NiVRRC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9hbGdvbGliNiJRNy91c3IvbG9jYWwvbWFwbGUxOC9saWJGJFEiLkYk

NiZRQS91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9nZnVuNiJRRC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9hbGdvbGliRiRRNy91c3IvbG9jYWwvbWFwbGUxOC9saWJGJFEiLkYk

restart:

sum(binomial(k+i,j),i=1..j);

LCYqKCwmSSJrRzYiIiIiRidGJ0YnLCZJImpHRiZGJ0YnRichIiItSSliaW5vbWlhbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkLChGJUYnRilGJ0YnRidGKUYnRicqKCwoRiVGJ0YpRipGJ0YnRidGKEYqLUYsNiRGJEYpRidGKg==

sum(%,j=1..k-1);

LCgiIiJGIy1JKWJpbm9taWFsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJiomIiIjRiNJImtHRilGI0YjRiNGI0YuRiMqJiIiJUYjKUYtLCZGLkYjRiMhIiJGI0Yz

sum(binomial(k+i,k),i=1..k-1);

LCYqKEkia0c2IiIiIiwmRiRGJkYmRiYhIiItSSliaW5vbWlhbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLCQqJiIiI0YmRiRGJkYmRiRGJkYmRiZGKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

System in Section 3.2.1 (the version that uses symmetry) eqK01:=-K01+t+t*L0; eqLe:=-L+1+2*t*L^2+t*(L-1)+2*t*(L0-K01); eqL0:=-L0+2*t*L*L0+t*L+t*L0+t*(L0-K01);

LCgqJkkjTDBHNiIiIiJJInRHRiVGJkYmSSRLMDFHRiUhIiJGJ0Ym

LCxJIkxHNiIhIiIiIiJGJiooIiIjRiZJInRHRiRGJilGI0YoRiZGJiomRilGJiwmRiNGJkYmRiVGJkYmKihGKEYmRilGJiwmSSNMMEdGJEYmSSRLMDFHRiRGJUYmRiY=

LCxJI0wwRzYiISIiKioiIiMiIiJJInRHRiRGKEkiTEdGJEYoRiNGKEYoKiZGKUYoRipGKEYoKiZGKUYoRiNGKEYoKiZGKUYoLCZGI0YoSSRLMDFHRiRGJUYoRig=

factor(resultant(subs(isolate(eqK01,K01),eqLe), subs(isolate(eqK01,K01),eqL0),L0));

LCQqJiwoKigiIiMiIiJJIkxHNiJGJ0kidEdGKUYnRidGKkYnRichIiJGJywwKihGJkYnKUYoRiZGJ0YqRidGJyomRihGJylGKkYmRidGK0YlRicqJEYwRidGK0YoRisqJkYmRidGKkYnRitGJ0YnRidGKw==

We choose the correct factor algL:=collect(op(3,%),L,factor);

LCwqKCIiIyIiIkkidEc2IkYlKUkiTEdGJ0YkRiVGJSomKSwmRiZGJUYlISIiRiRGJUYpRiVGLSokKUYmRiRGJUYtKiZGJEYlRiZGJUYtRiVGJQ==

The first few terms with(gfun): algeqtoseries(algL,t,L,10,true);

NyMrOUkidEc2IiIiIiIiISIiIyIiIiIjNSIiIyIjbSIiJCIkbSUiIiUiJWVNIiImIiZdbSMiIiciJ2U5QCIiKCIoVW1yIiIiKSIpI0ckPjkiIiotSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Positive system (Section 3.3) (the p stands for the plus) eqK01:=-K01+t+t*(K01+L0p): eqLp:=-Lp+2*t*(1+Lp)^2+t*Lp+2*t*L0p: eqL0p:=-L0p+2*t*(1+Lp)*(K01+L0p)+t*Lp+t*L0p: eqLp:=op(2,factor(resultant(subs(isolate(eqK01,K01),eqLp), subs(isolate(eqK01,K01),eqL0p),L0p)));

LC4qKCIiIyIiIilJI0xwRzYiRiRGJUkidEdGKEYlRiUqJkYnRiUpRilGJEYlISIiKigiIidGJUYnRiVGKUYlRiUqJkYkRiVGK0YlRixGJ0YsKiZGJEYlRilGJUYl

algeqtoseries(eqLp,t,Lp,10,true);

NyMrOUkidEc2IiIiIyIiIiIjNSIiIyIjbSIiJCIkbSUiIiUiJWVNIiImIiZdbSMiIiciJ2U5QCIiKCIoVW1yIiIiKSIpI0ckPjkiIioiKj1lNj4iIiM1LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNg==

We recover the equation for L collect(subs(Lp=-1+L,eqLp),L,factor);

LCwqKCIiIyIiIkkidEc2IkYlKUkiTEdGJ0YkRiVGJSomKSwmRiZGJUYlISIiRiRGJUYpRiVGLSokKUYmRiRGJUYtKiZGJEYlRiZGJUYtRiVGJQ==

The solution factor([solve(algL,L)]);

NyQsJCooIyIiIiIiJUYmSSJ0RzYiISIiLCoqJClGKCIiI0YmRiYqJkYuRiZGKEYmRipGJkYmKiQpLCwqJClGKEYnRiZGJiomRidGJilGKCIiJEYmRiYqJiIjQUYmRi1GJkYmKiYiIzdGJkYoRiZGKkYmRiYjRiZGLkYmRiZGJkYmLCQqKEYlRiYsKkYsRipGMEYmRi9GJkYmRipGJkYoRipGKg==

Lsol:=op(2,%);

LCQqKCMiIiIiIiVGJSwqKiQpSSJ0RzYiIiIjRiUhIiIqJCksLCokKUYqRiZGJUYlKiZGJkYlKUYqIiIkRiVGJSomIiNBRiVGKUYlRiUqJiIjN0YlRipGJUYtRiVGJSNGJUYsRiVGJSomRixGJUYqRiVGJUYlRi1GJUYqRi1GLQ==

series(Lsol,t);

Ky9JInRHNiIiIiIiIiEiIiNGJSIjNUYnIiNtIiIkIiRtJSIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

Asymptotics disc:=factor(discrim(algL,L));

LCwqJClJInRHNiIiIiUiIiJGKComRidGKClGJSIiJEYoRigqJiIjQUYoKUYlIiIjRihGKComIiM3RihGJUYoISIiRihGKA==

plot(disc,t=0..0.2);

