVcV0lMzkhIz0kIjA3d2pBOCcpMyMhIzg3JCQiMikpKSkpKSkpKSkpKSkpUSIhIz0kIjIlKipccFB5IyoqMyMhIzo3JCQiMGopcDhJJylwOCEjOyQiMUtJNXB0PyI0IyEjOTckJCIyOU5eOE5eOE4iISM9JCIybHFlaD9gQzQjISM6NyQkIjJNTExMTExMTCIhIz0kIjFOdFNpbW0kNCMhIzk3JCQiMjBAJW90JSp5OjghIz0kIjJBNyoqUikqW1s0IyEjOjckJCIyKSlILCgpSCwoKUgiISM9JCIxJVJjUk4sZzQjISM5NyQkIjEjR14/R14/RyIhIzwkIjJabz9oKls3KDQjISM6NyQkIjJtNzVbeUFlRSIhIz0kIjFtdCEpeTFBKTQjISM5NyQkIiREIiEiJSQiMnR4Iz49KCpHKjQjISM6NyQkIjJ5Y003IXpjTTchIz0kIjFCa1giKUhMK0AhIzk3JCQiMjgmPjcmPjcmPjchIz0kIjJYbSlwKlFeODUjISM6NyQkIjJRVjNyRj5bPyIhIz0kIjItUV8xI2VNLUAhIzo3JCQiMi8+dy8+dy8+IiEjPSQiMSpRMDc2PEw1IyEjOTckJCIyVEhOIyllcWs8IiEjPSQiMlUlKVsoZmdFL0AhIzo3JCQiMic9V24ocCF6aTYhIz0kIjJqWWwkR00+MEAhIzo3JCQiMj1LY3RHRCVcNiEjPSQiMnk+cVslKio0MUAhIzo3JCQiMmtqampqamo4IiEjPSQiMiZwWnIvailwNSMhIzo3JCQiMnYoemgwYmZCNiEjPSQiMV5td3NKJnk1IyEjOTckJCIyNzY2NjY2NjYiISM9JCIxUD4oXD0sKDNAISM5NyQkIjEqNCwqKTQsKik0IiEjPCQiMiY9NCIqXDRgNEAhIzo3JCQiMi84UjxfY3AzIiEjPSQiMUFWRF1JTTVAISM5NyQkIjI3SS9zIilvXzIiISM9JCIybGFGVC9RNjYjISM6NyQkIjJEL01zeUhRMSIhIz0kIjInMycqUW1rIj42IyEjOjckJCIyJW90JSp5OmpfNSEjPSQiMVRIJClIKXlFNiMhIzk3JCQiMm1tbW1tbTsvIiEjPSQiMTgoNGppRE02IyEjOTckJCIya2FeXSR5IzQuIiEjPSQiMiRvKSp5RnQ6OUAhIzo3JCQiMTFgRWoiMy8tIiEjPCQiMnhrYnRRdVs2IyEjOjckJCIyLSwsLCwsLCwiISM9JCIxPnE4U3NkOkAhIzk3JCQiIiIhIiMkIjImZj8+L2pFO0AhIzotJSZDT0xPUkc2JyUkUkdCRyQiKUMpZXElISIpJCIiISEiIiQiKCc+IVwmISIpLSUrX0FUVFJJQlVURUc2Iy8lJ3NvdXJjZUclLG1hdGhkZWZhdWx0Ry0lJVZJRVdHNiQ7JCIiISEiIiQiIiMhIiI7JCIkKyIhIiIkIiRTIyEiIi0mJSZfQVhJU0c2IyIiIjYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkUSE2IlEhNiItJSZTVFlMRUc2IyUmUE9JTlRHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiIhISIiLSUpQk9VTkRTX1lHNiMkIiIhISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlK10hIiItJS5CT1VORFNfSEVJR0hURzYjJCIlK10hIiItJSlDSElMRFJFTkc2Ii0lK0FOTk9UQVRJT05HNictJSlCT1VORFNfWEc2IyQiIiEhIiItJSlCT1VORFNfWUc2IyQiIiEhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUrXSEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrXSEiIi0lKUNISUxEUkVORzYiRzYi

evalf(subs(m=2,mup))

JCIrTFwjZT0jISIp

Estimates of the exponent

plot([seq([1/n,n^2*(1-coeff(ser,t,n-1)*coeff(ser,t,n+1)/coeff(ser,t,n)^2)],n=2..N-1)],view=[0..0.2,-3.3..-2.5],style=point)

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JSlDSElMRFJFTkc2Ig==Ig==

evalf(-1-Pi/arccos(-subs(m=2,cval)))

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