6*-%'CURVESG6$7dw7$$""!!""$"#5!""7$$"2c7D]+@:0"!#>$"19!H"z21u)*!#;7$$"2iAX!4QWm>!#>$"1t<]lx([w*!#;7$$"2Wxa4>q`*H!#>$"1YdEY0`U'*!#;7$$"1@T#['p6JS!#=$"1X([fcV)>&*!#;7$$"1x`2:5%>1&!#=$"1\e>(Q4#)R*!#;7$$"13;Kkok<g!#=$"0&e3ix&eG*!#:7$$"1lGd9\A2q!#=$"1Yw5h!\*p"*!#;7$$"1X([(\zkI!)!#=$"1%)*G84J00*!#;7$$"1d8Fa))y]!*!#=$"14WS_n&>$*)!#;7$$"2Z"He;H,55!#=$"12S!=fo/"))!#;7$$"2'3<Mo)QC5"!#=$"0h(ps_'Qq)!#:7$$"2_(\**)f)[17!#=$"1u+mkwI%e)!#;7$$"1'>Rycl4J"!#<$"1_8V7;uk%)!#;7$$"1h@V'[[;T"!#<$"1XL&yEv*\$)!#;7$$"1`06A#yI]"!#<$"0pJy%e9Y#)!#:7$$"1Gb5@qz6;!#<$"1WpU7`;B")!#;7$$"2MhAX5'*Qq"!#=$"02/V:&R>!)!#:7$$"2&He;L#H5"=!#=$"1^k&=yf"**y!#;7$$"2b7D]?ge!>!#=$"1aaZ[m:$z(!#;7$$"2mFb5"[!*4?!#=$"12yC=WJxw!#;7$$"2/,-/))z*3@!#=$"1Q)*H\GXnv!#;7$$"21/3;KaB@#!#=$"1Z(H;g*G`u!#;7$$"2l@V'GVG2B!#=$"1$)4#)*Hr)[t!#;7$$"1Y"He;z'4C!#<$"13!y3B#pOs!#;7$$"1.17C'Qg^#!#<$"1xspsYm?r!#;7$$"238E_%[i3E!#=$"1w^'\>t+-(!#;7$$"2G^-0]?'3F!#=$"1nom<9'="p!#;7$$"2[#\)pzD>"G!#=$"1mEMIq`+o!#;7$$"2Z)oPv)*)H"H!#=$"1`^V)H*3#p'!#;7$$"2i>Rycu2,$!#=$"10l#4@&f(e'!#;7$$"1D\)pzZ$>J!#<$"1\C!zZt?Z'!#;7$$"14=Osc!p@$!#<$"1vD@7@soj!#;7$$"2;>Qw_p5K$!#=$"1([<M)H%)ei!#;7$$"1e:JiuX:M!#<$"1D&zj/'pfh!#;7$$"1^,.1%['=N!#<$"1t:>1$f<0'!#;7$$"1b5@UIu:O!#<$"1Yn[Aaj]f!#;7$$"0%yc8\B<P!#;$"14>WTTQXe!#;7$$"1@U%)o<Y;Q!#<$"1(fZ1**GHu&!#;7$$"29E_/4V.#R!#=$"1.RSv>9Oc!#;7$$"2Xu[(\JR?S!#=$"1O'4\/_P`&!#;7$$"2%['Hfy3F7%!#=$"1_*[#e#4&Ha!#;7$$"2a#\)p>xTA%!#=$"1E2c-]fE`!#;7$$"2l@V'GhT<V!#=$"1.&Q^FOCB&!#;7$$"1f;Lm%yUU%!#<$"1$)*o)4:+D^!#;7$$"2wPv]Td)>X!#=$"0$yoYnMH]!#:7$$"1Rxa4Xw@Y!#<$"1Ec&3u7y#\!#;7$$"1^+,-EI>Z!#<$"1eH#Qvq5$[!#;7$$"2c'Hf=hgF[!#=$"2%3^B(faTs%!#<7$$"1^,.1!*Q@\!#<$"1w>"QC,?j%!#;7$$"1`06AM$y-&!#<$"2`S,!f"))y_%!#<7$$"1&ze<&f%[7&!#<$"2&4L9=oWLW!#<7$$"1F`18C'4B&!#<$"1,*HYlI1L%!#;7$$"1AV'G(H_A`!#<$"2:Oh#GvKUU!#<7$$"1rS"Gw*zEa!#<$"1xhMWKAUT!#;7$$"0&)pR*f_Fb!#;$"2Lc3*Qb*f/%!#<7$$"1zb6Bk=Gc!#<$"2;oQ7\!H]R!#<7$$"2&=NqSlZGd!#=$"2Dv;H*[RbQ!#<7$$"1pOtYR#[#e!#<$"1TxxD,mkP!#;7$$"116AW!y*Gf!#<$"1`Yf&*y/nO!#;7$$"1#Gc7XW#Gg!#<$"1A)=IRvWd$!#;7$$"2&f<NqwrKh!#=$"1C/$pLJvZ$!#;7$$"1(Gd94"HFi!#<$"1"Rc7.--R$!#;7$$"1a18EC"=L'!#<$"284vglgTH$!#<7$$"1#Gc7lD>V'!#<$"1@*3`^OE?$!#;7$$"1^**)z>&zJl!#<$"141tX/z6J!#;7$$"1nJjE$4hj'!#<$"1T&R7?(Q<I!#;7$$"1"=OsW'=Kn!#<$"1'**[owx3$H!#;7$$"1AS!3c#eIo!#<$"1->xKsrUG!#;7$$"1Y!4=cB#Rp!#<$"2lZ.PW"*eu#!#<7$$"1PsW*3*ePq!#<$"2/PP'G"*oeE!#<7$$"0mJjE'=Qr!#;$"11'f"fi'*pD!#;7$$"1,,-/5\Ss!#<$"0x"QvC@![#!#:7$$"1Z"He'*GXL(!#<$"1j!*z*4M")R#!#;7$$"/(RzQqZV(!#:$"2)[nHIu36B!#<7$$"1Y"Hew[U`(!#<$"226Q6%)e^A#!#<7$$"19Fa3(f1k(!#<$"2Z))\m^WP8#!#<7$$"1%*)ydviXt(!#<$"2k/kF02N0#!#<7$$"1tW*)yt!H%y!#<$"2001!))GVh>!#<7$$"127C[+oSz!#<$"0'H>7h!)y=!#:7$$"0&)pR*4UP!)!#;$"13%)zhQ[(z"!#;7$$"1BZ%*)e[99)!#<$"2jdryv;0r"!#<7$$"1E\)p*p*eC)!#<$"1=(*yV!*pB;!#;7$$"1tX"HeH4M)!#<$"1%R"4hV9X:!#;7$$"1)>Ry'*\=W)!#<$"2t'['fYy@Y"!#<7$$"1,+++%Q.a)!#<$"1=q7iYm"Q"!#;7$$"1)[(\**35Z')!#<$"2GF=>O#*[H"!#<7$$"1>OsW,[R()!#<$"0tVW#[B?7!#:7$$"1f;Lm5eX))!#<$"2"\W%yGu\8"!#<7$$"1kFb5:KX*)!#<$"2U.gwK*Hb5!#<7$$"1tU&3d)4W!*!#<$"1byTyKZo(*!#<7$$"1F_/4U&H9*!#<$"1o[%f(*[y)*)!#<7$$"1jD^-2yV#*!#<$"1ATgB>K'>)!#<7$$"18C['4b1N*!#<$"0o5#=xXit!#;7$$"1d7D]%['[%*!#<$"1-ctk#QDg'!#<7$$"1MoOt/QX&*!#<$"16Cq'znn&e!#<7$$"1_.29ay['*!#<$"1*yh&*z_V1&!#<7$$"1#Gc7Dl>v*!#<$"1iW)3"4hyU!#<7$$"1i?T#GvX%)*!#<$"1=R!*=CdxN!#<7$$"0d8FayU&**!#;$"2.Yy;")*H_F!#=7$$"2%f=P9tg/5!#<$"2'R#e'o40m?!#=7$$"205?SsV`,"!#<$"2Ajc4)3Qo7!#=7$$"2NnMpe!fD5!#<$"1$3gTkv37&!#=7$$"2X([(\")RZ."!#<$!2c2#*>k'e!f"!#>7$$"0(Rzy!H]/"!#:$!1,hJNKr"4*!#=7$$"2MmKla'Qb5!#<$!1bh*=Pp#f;!#<7$$"1w^.(y%pl5!#;$!2c"4y:<$3S#!#=7$$"2$Qw_X=Dv5!#<$!2Kt!R&[AR3$!#=7$$"1^,.Ew9&3"!#;$!2N#>D@ms'y$!#=7$$"2&4>Qc=Q&4"!#<$!1N4LvCv3X!#<7$$"2d8FaE$e06!#<$!1yzweteB_!#<7$$"2PrU&om2;6!#<$!1Et=V8!Q&f!#<7$$"1$f=PE>`7"!#;$!1#36JK.Ff'!#<7$$"2(>RyOUsN6!#<$!1bq%[TmrI(!#<7$$"2=MoO$><Y6!#<$!12UJ0kX>!)!#<7$$"2%Qw_D-Cc6!#<$!1uy(=5N5q)!#<7$$"2uZ&4*>$Ql6!#<$!1t?4q=&eJ*!#<7$$"1v\**y]Dw6!#;$!2mco!4>=/5!#<7$$"2NoOt)\Y&="!#<$!2'\hQMjCl5!#<7$$"2`+,-Iyh>"!#<$!20BUR-xd8"!#<7$$"2[$pQ(Rhc?"!#<$!2x74e:ex>"!#<7$$"2)\**)z&e1;7!#<$!2<5]#3VFl7!#<7$$"2KiC\OtfA"!#<$!22JZOx"4H8!#<7$$"2hAX!43JO7!#<$!2O,ca]&=&R"!#<7$$"1W([(4Q!eC"!#;$!2N3X\QOaX"!#<7$$"2oLnMHVgD"!#<$!2%e0$H\[*>:!#<7$$"2E['HR#zmE"!#<$!1VrE0WV'e"!#;7$$"2b.29'y$fF"!#<$!2_'Gk&fvQk"!#<7$$"2NnMpUPfG"!#<$!2&)fH-ieaq"!#<7$$"2Z"HeczE'H"!#<$!1#HW$pLeo<!#;7$$"226AWOujI"!#<$!1owg:+&)H=!#;7$$"2<MoO$G:;8!#<$!2.Xn**Ro'))=!#<7$$"2["Hec,,F8!#<$!2-*[4`bW`>!#<7$$"1.17WfwO8!#;$!1(=2\:u6,#!#;7$$"28C['HB=Z8!#<$!2(=*zw_8B2#!#<7$$"1yb6B6ic8!#;$!1`1]d-FF@!#;7$$"2tV([<-%pO"!#<$!2o5mZ[oo=#!#<7$$"1Gb5"o\mP"!#;$!1Z4)=(Q[UA!#;7$$"2k?T#o))z'Q"!#<$!2AqEn(*Q,I#!#<7$$"2V%)oPb@nR"!#<$!2)pmBsP.cB!#<7$$"2'['Hfo4rS"!#<$!2t,eTU[ST#!#<7$$"2nHf=p9rT"!#<$!1_#=NoP%pC!#;7$$"1(QxaDYtU"!#;$!1%)p\!)zeDD!#;7$$"2Z"He'4$\P9!#<$!1>jos/y!e#!#;7$$"2Pu[(*)p"oW"!#<$!2cO+ioj5j#!#<7$$"1)e<NA.vX"!#;$!1B#)pmO=)o#!#;7$$"2-'>R=61n9!#<$!2'*)>]z/")QF!#<7$$"1'>Ry#=Dx9!#;$!1)yx:V3Bz#!#;7$$"2tU&3Pc+(["!#<$!1r]F#zZI%G!#;7$$"2(=Pu))f$y\"!#<$!1%>X#GT&))*G!#;7$$"2sV([xU@2:!#<$!16&o=wCn%H!#;7$$"2wZ&4>(ey^"!#<$!2&\7)f+[0+$!#<7$$"29Ig?(*fv_"!#<$!1V`oX"H"\I!#;7$$"1b4>=;<Q:!#;$!1\;,@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rhoest:=fsolve(disc); 1/op(1,[%]);

NiQkIit0bWNLNSEjNSQiK3QzJXoqUkYl

JCIrWllnJW8qISIq

rhoest:=op(1,[fsolve(disc)]);

JCIrdG1jSzUhIzU=

With the number of vertices (variable u for non-root vertices) eqLe:=-L+1+2*t*L^2+t*u*(L-1)+2*t*u*(L0-K01): eqL0:=-L0+2*t*L*L0+t*L+t*u*L0+t*u*(L0-K01): eqK01:=-K01+t+t*L0: factor(resultant(subs(isolate(eqK01,K01),eqLe), subs(isolate(eqK01,K01),eqL0),L0));

LCQqJiwoKigiIiMiIiJJIkxHNiJGJ0kidEdGKUYnRicqJkYqRidJInVHRilGJ0YnRichIiJGJywwKihGKEYnKUYqRiZGJ0YsRidGLSooRiZGJylGKEYmRidGKkYnRicqKkYmRidGKEYnRipGJ0YsRidGJyomRjBGJ0YsRidGLSooRiZGJ0YqRidGLEYnRi1GKEYtRidGJ0YnRi0=

We select the quadratic factor (careful here, sometimes it is the second, sometimes the third...) algLu:=collect(op(3,%),L,factor);

LCwqKCIiIyIiIkkidEc2IkYlKUkiTEdGJ0YkRiVGJSomLCgqJilGJkYkRiVJInVHRidGJSEiIiooRiRGJUYmRiVGLkYlRiVGJUYvRiVGKUYlRiVGLEYvRjBGL0YlRiU=

factor([solve(algLu,L)]);

NyQsJCooIyIiIiIiJUYmSSJ0RzYiISIiLCoqJilGKCIiI0YmSSJ1R0YpRiZGJiooRi5GJkYoRiZGL0YmRipGJkYmKiQpLDIqJilGKEYnRiYpRi9GLkYmRiYqKEYnRiYpRigiIiRGJkY2RiZGKiooIiIpRiZGOEYmRi9GJkYmKihGJ0YmRi1GJkY2RiZGJiooIiM9RiZGLUYmRi9GJkYmKihGJ0YmRihGJkYvRiZGKiomRjtGJkYoRiZGKkYmRiYjRiZGLkYmRiZGJkYmLCQqKEYlRiYsKkYsRipGMEYmRjFGJkYmRipGJkYoRipGKg==

Lusol:=op(2,%);

LCQqKCMiIiIiIiVGJSwqKiYpSSJ0RzYiIiIjRiVJInVHRitGJSEiIiooRixGJUYqRiVGLUYlRiUqJCksMiomKUYqRiZGJSlGLUYsRiVGJSooRiZGJSlGKiIiJEYlRjVGJUYuKigiIilGJUY3RiVGLUYlRiUqKEYmRiVGKUYlRjVGJUYlKigiIz1GJUYpRiVGLUYlRiUqKEYmRiVGKkYlRi1GJUYuKiZGOkYlRipGJUYuRiVGJSNGJUYsRiVGJUYlRi5GJUYqRi5GLg==

Estimating the average number of vertices (linear in n) N:=100: ser0:=series(subs(u=1,Lsol),t,N): ser1:=series(subs(u=1,diff(Lusol,u)),t,N): plot([seq([1/n,evalf(coeff(ser1,t,n)/n/coeff(ser0,t,n))],n=2..N-2)],x=0..1/10,y=0.17..0.19,style=point);

NistJSdDVVJWRVNHNiQ3XXE3JCQiIiYhIiIkIiIiISIiNyQkIjFMTExMTExMTCEjOyQiKzo6Ojo6ISM1NyQkIiNEISIjJCIqKCp5X3AiISIqNyQkIiIjISIiJCIrJz0jPWw8ISM1NyQkIjJtbW1tbW1tbSIhIzwkIit0bypceiIhIzU3JCQiMiZHOWRHOWRHOSEjPCQiKyJmcyYzPSEjNTckJCIkRCIhIiQkIit3aCZcIj0hIzU3JCQiMTY2NjY2NjY2ISM7JCIrWzUheiI9ISM1NyQkIiIiISIiJCIqKnA0Pj0hIio3JCQiMSI0NDQ0NDQ0KiEjPCQiK3BITz49ISM1NyQkIjFMTExMTExMJCkhIzwkIis4PTc+PSEjNTckJCIxJHAyQnAyQnAoISM8JCIrIkguJz09ISM1NyQkIjFVciZHOWRHOSghIzwkIitbcyR6Ij0hIzU3JCQiMW5tbW1tbW1tISM8JCIqOSo+PD0hIio3JCQiJEQnISIlJCIrc05WOz0hIzU3JCQiMjBadzYlSE4jKWUhIz0kIit2cW06PSEjNTckJCIxYmJiYmJiYmIhIzwkIis+YCJcIj0hIzU3JCQiMVVvdCUqeTpqXyEjPCQiK0N0PTk9ISM1NyQkIiImISIjJCIrJCl6Wzg9ISM1NyQkIjI7dy8+dy8+dyUhIz0kIis5Jz5HIj0hIzU3JCQiMmNhYWFhYWFhJSEjPSQiK0ZIPTc9ISM1NyQkIjI7X2NwM0V5TSUhIz0kIitNd2Q2PSEjNTckJCIya21tbW1tbTslISM9JCIrWkcrNj0hIzU3JCQiIiUhIiMkIitFdFg1PSEjNTckJCIyayVROllROllRISM9JCIrWCdSKjQ9ISM1NyQkIjJOcS5QcS5QcSQhIz0kIismSFslND0hIzU3JCQiMXImRzlkRzlkJCEjPCQiK2Q8KSozPSEjNTckJCIyYicqbz8nZUZbTSEjPSQiK08mUSYzPSEjNTckJCIxTExMTExMTEwhIzwkIisnPjwiMz0hIzU3JCQiMS5IaF5rIWVBJCEjPCQiK25qcjI9ISM1NyQkIiVESiEiJiQiK2NaTDI9ISM1NyQkIjIvLi4uLi4uLiQhIz0kIio6cnAhPSEiKjckJCIyYEIpZXFrPFRIISM9JCIrPFdpMT0hIzU3JCQiMWRHOWRHOWRHISM8JCIpTkgxPSEiKTckJCIyd3h4eHh4eHgjISM9JCIrPHUoZiE9ISM1NyQkIjEuRnEtRnEtRiEjPCQiK2BfbjA9ISM1NyQkIjFAJW90JSp5OmojISM8JCIramhRMD0hIzU3JCQiMWtENWtENWtEISM8JCIrZSQ0XiE9ISM1NyQkIiNEISIkJCIqNldbIT0hIio3JCQiMkQhUkMhUkMhUkMhIz0kIip1KmUvPSEiKjckJCIyM1FfNFFfNFEjISM9JCIrQGNNLz0hIzU3JCQiMnMkKVtgUiJlREIhIz0kIittNjYvPSEjNTckJCIyR0ZGRkZGRkYjISM9JCIrTGUpUSE9ISM1NyQkIjJCQUFBQUFBQSMhIz0kIis9InBPIT0hIzU3JCQiMjNFeU0vOFI8IyEjPSQiK1swWS49ISM1NyQkIjEmM29XZGZ3NyMhIzwkIisob2ZLIT0hIzU3JCQiMktMTExMTEwzIyEjPSQiK0RoMS49ISM1NyQkIjE3MWBFaiIzLyMhIzwkIisjW3pHIT0hIzU3JCQiIiMhIiMkIissJSpwLT0hIzU3JCQiMFxEUEoleWc+ISM7JCIqYkREIT0hIio3JCQiMksjcDJCcDJCPiEjPSQiKz13Ti09ISM1NyQkIjInKT1JR1gjeicpPSEjPSQiKzlgPi09ISM1NyQkIjI8Jj0mPSY9Jj0mPSEjPSQiK2okUT8hPSEjNTckJCIxPT09PT09PT0hIzwkIis2bCk9IT0hIzU3JCQiMmNHOWRHOWR5IiEjPSQiKzsmUjwhPSEjNTckJCIyMUc3XCdmUWE8ISM9JCIrYXJmLD0hIzU3JCQiMkZbTTUkejhDPCEjPSQiKzYjZjkhPSEjNTckJCIxKUdQVUQ6XHAiISM8JCIrKVtEOCE9ISM1NyQkIjJtbW1tbW1tbSIhIz0kIisoeiY+LD0hIzU3JCQiMSMzJkhpVU1SOyEjPCQiK2gqcDUhPSEjNTckJCIyO1gxZUEuSGgiISM9JCIrNnklNCE9ISM1NyQkIjJzZSx0ZSx0ZSIhIz0kIisjPkgzIT0hIzU3JCQiJkRjIiEiJyQiK19Scis9ISM1NyQkIjImUTpZUTpZUTohIz0kIitgPmcrPSEjNTckJCIyX15eXl5eXl4iISM9JCIrZklcKz0hIzU3JCQiMmUkR1Y4dGAjXCIhIz0kIitZclErPSEjNTckJCIydzYlSE4jKWVxOSEjPSQiKyQ0JUcrPSEjNTckJCIyMSUpPUJPdiNcOSEjPSQiKyh5JD0rPSEjNTckJCIyJkc5ZEc5ZEc5ISM9JCIrQWgzKz0hIzU3JCQiMkBORFVxXSUzOSEjPSQiKyUqNCoqKnoiISM1NyQkIjIpKSkpKSkpKSkpKSkpKVEiISM9JCIrMiQpKikqeiIhIzU3JCQiMGopcDhJJylwOCEjOyQiK3B6ISkqeiIhIzU3JCQiMjlOXjhOXjhOIiEjPSQiKyQqKT4oKnoiISM1NyQkIjJNTExMTExMTCIhIz0kIismKlJqKnoiISM1NyQkIjIwQCVvdCUqeTo4ISM9JCIrJz5dJip6IiEjNTckJCIyKSlILCgpSCwoKUgiISM9JCIrQCVvJSp6IiEjNTckJCIxI0deP0deP0ciISM8JCIrKGYpUSp6IiEjNTckJCIybTc1W3lBZUUiISM9JCIrZDFKKnoiISM1NyQkIiREIiEiJSQiK01YQip6IiEjNTckJCIyeWNNNyF6Y003ISM9JCIrbSw7KnoiISM1NyQkIjI4Jj43Jj43Jj43ISM9JCIrJVwoMyp6IiEjNTckJCIyUVYzckY+Wz8iISM9JCIqWTshKnoiISIqNyQkIjIvPncvPncvPiIhIz0kIiosWiopeiIhIio3JCQiMlRITiMpZXFrPCIhIz0kIisjNHopKXoiISM1NyQkIjInPVduKHAhemk2ISM9JCIrY0UiKSl6IiEjNTckJCIyPUtjdEdEJVw2ISM9JCIrYnd1KXoiISM1NyQkIjJrampqampqOCIhIz0kIitVU28peiIhIzU3JCQiMnYoemgwYmZCNiEjPSQiK3U8aSl6IiEjNTckJCIyNzY2NjY2NjYiISM9JCIrNDNjKXoiISM1NyQkIjEqNCwqKTQsKik0IiEjPCQiKzE2XSl6IiEjNTckJCIyLzhSPF9jcDMiISM9JCIrRkVXKXoiISM1NyQkIjI3SS9zIilvXzIiISM9JCIrTmBRKXoiISM1NyQkIjJEL01zeUhRMSIhIz0kIislPkgkKXoiISM1NyQkIjIlb3QlKnk6al81ISM9JCIrcFRGKXoiISM1NyQkIjJtbW1tbW07LyIhIz0kIitILUEpeiIhIzU3JCQiMmthXl0keSM0LiIhIz0kIitVdDspeiIhIzU3JCQiMTFgRWoiMy8tIiEjPCQiK3hhNil6IiEjNS0lJkNPTE9SRzYnJSRSR0JHJCIpQyllcSUhIikkIiIhISIiJCIoJz4hXCYhIiktJStfQVRUUklCVVRFRzYjLyUnc291cmNlRyUsbWF0aGRlZmF1bHRHLSUlVklFV0c2JDskIiIhISIiJCIiIiEiIjskIiM8ISIjJCIjPiEiIy0mJSZfQVhJU0c2IyIiIjYnLSUrX0dSSURMSU5FU0c2Jy0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lKV9WSVNJQkxFRzYjIiIhLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNictJStfR1JJRExJTkVTRzYnLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSUpX1ZJU0lCTEVHNiMiIiEtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieDYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieTYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLSUmU1RZTEVHNiMlJlBPSU5URy0lKV9WSVNJQkxFRzYjIiIiLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkSSchIiItJSlCT1VORFNfWUc2IyQiJEkiISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlK0shIiItJS5CT1VORFNfSEVJR0hURzYjJCIlU00hIiItJSlDSElMRFJFTkc2Ii0lK0FOTk9UQVRJT05HNictJSlCT1VORFNfWEc2IyQiIiEhIiItJSlCT1VORFNfWUc2IyQiIiEhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUrUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrUyEiIi0lKUNISUxEUkVORzYiRzYi

Lsing:=op(algeqtoseries(subs(t=RootOf(_Z^4+4*_Z^3+22*_Z^2-12*_Z+1)*(1-epsilon),algL),epsilon,L,2,true));

LC4qJiMiIiIiIiVGJSktSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywsKiQpSSNfWkdGKkYmRiVGJSomRiZGJSlGMiIiJEYlRiUqJiIjQUYlKUYyIiIjRiVGJSomIiM3RiVGMkYlISIiRiVGJUY1RiVGPCokKUYoRjlGJUY8KiYjIiNARiZGJUYoRiVGPCMiIiZGOUYlKiYtRik2IywsKiZGNUYlRidGJUYlKiYiIzhGJUY+RiVGJSomRiZGJUY4RiVGJSomIiNwRiVGKEYlRiUiI0RGPEYlKUkoZXBzaWxvbkdGLSNGJUY5RiVGJS1JIk9HRis2I0ZQRiU=

Ldiff:=solve(limit(diff(subs(L=L(u),algLu),u),u=1),(D(L))(1));

KihJInRHNiIiIiIsKiomRiNGJS1JIkxHRiQ2I0YlRiVGJSomIiIjRiVGKEYlISIiRiNGJUYsRiVGJSwqKigiIiVGJUYjRiVGKEYlRiUqJClGI0YsRiVGLSomRixGJUYjRiVGJUYlRi1GLQ==

Equation for Ldiff factor(resultant(subs(L(1)=L, numer(Ldiff-Ld)), algL,L));

LCQqJkkidEc2IiIiIiw+KigiIiNGJilJI0xkR0YlRilGJilGJCIiJUYmRiYqJkYrRiYpRiQiIiZGJiEiIiooIiIpRiZGKkYmKUYkIiIkRiZGJiooRilGJkYrRiZGLEYmRjEqKCIjV0YmRipGJilGJEYpRiZGJiooIiM5RiZGK0YmRjRGJkYxKiZGKUYmRixGJkYxKigiI0NGJkYqRiZGJEYmRjEqKCIjY0YmRitGJkY5RiZGJiomIiM1RiZGNEYmRjEqJkYpRiZGKkYmRiYqKCIjREYmRitGJkYkRiZGMSomRi1GJkY5RiZGMSomRilGJkYrRiZGJkYmRjE=

algLd:=op(3,%);

LD4qKCIiIyIiIilJI0xkRzYiRiRGJSlJInRHRigiIiVGJUYlKiZGJ0YlKUYqIiImRiUhIiIqKCIiKUYlRiZGJSlGKiIiJEYlRiUqKEYkRiVGJ0YlRilGJUYvKigiI1dGJUYmRiUpRipGJEYlRiUqKCIjOUYlRidGJUYyRiVGLyomRiRGJUYpRiVGLyooIiNDRiVGJkYlRipGJUYvKigiI2NGJUYnRiVGN0YlRiUqJiIjNUYlRjJGJUYvKiZGJEYlRiZGJUYlKigiI0RGJUYnRiVGKkYlRi8qJkYrRiVGN0YlRi8qJkYkRiVGJ0YlRiU=

Ldsing:=op(algeqtoseries(subs(t=RootOf(_Z^4+4*_Z^3+22*_Z^2-12*_Z+1)*(1-epsilon),algLd),epsilon,Ld,1));

LCYqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCwqJiIjXiIiIiktRiU2IywsKiQpSSNfWkdGJiIiJUYuRi4qJkY2Ri4pRjUiIiRGLkYuKiYiI0FGLilGNSIiI0YuRi4qJiIjN0YuRjVGLiEiIkYuRi5GOUYuRi4qJiIkNCNGLilGMEY9Ri5GLiomIiV3OEYuRjxGLkYuKiYiJVY1Ri5GMEYuRi4iJGYiRkBGLilJKGVwc2lsb25HRikjRi5GPUZARi4tSSJPR0YnNiNGLkYu

subs(RootOf(_Z^4+4*_Z^3+22*_Z^2-12*_Z+1)=R,op(1,Ldsing)*sqrt(epsilon));

LUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsLComIiNeIiIiKUkiUkdGJyIiJEYsRiwqJiIkNCNGLClGLiIiI0YsRiwqJiIldzhGLClJI19aR0YkRjNGLEYsKiYiJVY1RixGLkYsRiwiJGYiISIi

isolate(51*R^3+209*R^2+1376*_Z^2+1043*R-159,_Z^2);

LyokKUkjX1pHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiMiIiIsKiomIyIjXiIldzhGKylJIlJHRikiIiRGKyEiIiomIyIkNCNGMEYrKUYyRipGK0Y0KiYjIiVWNUYwRitGMkYrRjQjIiRmIkYwRis=

c:=sqrt(op(2,%));

LCQqJiMiIiIiJFckRiUpLCoqJiIlJ1ElRiUpSSJSRzYiIiIkRiUhIiIqJiImdXoiRiUpRiwiIiNGJUYvKiYiJilwKilGJUYsRiVGLyImdU8iRiUjRiVGM0YlRiU=

subs(RootOf(_Z^4+4*_Z^3+22*_Z^2-12*_Z+1)=R,op(5,Lsing)/sqrt(epsilon));

LUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsLComIiIkIiIiKUkiUkdGJ0YrRixGLComIiM4RiwpRi4iIiNGLEYsKiYiIiVGLClJI19aR0YkRjJGLEYsKiYiI3BGLEYuRixGLCIjRCEiIg==

isolate(3*R^3+13*R^2+4*_Z^2+69*R-25,_Z^2);

LyokKUkjX1pHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiMiIiIsKiomIyIiJCIiJUYrKUkiUkdGKUYvRishIiIqJiMiIzhGMEYrKUYyRipGK0YzKiYjIiNwRjBGK0YyRitGMyMiI0RGMEYr

beta:=sqrt(op(2,%));

LCQqJiMiIiIiIiNGJSksKiomIiIkRiUpSSJSRzYiRipGJSEiIiomIiM4RiUpRixGJkYlRi4qJiIjcEYlRixGJUYuIiNERiVGJEYlRiU=

(subs(R=rhoest,2*c/beta));

JCIrImU9R3oiISM1

The system of Section 3.2.2 Here, we have letters a and b rather than 0 and 1 exKab:=t+t*La; exKbbaa:=t*Kba+t*Lbaa; exKbaba:=2*t*Kba+t*Laba; exKabba:=t*Kba+t*Lbba; exL:=1+2*t*L^2+t*(L-1)+2*t*(La-Kab+Lbaa-Kbbaa+Laba-Kbaba+Lbba-Kabba); exLa:=2*t*L*La+t*L+t*La+t*(La-Kab+Laba-Kbaba+Lbaa-Kbbaa+Lbba-Kabba); exLbb:=2*t*L*Lbb+t*Lb+t*Lbb+t*(Labb-Kaabb); exLab:=2*t*L*Lab+t*La+t+t*Lab+t*(Lb-Kab+Lbab-Kabab+Laab-Kbaab); exLbaa:=2*t*L*Lbaa+t*Lba+t*Lbaa+t*(Lbaa-Kbbaa); exLaba:=2*t*L*Laba+t*(Lab+L*Kba)+t*Laba+t*(Laba-Kbaba); exLbba:=2*t*L*Lbba+t*(Lbb+L*Kba)+t*Lbba+t*(Lbba-Kabba);

LCYqJkkjTGFHNiIiIiJJInRHRiVGJkYmRidGJg==

LCYqJkkkS2JhRzYiIiIiSSJ0R0YlRiZGJiomSSVMYmFhR0YlRiZGJ0YmRiY=

LCYqKCIiIyIiIkkkS2JhRzYiRiVJInRHRidGJUYlKiZJJUxhYmFHRidGJUYoRiVGJQ==

LCYqJkkkS2JhRzYiIiIiSSJ0R0YlRiZGJiomSSVMYmJhR0YlRiZGJ0YmRiY=

LCoiIiJGIyooIiIjRiNJInRHNiJGIylJIkxHRidGJUYjRiMqJkYmRiMsJkYpRiNGIyEiIkYjRiMqKEYlRiNGJkYjLDJJI0xhR0YnRiNJJEthYkdGJ0YsSSVMYmFhR0YnRiNJJktiYmFhR0YnRixJJUxhYmFHRidGI0kmS2JhYmFHRidGLEklTGJiYUdGJ0YjSSZLYWJiYUdGJ0YsRiNGIw==

LCoqKiIiIyIiIkkidEc2IkYlSSJMR0YnRiVJI0xhR0YnRiVGJSomRiZGJUYoRiVGJSomRiZGJUYpRiVGJSomRiZGJSwyRilGJUkkS2FiR0YnISIiSSVMYmFhR0YnRiVJJktiYmFhR0YnRi9JJUxhYmFHRidGJUkmS2JhYmFHRidGL0klTGJiYUdGJ0YlSSZLYWJiYUdGJ0YvRiVGJQ==

LCoqKiIiIyIiIkkidEc2IkYlSSJMR0YnRiVJJExiYkdGJ0YlRiUqJkYmRiVJI0xiR0YnRiVGJSomRiZGJUYpRiVGJSomRiZGJSwmSSVMYWJiR0YnRiVJJkthYWJiR0YnISIiRiVGJQ==

LCwqKiIiIyIiIkkidEc2IkYlSSJMR0YnRiVJJExhYkdGJ0YlRiUqJkYmRiVJI0xhR0YnRiVGJUYmRiUqJkYmRiVGKUYlRiUqJkYmRiUsLkkjTGJHRidGJUkkS2FiR0YnISIiSSVMYmFiR0YnRiVJJkthYmFiR0YnRjFJJUxhYWJHRidGJUkmS2JhYWJHRidGMUYlRiU=

LCoqKiIiIyIiIkkidEc2IkYlSSJMR0YnRiVJJUxiYWFHRidGJUYlKiZGJkYlSSRMYmFHRidGJUYlKiZGJkYlRilGJUYlKiZGJkYlLCZGKUYlSSZLYmJhYUdGJyEiIkYlRiU=

LCoqKiIiIyIiIkkidEc2IkYlSSJMR0YnRiVJJUxhYmFHRidGJUYlKiZGJkYlLCYqJkkkS2JhR0YnRiVGKEYlRiVJJExhYkdGJ0YlRiVGJSomRiZGJUYpRiVGJSomRiZGJSwmRilGJUkmS2JhYmFHRichIiJGJUYl

LCoqKiIiIyIiIkkidEc2IkYlSSJMR0YnRiVJJUxiYmFHRidGJUYlKiZGJkYlLCYqJkkkS2JhR0YnRiVGKEYlRiVJJExiYkdGJ0YlRiVGJSomRiZGJUYpRiVGJSomRiZGJSwmRilGJUkmS2FiYmFHRichIiJGJUYl

Checking a forward equation factor(subs(L=1+2*La,exL-2*exLa));

IiIi

Using symmetries syst:=subs(Kba=Kab,Kbbaa=Kaabb,Kbaab=Kabba,Kbaba=Kabab,Lb=La,Lba=Lab,Lbaa=Labb,Lbb=Laa,Lbba=Laab,Lbab=Laba,[exL-L,exLa-La,exKab-Kab,exLaba-Laba,exKbaba-Kbaba,exLbaa-Lbaa,exKbbaa-Kbbaa,exLbba-Lbba,exKabba-Kabba,exLab-Lab,exLbb-Lbb]): eliminate(syst,{Kaabb, Kab, Kabab, Kabba, La, Laab, Lab, Laba, Labb, Laa}): factor(op(2,%));

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

algL:=collect(op(%),L,factor);

LCwqKCIiKSIiIilJIkxHNiIiIiVGJSlJInRHRigiIiRGJUYlKipGKUYlKUYrIiIjRiUsKiomRixGJUYqRiVGJSomRilGJUYuRiVGJSomIiInRiVGK0YlISIiRixGJUYlKUYnRixGJUY1KipGL0YlRitGJSwuKiZGLEYlKUYrIiImRiVGJSomIiM3RiUpRitGKUYlRjUqJiIjNUYlRipGJUY1KiYiIzlGJUYuRiVGJSomRkBGJUYrRiVGNUYsRiVGJSlGJ0YvRiVGJSooLCZGK0YlRiVGNUYlLC4qJiIjNkYlRjpGJUYlKiZGQEYlRj5GJUY1KiZGNEYlRipGJUY1KiZGLEYlRi5GJUY1RitGNUYlRiVGJUYnRiVGJSomRkZGJSwuKiZGO0YlRjpGJUYlKiZGKUYlRj5GJUY1RktGJSomIiIoRiVGLkYlRjUqJkY7RiVGK0YlRiVGJUY1RiVGJQ==

with(gfun): algeqtoseries(algL,t,L,10,true);

NyMrOUkidEc2IiIiIiIiISIiIyIiIiIjNSIiIyIjbSIiJCIkLyYiIiUiJTNTIiImIiZNRyQiIiciJzNjRiIiKCIoIVtnQiIiKSIpP21iPyIiKi1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzU=

Positive system of Section 3.3 (the p stands for plus) exKab:=t+t*(Kba+Lap); exKbbaa:=t*Kba+t*(Kbbaa+Lbaap); exKbaba:=2*t*Kba+t*(Kbaba+Labap); exKabba:=t*Kba+t*(Kabba+Lbbap); exLp:=2*t*(1+Lp)^2+t*Lp+2*t*(Lap+Lbaap+Labap+Lbbap); exLap:=2*t*(1+Lp)*(Kba+Lap)+t*Lp+t*Lap+t*(Lbaap+Labap+Lbbap); exLbb:=2*t*(1+Lp)*Lbb+t*(Kab+Lbp)+t*Lbb+t*Labbp; exLabp:=2*t*(1+Lp)*(Kab+Lbap)+t*(Kba+Lap)+t*Labp+t*(Laabp+Lbabp); exLbaap:=2*t*(1+Lp)*(Kbbaa+Lbaap)+t*Lbap+t*Lbaap; exLabap:=2*t*(1+Lp)*(Kbaba+Labap)+t*(Labp+Lp*Kba)+t*Labap; exLbbap:=2*t*(1+Lp)*(Kabba+Lbbap)+t*(Lbb+Lp*Kba)+t*Lbbap;

LCZJInRHNiIiIiIqJkYjRiUsJkkkS2JhR0YkRiVJJExhcEdGJEYlRiVGJQ==

LCYqJkkidEc2IiIiIkkkS2JhR0YlRiZGJiomRiRGJiwmSSZLYmJhYUdGJUYmSSZMYmFhcEdGJUYmRiZGJg==

LCYqKCIiIyIiIkkidEc2IkYlSSRLYmFHRidGJUYlKiZGJkYlLCZJJktiYWJhR0YnRiVJJkxhYmFwR0YnRiVGJUYl

LCYqJkkidEc2IiIiIkkkS2JhR0YlRiZGJiomRiRGJiwmSSZLYWJiYUdGJUYmSSZMYmJhcEdGJUYmRiZGJg==

LCgqKCIiIyIiIkkidEc2IkYlKSwmRiVGJUkjTHBHRidGJUYkRiVGJSomRiZGJUYqRiVGJSooRiRGJUYmRiUsKkkkTGFwR0YnRiVJJkxiYWFwR0YnRiVJJkxhYmFwR0YnRiVJJkxiYmFwR0YnRiVGJUYl

LCoqKiIiIyIiIkkidEc2IkYlLCZGJUYlSSNMcEdGJ0YlRiUsJkkkS2JhR0YnRiVJJExhcEdGJ0YlRiVGJSomRiZGJUYpRiVGJSomRiZGJUYsRiVGJSomRiZGJSwoSSZMYmFhcEdGJ0YlSSZMYWJhcEdGJ0YlSSZMYmJhcEdGJ0YlRiVGJQ==

LCoqKiIiIyIiIkkidEc2IkYlLCZGJUYlSSNMcEdGJ0YlRiVJJExiYkdGJ0YlRiUqJkYmRiUsJkkkS2FiR0YnRiVJJExicEdGJ0YlRiVGJSomRiZGJUYqRiVGJSomRiZGJUkmTGFiYnBHRidGJUYl

LCoqKiIiIyIiIkkidEc2IkYlLCZGJUYlSSNMcEdGJ0YlRiUsJkkkS2FiR0YnRiVJJUxiYXBHRidGJUYlRiUqJkYmRiUsJkkkS2JhR0YnRiVJJExhcEdGJ0YlRiVGJSomRiZGJUklTGFicEdGJ0YlRiUqJkYmRiUsJkkmTGFhYnBHRidGJUkmTGJhYnBHRidGJUYlRiU=

LCgqKiIiIyIiIkkidEc2IkYlLCZGJUYlSSNMcEdGJ0YlRiUsJkkmS2JiYWFHRidGJUkmTGJhYXBHRidGJUYlRiUqJkYmRiVJJUxiYXBHRidGJUYlKiZGJkYlRixGJUYl

LCgqKiIiIyIiIkkidEc2IkYlLCZGJUYlSSNMcEdGJ0YlRiUsJkkmS2JhYmFHRidGJUkmTGFiYXBHRidGJUYlRiUqJkYmRiUsJiomSSRLYmFHRidGJUYpRiVGJUklTGFicEdGJ0YlRiVGJSomRiZGJUYsRiVGJQ==

LCgqKiIiIyIiIkkidEc2IkYlLCZGJUYlSSNMcEdGJ0YlRiUsJkkmS2FiYmFHRidGJUkmTGJiYXBHRidGJUYlRiUqJkYmRiUsJiomSSRLYmFHRidGJUYpRiVGJUkkTGJiR0YnRiVGJUYlKiZGJkYlRixGJUYl

syst:=subs(Kba=Kab,Kbbaa=Kaabb,Kbaab=Kabba,Kbaba=Kabab,Lbp=Lap,Lbap=Labp,Lbaap=Labbp,Lbbp=Laap,Lbbap=Laabp,Lbabp=Labap,Lbb=Laa,[exLp-Lp,exLap-Lap,exKab-Kab,exLabap-Labap,exKbaba-Kbaba,exLbaap-Lbaap,exKbbaa-Kbbaa,exLbbap-Lbbap,exKabba-Kabba,exLabp-Labp,exLbb-Lbb]): eliminate(syst,{Kaabb, Kab, Kabab, Kabba, Lap, Laabp, Labp, Labap, Labbp, Laa}): factor(op(2,%));

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

collect(subs(Lp=L-1,op(%)),L,factor);

LCwqKCIiKSIiIilJIkxHNiIiIiVGJSlJInRHRigiIiRGJSEiIioqRilGJSlGKyIiI0YlLCoqJkYsRiVGKkYlRiUqJkYpRiVGL0YlRiUqJiIiJ0YlRitGJUYtRixGJUYlKUYnRixGJUYlKipGMEYlRitGJSwuKiZGLEYlKUYrIiImRiVGJSomIiM3RiUpRitGKUYlRi0qJiIjNUYlRipGJUYtKiYiIzlGJUYvRiVGJSomRkBGJUYrRiVGLUYsRiVGJSlGJ0YwRiVGLSooLCZGK0YlRiVGLUYlLC4qJiIjNkYlRjpGJUYlKiZGQEYlRj5GJUYtKiZGNUYlRipGJUYtKiZGLEYlRi9GJUYtRitGLUYlRiVGJUYnRiVGLSomRkZGJSwuKiZGO0YlRjpGJUYlKiZGKUYlRj5GJUYtRktGJSomIiIoRiVGL0YlRi0qJkY7RiVGK0YlRiVGJUYtRiVGLQ==

Checking factor(%/algL);

ISIi

Asymptotics disc:=factor(discrim(algL,L));

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

fsolve(disc);

NjIkIStuJlF4ZCIhIikkIiIhRidGJkYmRiZGJkYmRiZGJkYmRiZGJkYmJCIrbW5HVSkqISM2JCIrXlNbOUYhIzUkIiIiRic=

{fsolve(disc,t=0..1)};

PCYkIiIhRiQkIittbkdVKSohIzYkIiteU1s5RiEjNSQiIiJGJA==

1/op(2,%);

JCIrV1MtOzUhIik=

1/op(3,%%);

JCIrTjglUm8kISIq

The system of Section 4.2.1 (p stands for the prime, and there is a factor u for each non-root vertex) eqKp01:=-Kp01+t+t*u*Le*Lp0: eqLe:=-Le+1+Le*Lpe: eqLpe:=-Lpe+2*t*Le+t*u*Le*(Lpe+2*Lp0-2*Kp01): eqLp0:=-Lp0+t*Le+t*u*Le*(2*Lp0-Kp01): syst:={eqLe,eqLpe,eqLp0,eqKp01}:indets(syst);

PChJJUtwMDFHNiJJI0xlR0YkSSRMcDBHRiRJJExwZUdGJEkidEdGJEkidUdGJA==

eliminate(syst,{Lpe,Lp0,Kp01}): factor(op(2,%));

PCMsMiooKUkjTGVHNiIiIiQiIiIpSSJ0RzYiIiIjIiIiKUkidUc2IiIiIyIiIiEiIiooKUkjTGVHNiIiIiMiIiIpSSJ0RzYiIiIjIiIiKUkidUc2IiIiIyIiIiIiIioqIiIjIiIiKUkjTGVHNiIiIiMiIiIpSSJ0RzYiIiIjIiIiSSJ1RzYiIiIiISIiKioiIiMiIiIpSSNMZUc2IiIiIyIiIkkidEc2IiIiIkkidUc2IiIiIiIiIiooIiIjIiIiKUkjTGVHNiIiIiMiIiJJInRHNiIiIiIiIiIqKiIiIyIiIkkjTGVHNiIiIiJJInRHNiIiIiJJInVHNiIiIiIhIiJJI0xlRzYiISIiIiIiIiIi

algLu:=collect(subs(Le=L,op(%)),L,factor);

LCoiIiJGIyooKUkidEc2IiIiI0YjKUkidUdGJ0YoRiMpSSJMR0YnIiIkRiMhIiIqKEYmRiMsKiomRiZGI0YpRiNGIyooRihGI0YmRiNGKkYjRi4qJkYoRiNGKkYjRiNGKEYjRiMpRixGKEYjRiMqJiwmRjJGLkYjRi5GI0YsRiNGIw==

algL:=map(factor,subs(u=1,%));

LCoiIiJGIyomKUkiTEc2IiIiJEYjKUkidEdGJyIiI0YjISIiKihGKkYjLCZGKkYjIiIlRixGIylGJkYrRiNGLComLCYqJkYrRiNGKkYjRiNGI0YjRiNGJkYjRiw=

with(gfun):algeqtoseries(algL,t,L,10,true);

NyMrOUkidEc2IiIiIiIiISIiIyIiIiIjNSIiIyIjbSIiJCIkIVwiIiUiJSkqUSIiJiImI1tLIiInIicjKSl6IyIiKCIoaUxaIyIiKSIpJT4lSEEiIiotSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Positive system (Section 4.3). The p still stands for the prime (we do NOT indicate the plus). No series K nor K' is involved eqLe:=-Le+Le*Lep+Lep; eqLep:=-Lep+2*t*(1+Le)+t*(1+Le)*(Lep+2*L0p); eqL0p:=-L0p+t*Le+t*(1+Le)*L0p;

LCgqJkkjTGVHNiIiIiJJJExlcEdGJUYmRiZGJCEiIkYnRiY=

LChJJExlcEc2IiEiIiooIiIjIiIiSSJ0R0YkRigsJkYoRihJI0xlR0YkRihGKEYoKihGKUYoRipGKCwmRiNGKComRidGKEkkTDBwR0YkRihGKEYoRig=

LChJJEwwcEc2IiEiIiomSSJ0R0YkIiIiSSNMZUdGJEYoRigqKEYnRigsJkYoRihGKUYoRihGI0YoRig=

syst:={eqLe,eqLep,eqL0p}:indets(syst);

PCZJJEwwcEc2IkkjTGVHRiRJJExlcEdGJEkidEdGJA==

eliminate(syst,{Lep,L0p}): factor(op(2,%));

PCMsMiomKUkjTGVHNiIiIiQiIiIpSSJ0RzYiIiIjIiIiIiIiKigiIiUiIiIpSSNMZUc2IiIiIyIiIilJInRHNiIiIiMiIiIiIiIqKCIiJSIiIilJI0xlRzYiIiIjIiIiSSJ0RzYiIiIiISIiKigiIiYiIiJJI0xlRzYiIiIiKUkidEc2IiIiIyIiIiIiIiooIiInIiIiSSNMZUc2IiIiIkkidEc2IiIiIiEiIiomIiIjIiIiKUkidEc2IiIiIyIiIiIiIkkjTGVHNiIiIiIqJiIiIyIiIkkidEc2IiIiIiEiIg==

collect(subs(Le=L-1,op(%)),L,factor);

LCoiIiIhIiIqJilJIkxHNiIiIiRGIylJInRHRigiIiNGI0YjKihGK0YjLCZGK0YjIiIlRiRGIylGJ0YsRiNGIyomLCYqJkYsRiNGK0YjRiNGI0YjRiNGJ0YjRiM=

Checking factor(%/algL);

ISIi

Asymptotics disc:=factor(discrim(algL,L));

LCQqKiIiJSIiIilJInRHNiIiIiNGJSwoKiZGKUYlRiZGJUYlKiYiI0tGJUYnRiUhIiIiIiRGJUYlKSwmRidGJUYlRi5GKUYlRiU=

{fsolve(%,t)};

PCYkIiIhRiQkIitlXGVJJSohIzYkIiIiRiQkIis6JXAwZiIhIik=

1/op(2,%);

JCIrNSd6LjEiISIp

[solve(2*t^2-32*t+3)];evalf(%);

NyQsJiIiKSIiIiomIyIiJiIiI0YlKSIjNSNGJUYpRiVGJSwmRiRGJUYmISIi

NyQkIis6JXAwZiIhIikkIildZUklKiEiKg==

rho:=8-(5/2)*sqrt(10);

LCYiIikiIiIqJiMiIiYiIiNGJCkiIzUjRiRGKEYkISIi

Expansion at the singular point algeqtoseries(subs(t=rho*(1-epsilon),algL),epsilon,L,3);

NyQrK0koZXBzaWxvbkc2IiwmIiM+IiIiKiYiIidGKCkiIzUjRigiIiNGKEYoIiIhLCQqKiNGLiIiJkYoRiZGKCwmKiZGM0YoRitGKEYoIiNHISIiRigsJiomRi5GKEYrRihGKCIjNkY3RjdGKEYoLCQqKiMiIz0iJEQiRihGJkYoLCYqJiIlRmRGKEYrRihGKCImXSk9RjdGKClGOCIiJEY3RihGLi1JIk9HJSpwcm90ZWN0ZWRHNiNGKEZFLCwjRi5GRUYoKiYjRihGRUYoRitGKEYoKiYtSSdSb290T2ZHNiRGSEkoX3N5c2xpYkdGJTYjLCgqJiIjWEYoKUkjX1pHRlFGLkYoRigqJiIjO0YoRitGKEY3IiNTRjdGKClGJEYtRihGKComLCYjIiQzJCIkMCVGKComIyIjJClGW29GKEYrRihGKEYoRiRGKEYoLUZHNiMqJClGJCNGRUYuRihGKA==

Lsing:=op(2,%);

LCwjIiIjIiIkIiIiKiYjRiZGJUYmKSIjNSNGJkYkRiZGJiomLUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsKComIiNYRiYpSSNfWkdGL0YkRiZGJiomIiM7RiZGKUYmISIiIiNTRjtGJilJKGVwc2lsb25HRjJGK0YmRiYqJiwmIyIkMyQiJDAlRiYqJiMiIyQpRkNGJkYpRiZGJkYmRj5GJkYmLUkiT0dGMDYjKiQpRj4jRiVGJEYmRiY=

Average number of non-root vertices factor(limit(diff(subs(L=L(u),algLu),u),u=1));

LDIqKiIiJCIiIilJInRHNiIiIiNGJSktSSJMR0YoNiNGJUYpRiUtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRixGLUYlISIiKihGKUYlRiZGJSlGK0YkRiVGNSoqRilGJUYmRiVGK0YlRi5GJUY1KioiIilGJUYnRiVGK0YlRi5GJUYlKihGKUYlRidGJUYqRiVGJSooRilGJUYuRiVGJ0YlRjUqKEYpRiVGJ0YlRitGJUY1Ri5GNQ==

Ldsol:=op(2,subs(L(1)=L,factor(isolate(%,(D(L))(1)))));

LCQqLCIiIyIiIkkidEc2IkYlSSJMR0YnRiUsKComKUYoRiRGJUYmRiVGJUYoISIiRiVGJUYlLCwqKCIiJEYlRitGJSlGJkYkRiVGJSooRiRGJUYoRiVGMEYlRiUqKCIiKUYlRihGJUYmRiVGLComRiRGJUYmRiVGJUYlRiVGLEYs

algLd:=op(3,factor(resultant(algL,numer(Ld-Ldsol),L)));

LEoqKCIiIyIiIilJI0xkRzYiIiIkRiUpSSJ0R0YoIiInRiVGJSooIiNPRiVGJkYlKUYrIiImRiUhIiIqKCIiJUYlKUYnRiRGJUYqRiVGMSooIiNwRiVGJkYlKUYrRjNGJUYlKigiIyUpRiVGNEYlRi9GJUYlKigiI1FGJUYmRiUpRitGKUYlRjEqKCIkYSRGJUY0RiVGN0YlRjEqKCIiKUYlRidGJUYvRiVGMSooRilGJUYmRiUpRitGJEYlRiUqKCIkIVxGJUY0RiVGPEYlRiUqKCIjVEYlRidGJUY3RiVGJSooIiRNI0YlRjRGJUZCRiVGMSooIiQzIkYlRidGJUY8RiVGMSomRiRGJUY3RiVGMSooIiM9RiVGNEYlRitGJUYlKigiJC4iRiVGJ0YlRkJGJUYlKiZGM0YlRjxGJUYlKigiI0lGJUYnRiVGK0YlRjEqJkYzRiVGQkYlRjEqJkYkRiVGJ0YlRiU=

Ldsing:=map(factor,series(subs(L=Lsing,t=rho*(1-epsilon),Ldsol),epsilon,1));

LCYqLiMiIiMiIioiIiIsJiomRiVGJykiIzUjRidGJUYnRiciIiYhIiJGJywmRiVGJyokRipGJ0YnRicpSShlcHNpbG9uRzYiRixGLiwmRitGLkYwRidGLi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGMzYjLCgqJiIjWEYnKUkjX1pHRjdGJUYnRicqJiIjO0YnRipGJ0YuIiNTRi5GLkYnLUkiT0dGODYjRidGJw==

beta:=-op(3,Lsing)/sqrt(epsilon);;

LCQtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywoKiYiI1giIiIpSSNfWkdGJSIiI0YtRi0qJiIjO0YtKSIjNSNGLUYwRi0hIiIiI1NGNkY2

c:=op(1,Ldsing)*sqrt(epsilon);

LCQqLCMiIiMiIioiIiIsJiomRiVGJykiIzUjRidGJUYnRiciIiYhIiJGJywmRiVGJyokRipGJ0YnRicsJkYrRi5GMEYnRi4tSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywoKiYiI1hGJylJI19aR0Y0RiVGJ0YnKiYiIztGJ0YqRidGLiIjU0YuRi5GJw==

av:=simplify(2*c/beta);

LCYqJiMiIiIiIiRGJSkiIzUjRiUiIiNGJUYlIyIiJiIiJyEiIg==

evalf(%);

JCIrKD4jZjJBISM1

Numerical check N:=50:Lser:=op(algeqtoseries(algL,t,L,N,true)): Ldser:=series(subs(L=Lser,Ldsol),t,N+1): plot([seq([1/n,evalf(coeff(Ldser,t,n)/coeff(Lser,t,n)/n)],n=2..N-1)],x=0..0.1,y=0.2..0.24,style=POINT);

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NiI=

The system of Section 4.2.2 (the p stands for prime); or rather, the right hand side of the equations of the system exKab:=Kabp; exKabp:=t+t*L*Lap; exKbbaap:=t*Kba+t*Lbaap*L; exKbabap:=t*Kba+t*Labap*L; exKabbap:=t*Lbbap*L; exL:=1+L*Lp; exLa:=L*Lap; exLaa:=L*Laap; exLab:=L*Labp; exLp:=2*t*L+t*L*(Lp+2*(Lap-Kabp+Labbp-Kaabbp+Labap-Kababp+Laabp-Kabbap)); exLap:=t*L+t*L*(Lap+Lap-Kabp+Lbaap-Kbbaap+Labap-Kbabap+Lbbap-Kabbap); exLbbp:=t*Lb+t*L*(Laap+Labbp-Kaabbp); exLabp:=t*La+t+t*L*(Labp+Lap-Kabp+Lbabp-Kababp+Laabp-Kbaabp); exLbaap:=t*Lba+L*t*(Lbaap+Lbaap-Kbbaap); exLabap:=t*Lab+L*t*(Labap+Labap-Kbabap); exLbbap:=t*Lbb+L*t*(Lbbap+Lbbap-Kabbap);

SSVLYWJwRzYi

LCYqKEkiTEc2IiIiIkkkTGFwR0YlRiZJInRHRiVGJkYmRihGJg==

LCYqKEkiTEc2IiIiIkkmTGJhYXBHRiVGJkkidEdGJUYmRiYqJkkkS2JhR0YlRiZGKEYmRiY=

LCYqKEkiTEc2IiIiIkkmTGFiYXBHRiVGJkkidEdGJUYmRiYqJkkkS2JhR0YlRiZGKEYmRiY=

KihJInRHNiIiIiJJJkxiYmFwR0YkRiVJIkxHRiRGJQ==

LCYqJkkiTEc2IiIiIkkjTHBHRiVGJkYmRiZGJg==

KiZJIkxHNiIiIiJJJExhcEdGJEYl

KiZJIkxHNiIiIiJJJUxhYXBHRiRGJQ==

KiZJIkxHNiIiIiJJJUxhYnBHRiRGJQ==

LCYqKCIiIyIiIkkidEc2IkYlSSJMR0YnRiVGJSooRiZGJUYoRiUsNEkjTHBHRidGJSomRiRGJUkkTGFwR0YnRiVGJSomRiRGJUklS2FicEdGJ0YlISIiKiZGJEYlSSZMYWJicEdGJ0YlRiUqJkYkRiVJJ0thYWJicEdGJ0YlRjAqJkYkRiVJJkxhYmFwR0YnRiVGJSomRiRGJUknS2FiYWJwR0YnRiVGMComRiRGJUkmTGFhYnBHRidGJUYlKiZGJEYlSSdLYWJiYXBHRidGJUYwRiVGJQ==

LCYqJkkidEc2IiIiIkkiTEdGJUYmRiYqKEYkRiZGJ0YmLDIqJiIiI0YmSSRMYXBHRiVGJkYmSSVLYWJwR0YlISIiSSZMYmFhcEdGJUYmSSdLYmJhYXBHRiVGLkkmTGFiYXBHRiVGJkknS2JhYmFwR0YlRi5JJkxiYmFwR0YlRiZJJ0thYmJhcEdGJUYuRiZGJg==

LCYqJkkidEc2IiIiIkkjTGJHRiVGJkYmKihGJEYmSSJMR0YlRiYsKEklTGFhcEdGJUYmSSZMYWJicEdGJUYmSSdLYWFiYnBHRiUhIiJGJkYm

LCgqJkkidEc2IiIiIkkjTGFHRiVGJkYmRiRGJiooRiRGJkkiTEdGJUYmLDBJJUxhYnBHRiVGJkkkTGFwR0YlRiZJJUthYnBHRiUhIiJJJkxiYWJwR0YlRiZJJ0thYmFicEdGJUYuSSZMYWFicEdGJUYmSSdLYmFhYnBHRiVGLkYmRiY=

LCYqJkkidEc2IiIiIkkkTGJhR0YlRiZGJiooSSJMR0YlRiZGJEYmLCYqJiIiI0YmSSZMYmFhcEdGJUYmRiZJJ0tiYmFhcEdGJSEiIkYmRiY=

LCYqJkkidEc2IiIiIkkkTGFiR0YlRiZGJiooSSJMR0YlRiZGJEYmLCYqJiIiI0YmSSZMYWJhcEdGJUYmRiZJJ0tiYWJhcEdGJSEiIkYmRiY=

LCYqJkkidEc2IiIiIkkkTGJiR0YlRiZGJiooSSJMR0YlRiZGJEYmLCYqJiIiI0YmSSZMYmJhcEdGJUYmRiZJJ0thYmJhcEdGJSEiIkYmRiY=

syst:=subs(Kba=Kab,Kbbaa=Kaabb,Kbbaap=Kaabbp,Kbaabp=Kabbap,Kbaba=Kabab,Kbabap=Kababp,Lb=La,Lba=Lab,Lbaa=Labb,Lbaap=Labbp,Lbb=Laa,Lbbp=Laap,Lbba=Laab,Lbbap=Laabp,Lbabp=Labap,[exL-L,exLp-Lp,exLa-La,exLap-Lap,exKab-Kab,exKabp-Kabp,exLabap-Labap,exKbabap-Kbabap,exLbaap-Lbaap,exKbbaap-Kbbaap,exLbbap-Lbbap,exKabbap-Kabbap,exLab-Lab,exLabp-Labp,exLaa-Lbb,exLbbp-Lbbp]);

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

nops(%);indets(syst); nops(%);

IiM7

PDNJJ0thYWJicEc2IkkkS2FiR0YkSSdLYWJhYnBHRiRJJ0thYmJhcEdGJEklS2FicEdGJEkiTEdGJEkjTGFHRiRJJExhYUdGJEkmTGFhYnBHRiRJJUxhYXBHRiRJJExhYkdGJEkmTGFiYXBHRiRJJkxhYmJwR0YkSSVMYWJwR0YkSSRMYXBHRiRJI0xwR0YkSSJ0R0Yk

IiM8

eliminate(syst,indets(syst) minus {t,L}): factor(op(1,op(2,%)));

LD4qKCIiIyIiIilJIkxHNiIiIidGJSlJInRHRigiIiZGJUYlKiYpRidGLEYlRipGJSEiIiooIiIpRiVGLkYlKUYrIiIlRiVGLyooIiIkRiUpRidGM0YlRipGJUYvKigiIztGJUY2RiUpRitGNUYlRiUqKEYzRiUpRidGNUYlRjJGJUYlKihGM0YlRjtGJUY5RiVGLyooIiM6RiVGO0YlKUYrRiRGJUYvKihGJEYlKUYnRiRGJUY5RiVGLyooIiIoRiVGQUYlRj9GJUYlKihGQ0YlRkFGJUYrRiVGJSooRixGJUYnRiVGK0YlRi9GJ0YvRiVGJQ==

algL:=collect(%,L,factor);

LDAiIiJGIyooIiIjRiMpSSJ0RzYiIiImRiMpSSJMR0YoIiInRiNGIyooKUYnIiIlRiMsJkYnRiMiIilGI0YjKUYrRilGIyEiIiooKUYnIiIkRiMsJiomRjZGIylGJ0YlRiNGIyIjO0YzRiMpRitGL0YjRjMqKkY5RiMsJiomRiVGI0YnRiNGI0Y2RiNGIywmRj5GI0YpRjNGIylGK0Y2RiNGIyooRidGIywoKiZGJUYjRjlGI0YjKiYiIihGI0YnRiNGM0ZFRjNGIylGK0YlRiNGMyomLCYqJkYpRiNGJ0YjRjNGI0YzRiNGK0YjRiM=

with(gfun):algeqtoseries(algL,t,L,10,true);

NyMrOUkidEc2IiIiIiIiISIiIyIiIiIjNSIiIyIjbSIiJCIkLyYiIiUiJVtUIiImIiYlek4iIiciJyVRPiQiIigiKHNDI0giIikiKW9ORkYiIiotSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Positive system of Section 4.3 (the p still stands for prime, not for plus) exKab:=Kabp; exKabp:=t+t*(1+L)*(Kabp+Lap); exL:=L*Lp+Lp; exLa:=(1+L)*(Kabp+Lap); exLaa:=(1+L)*Laap; exLab:=L*(Kabp+Labp)+Labp; exLp:=2*t*(1+L)+t*(1+L)*(Lp+2*(Lap+Labbp+Labap+Laabp)); exLap:=t*L+t*(1+L)*(Lap+Labap+Lbaap+Lbbap); exLbbp:=t*Lb+t*(1+L)*(Laap+Labbp); exLabp:=t*La+t*(1+L)*(Labp+Lbabp+Laabp); exLbaap:=t*Lba+t*(1+L)*Lbaap; exLabap:=t*Lab+t*(1+L)*Labap; exLbbap:=t*Lbb+t*(1+L)*Lbbap;

SSVLYWJwRzYi

LCZJInRHNiIiIiIqKEYjRiUsJkkiTEdGJEYlRiVGJUYlLCZJJUthYnBHRiRGJUkkTGFwR0YkRiVGJUYl

LCYqJkkiTEc2IiIiIkkjTHBHRiVGJkYmRidGJg==

KiYsJkkiTEc2IiIiIkYmRiZGJiwmSSVLYWJwR0YlRiZJJExhcEdGJUYmRiY=

KiYsJkkiTEc2IiIiIkYmRiZGJkklTGFhcEdGJUYm

LCYqJkkiTEc2IiIiIiwmSSVLYWJwR0YlRiZJJUxhYnBHRiVGJkYmRiZGKUYm

LCYqKCIiIyIiIkkidEc2IkYlLCZJIkxHRidGJUYlRiVGJUYlKihGJkYlRihGJSwsSSNMcEdGJ0YlKiZGJEYlSSRMYXBHRidGJUYlKiZGJEYlSSZMYWJicEdGJ0YlRiUqJkYkRiVJJkxhYmFwR0YnRiVGJSomRiRGJUkmTGFhYnBHRidGJUYlRiVGJQ==

LCYqJkkidEc2IiIiIkkiTEdGJUYmRiYqKEYkRiYsJkYnRiZGJkYmRiYsKkkkTGFwR0YlRiZJJkxiYWFwR0YlRiZJJkxhYmFwR0YlRiZJJkxiYmFwR0YlRiZGJkYm

LCYqJkkidEc2IiIiIkkjTGJHRiVGJkYmKihGJEYmLCZJIkxHRiVGJkYmRiZGJiwmSSVMYWFwR0YlRiZJJkxhYmJwR0YlRiZGJkYm

LCYqJkkidEc2IiIiIkkjTGFHRiVGJkYmKihGJEYmLCZJIkxHRiVGJkYmRiZGJiwoSSVMYWJwR0YlRiZJJkxiYWJwR0YlRiZJJkxhYWJwR0YlRiZGJkYm

LCYqJkkidEc2IiIiIkkkTGJhR0YlRiZGJiooRiRGJiwmSSJMR0YlRiZGJkYmRiZJJkxiYWFwR0YlRiZGJg==

LCYqJkkidEc2IiIiIkkkTGFiR0YlRiZGJiooRiRGJiwmSSJMR0YlRiZGJkYmRiZJJkxhYmFwR0YlRiZGJg==

LCYqJkkidEc2IiIiIkkkTGJiR0YlRiZGJiooRiRGJiwmSSJMR0YlRiZGJkYmRiZJJkxiYmFwR0YlRiZGJg==

syst:=subs(Kba=Kab,Kbbaa=Kaabb,Kbbaap=Kaabbp,Kbaabp=Kabbap,Kbaba=Kabab,Kbabap=Kababp,Lb=La,Lba=Lab,Lbaa=Labb,Lbaap=Labbp,Lbb=Laa,Lbbp=Laap,Lbba=Laab,Lbbap=Laabp,Lbabp=Labap,[exL-L,exLp-Lp,exLa-La,exLap-Lap,exKab-Kab,exKabp-Kabp,exLabap-Labap,exLbaap-Lbaap,exLbbap-Lbbap,exLab-Lab,exLabp-Labp,exLaa-Lbb,exLbbp-Lbbp]);

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

nops(syst);

IiM4

indets(syst) minus {t}; nops(%);

PC9JJEthYkc2IkklS2FicEdGJEkiTEdGJEkjTGFHRiRJJExhYUdGJEkmTGFhYnBHRiRJJUxhYXBHRiRJJExhYkdGJEkmTGFiYXBHRiRJJkxhYmJwR0YkSSVMYWJwR0YkSSRMYXBHRiRJI0xwR0Yk

IiM4

eliminate(syst,indets(syst) minus {t,L}): factor(op(1,op(2,%)));

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

algLbis:=collect(subs(L=L-1,%),L,factor);

LDAiIiIhIiIqKCIiI0YjKUkidEc2IiIiJkYjKUkiTEdGKSIiJ0YjRiQqKClGKCIiJUYjLCZGKEYjIiIpRiNGIylGLEYqRiNGIyooKUYoIiIkRiMsJiomRjZGIylGKEYmRiNGIyIjO0YkRiMpRixGMEYjRiMqKkY5RiMsJiomRiZGI0YoRiNGI0Y2RiNGIywmRj5GI0YqRiRGIylGLEY2RiNGJCooRihGIywoKiZGJkYjRjlGI0YjKiYiIihGI0YoRiNGJEZFRiRGIylGLEYmRiNGIyomLCYqJkYqRiNGKEYjRiNGI0YjRiNGLEYjRiM=

factor(algLbis/algL);

ISIi

Asymptotics disc:=factor(discrim(algL,L));

LCQqKClJInRHNiIiIz8iIiIsQiomIiYrdyZGKClGJSIjOkYoRigqJiInX15bRigpRiUiIzlGKEYoKiYiJ0MpXCpGKClGJSIjOEYoRigqJiIoJjNHTkYoKUYlIiM3RighIiIqJiIpJTQpSD5GKClGJSIjNkYoRjoqJiIpYC8tR0YoKUYlIiM1RihGOiomIig9IjR6RigpRiUiIipGKEYoKiYiKThgYl1GKClGJSIiKUYoRigqJiIoPW1tJ0YoKUYlIiIoRihGKComIil3cDxcRigpRiUiIidGKEY6KiYiKDMoelpGKClGJSIiJkYoRigqJiIpUGhhRkYoKUYlIiIlRihGKComIilfJzMoPUYoKUYlIiIkRihGOiomIihXIWZfRigpRiUiIiNGKEYoKiYiJ1tDbkYoRiVGKEY6IiZnKUhGKEYoLCgqJEZbb0YoRigqJkZcb0YoRiVGKEYoRihGOkYoRjo=

{fsolve(%,t=0..1)};

PCUkIiIhRiQkIitKKGYzNiohIzYkIitDYzhVVCEjNQ==

1/op(2,%);

JCIrZjdmKDQiISIp

read('syst_inf_prime_3'); Elimination to get an equation for L eliminate(syst,indets(syst) minus {t,L}): factor(op(1,op(2,%)));

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

algL:=collect(%,L,factor);

LEwiIiUiIiIqKiIiI0YkKUkidEc2IiIjPkYkLCZGKEYkIiIpISIiRiQpSSJMR0YpIiM/RiRGLSoqKUYoIiM9RiQsJkYoRiQiIzdGJEYkLCZGKEYkRipGLUYkKUYvRipGJEYkKigpRigiIzxGJCwqKiQpRigiIiRGJEYkKiYiI0ZGJClGKEYmRiRGJComIiQ3I0YkRihGJEYkIiVeO0YkRiQpRi9GM0YkRiQqKClGKCIjO0YkLCwqJiIjOEYkKUYoRiNGJEYkKiYiI2pGJEY9RiRGLSomIiR6I0YkRkFGJEYkKiYiJXk5RiRGKEYkRiQiJTIhKUYkRiQpRi9GOkYkRi0qKClGKCIjOkYkLC4qJkZXRiQpRigiIiZGJEYkKiYiIyQqRiRGTEYkRi0qJiIkbilGJEY9RiRGJComIiUvOEYkRkFGJEYtKiYiJUlrRiRGKEYkRi0iJiwoR0YtRiQpRi9GSEYkRi0qKClGKCIjOUYkLDAqJkYjRiQpRigiIidGJEYkKiYiIyMqRiRGWkYkRi0qJiIkZCdGJEZMRiRGJComIiVbYEYkRj1GJEYtKiYiJSI+JUYkRkFGJEYkKiYiJm0vI0YkRihGJEYkIiZNKHpGJEYkKUYvRldGJEYtKigpRihGS0YkLDAqJkY1RiRGZW9GJEYkKiYiJDwlRiRGWkYkRi0qJiIlREdGJEZMRiRGJComIiYqUj9GJEY9RiRGLSomIiY9LSJGJEZBRiRGJComIiZULiZGJEYoRiRGJCInLmo8RiRGJClGL0Zib0YkRiQqKkYmRiQpRihGNUYkLDAqJkZmb0YkRmVvRiRGJComIiRgJUYkRlpGJEYtKiYiJUJSRiRGTEYkRiQqJiImLHQjRiRGPUYkRi0qJiIlJj4qRiRGQUYkRiQqJiImViZcRiRGKEYkRiQiJzh3OkYkRiQpRi9GS0YkRi0qKClGKCIjNkYkLDBGZG9GJComIiVyNkYkRlpGJEYtKiYiJiJHOkYkRkxGJEYkKiYiJyEpKjMiRiRGPUYkRi0qJiImKVtCRiRGQUYkRiQqJiInKkhnIkYkRihGJEYkIidsK1lGJEYkKUYvRjVGJEYkKigpRigiIzVGJCwuKiYiJGcqRiRGWkYkRiQqJiImJVtARiRGTEYkRi0qJiInR3Y7RiRGPUYkRiQqJiImbCo9RiRGQUYkRi0qJiInWiI9I0YkRihGJEYtIidtMWJGLUYkKUYvRmVyRiRGJCooKUYoIiIqRiQsLiomIiQoXEYkRlpGJEYkKiYiJlRBI0YkRkxGJEYtKiYiJ1Y6P0YkRj1GJEYkKiYiJTBYRiRGQUYkRi0qJiInLktERiRGKEYkRi0iJ201YUYtRiQpRi9GZXNGJEYtKigpRihGLEYkLC4qJiIkXSJGJEZaRiRGJComIiY4ciJGJEZMRiRGLSomIid2MT5GJEY9RiRGJComIiZmNyJGJEZBRiRGJComIidXN0RGJEYoRiRGLSInPGBWRi1GJClGL0ZldEYkRiQqKClGKCIiKEYkLC4qJkYwRiRGWkYkRiQqJiIlXSgqRiRGTEYkRi0qJiInQjk5RiRGPUYkRiQqJiImOyc9RiRGQUYkRiQqJiInWjhARiRGKEYkRi0iJytdR0YtRiQpRi9GLEYkRi0qKEZlb0YkLCwqJiIlQFNGJEZMRiRGJComIiYjUiIpRiRGPUYkRi0qJiImYmYiRiRGQUYkRi0qJiInSyRbIkYkRihGJEYkIidJLTpGJEYkKUYvRmR2RiRGLSooRlpGJCwsKiYiJVE2RiRGTEYkRiQqJiImbmMkRiRGPUYkRi0qJiIlQyEqRiRGQUYkRi0qJiImITQmKUYkRihGJEYkIiZDRidGJEYkKUYvRmZvRiRGJCoqRiZGJEZMRiQsLComIiMqKkYkRkxGJEYkKiYiJWlkRiRGPUYkRi0qJiIlWjxGJEZBRiRGLSomIiZxJT5GJEYoRiRGJCImPSwiRiRGJClGL0ZlbkYkRi0qKkYmRiRGPUYkLCwqJkYsRiRGTEYkRiQqJiIlKEgiRiRGPUYkRi0qJiIkXSVGJEZBRiRGLSomIiUrcEYkRihGJEYkIiVGQ0YkRiQpRi9GI0YkRiQqKkYjRiRGQUYkLCoqJiIjIipGJEY9RiRGJComIiNORiRGQUYkRiQqJiIkNCpGJEYoRiRGLSIkLiNGLUYkKUYvRj5GJEYkKipGJkYkRihGJCwqKiZGNUYkRj1GJEYkKiZGZW5GJEZBRiRGJComIiRNJEYkRihGJEYtIiNVRi1GJClGL0YmRiRGLSomLCYqJiIjd0YkRihGJEYtRiNGLUYkRi9GJEYk

with(gfun):algeqtoseries(algL,t,L,10,true);

NyMrOUkidEc2IiIiIiIiISIiIyIiIiIjNSIiIyIjbSIiJCIkLyYiIiUiJTtVIiImIiZzciQiIiciJzElUiQiIigiKCshekoiIikiKSUqcE9JIiIqLUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNQ==

Asymptotics disc:=factor(discrim(algL,L)): nops(disc);

IiIl

seq(indets(op(i,disc)),i=1..nops(disc)); seq(nops(op(i,disc)),i=1..nops(disc));

NiY8IjwjSSJ0RzYiRiRGJA==

NiYiIiIiJGsiIiIjRiU=

d2:=op(2,disc): d3:=op(3,disc); d4:=op(4,disc);

KiQpLCZJInRHNiIiIiJGJyEiIiIiJUYn

KiQpSSJ0RzYiIiRVJCIiIg==

{fsolve(d2,t=1/13..1/8)};

PCMkIiskNCJmMCopISM2

1/op(%);

JCIrIzQhKkc3IiEiKQ==

We write a positive version of the system and use the "combstruct" and "NewtonGF" packages to compute coefficients.The reader should download this package, and the following line should be modified accordingly. libname:="/usr/labri/bousquet/algolib/NewtonGF1.3",libname;

NiZRSC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9OZXd0b25HRjEuMzYiUUQvdXNyL2xhYnJpL2JvdXNxdWV0L2FsZ29saWIvYWxnb2xpYkYkUTovdXNyL2xvY2FsL21hcGxlL2xpYi9nZnVuRiRRNS91c3IvbG9jYWwvbWFwbGUvbGliRiQ=

with(combstruct):with(NewtonGF): read('syst_inf_prime_positive_4'): ser:=1+subs(GFSeries(eulerian_prim,unlabelled,z,30),L);

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

with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],z,29): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,z=1/13..1/8);

JCIrU2s+ZigpISM2

1/%;

JCIrT3NsVDYhIik=

ser:=1+subs(GFSeries(eulerian_prim,unlabelled,z,50),L): We check how the order of the expansion affects the estimate of the radius for i from 1 to 5 do res:=hermite_pade( [1,ser,ser^2],z,10*i-1):dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)):fsolve(dis,z=1/13..1/8);print(map(Z->1/Z,[%]));od:

NyMkIitAVSwnPiIhIik=

NyMkIisncGY7OSIhIik=

NyMkIitPc2xUNiEiKQ==

NyMkIitSc2xUNiEiKQ==

NyMkIitSc2xUNiEiKQ==

We write a positive version of the system and use the combstruct and NewtonGF packages to compute coefficients. Same remarks as above regarding the package. libname:="/usr/labri/bousquet/algolib/NewtonGF1.3",libname;

NidRSC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9OZXd0b25HRjEuMzYiRiNRRC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9hbGdvbGliRiRROi91c3IvbG9jYWwvbWFwbGUvbGliL2dmdW5GJFE1L3Vzci9sb2NhbC9tYXBsZS9saWJGJA==

with(combstruct):with(NewtonGF): read('syst_inf_prime_positive_5'): ser:=1+subs(GFSeries(eulerian_prim,unlabelled,z,50),L);

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

with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],z,49): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,z=1/13..1/8);

JCIrUixiWycpISM2

1/%;

JCIrcUpFYzYhIik=

We check how the order of the expansion affects the estimate of the radius for i from 1 to 5 do res:=hermite_pade( [1,ser,ser^2],z,10*i-1):dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)):fsolve(dis,z=1/13..1/8);print(map(Z->1/Z,[%]));od:

NyMkIisnPmU4PSIhIik=

NyMkIisxPEhjNiEiKQ==

NyUkIitgPkVjNiEiKSQiKzImKjQyNkYlJCIrTXE0MjZGJQ==

NyMkIitxSkVjNiEiKQ==

NyMkIitxSkVjNiEiKQ==

We write a positive version of the system and use the combstruct and NewtonGF packages to compute coefficients.The reader should download thsi package as well and add it to libname: the following line in NOT intended to work for anyone else than mbm! libname:="/usr/labri/bousquet/algolib/NewtonGF1.3",libname;

NihRSC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9OZXd0b25HRjEuMzYiRiNGI1FEL3Vzci9sYWJyaS9ib3VzcXVldC9hbGdvbGliL2FsZ29saWJGJFE6L3Vzci9sb2NhbC9tYXBsZS9saWIvZ2Z1bkYkUTUvdXNyL2xvY2FsL21hcGxlL2xpYkYk

with(combstruct):with(NewtonGF): read('syst_inf_prime_positive_6'): ser:=1+subs(GFSeries(eulerian_prim,unlabelled,z,50),L);

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

with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],z,49): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,z=1/13..1/8);

JCIrbUFxaCYpISM2

1/%;

JCIrSD8qejsiISIp

We check how the order of the expansion affects the estimate of the radius for i from 1 to 5 do res:=hermite_pade( [1,ser,ser^2],z,10*i-1):dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)):fsolve(dis,z=1/13..1/8);print(map(Z->1/Z,[%]));od:

NyMkIisiKmY0J0ciISIp

NyMkIit2KEhIPCIhIik=

NyMkIisqKT0qejsiISIp

NyMkIitHPyp6OyIhIik=

NyMkIitIPyp6OyIhIik=

System of Section 5.2 eqU:=-U(x)+1+2*t*x*U(x)^2+2*t*x*U0(1)+t*x/(x-1)*(U(x)-x*U(1))+t*x; eqU0:=-U0(x)+2*t*x*U0(x)*U(x)+t*x*U(x)+t*x*U0(1)+t*x/(x-1)*(U0(x)-x*U0(1));

LC4tSSJVRzYiNiNJInhHRiUhIiIiIiJGKSoqIiIjRilJInRHRiVGKUYnRikpRiNGK0YpRikqKkYrRilGLEYpRidGKS1JI1UwR0YlNiNGKUYpRikqKkYsRilGJ0YpLCZGJ0YpRilGKEYoLCZGI0YpKiZGJ0YpLUYkRjFGKUYoRilGKSomRixGKUYnRilGKQ==

LCwtSSNVMEc2IjYjSSJ4R0YlISIiKiwiIiMiIiJJInRHRiVGK0YnRitGI0YrLUkiVUdGJUYmRitGKyooRixGK0YnRitGLUYrRisqKEYsRitGJ0YrLUYkNiNGK0YrRisqKkYsRitGJ0YrLCZGJ0YrRitGKEYoLCZGI0YrKiZGJ0YrRjFGK0YoRitGKw==

Checking the forward equation factor(subs(U0(x)=(U(x)-1)/2,U0(1)=(U(1)-1)/2,eqU-2*eqU0));

IiIh

Solution of the system (Section 5.4.1) An equation for U. eq:=subs(U0(x)=(U(x)-1)/2,U0(1)=(U(1)-1)/2,U(x)=U,U(1)=U1,eqU);

LC5JIlVHNiIhIiIiIiJGJioqIiIjRiZJInRHRiRGJkkieEdGJEYmKUYjRihGJkYmKipGKEYmRilGJkYqRiYsJiomI0YmRihGJkkjVTFHRiRGJkYmRi9GJUYmRiYqKkYpRiZGKkYmLCZGKkYmRiZGJUYlLCYqJkYwRiZGKkYmRiVGI0YmRiZGJiomRilGJkYqRiZGJg==

The quadratic method numer(factor(discrim(eq,U)));

LDQqKiIiKSIiIkkjVTFHNiJGJSlJInRHRiciIiNGJSlJInhHRiciIiRGJUYlKipGJEYlRiZGJUYoRiUpRixGKkYlISIiKiZGKEYlRi9GJUYlKihGJEYlRilGJUYrRiVGMCooIiM5RiVGKUYlRi9GJUYlKigiIidGJUYpRiVGLEYlRjAqJEYvRiVGJSomRipGJUYsRiVGMEYlRiU=

factor(discrim(%,x));

LCQqKCIjSyIiIiw6KigiI2tGJSlJI1UxRzYiIiIkRiUpSSJ0R0YrRixGJUYlKigiI1tGJSlGKiIiI0YlRi1GJUYlKigiI3NGJUYxRiUpRi5GMkYlISIiKigiIzpGJUYqRiVGLUYlRjYqKEYyRiVGMUYlRi5GJUYlKigiIipGJUYqRiVGNUYlRiUqJEYtRiVGJSooIiM+RiVGKkYlRi5GJUYlKiYiI0ZGJUY1RiVGJUYqRjYqJkY+RiVGLkYlRjZGJUYlRiVGLUYlRiU=

the algebraic equation satisfied by U1 algU:=collect(op(2,%),U1,factor);

LDAqKCIjayIiIilJI1UxRzYiIiIkRiUpSSJ0R0YoRilGJUYlKioiIiNGJUYrRiUsKComIiNDRiUpRitGLUYlRiUqJiIjT0YlRitGJSEiIkYlRiVGJSlGJ0YtRiVGJSomLCoqJiIjOkYlRipGJUY0KiYiIipGJUYxRiVGJSomIiM+RiVGK0YlRiVGJUY0RiVGJ0YlRiUqJEYqRiVGJSomIiNGRiVGMUYlRiVGPEY0RiVGJQ==

with(gfun): algeqtoseries(algU,t,U1,10,true);

NyMrOUkidEc2IiIiIiIiISIiIyIiIiIjNSIiIyIjbSIiJCIkMSYiIiUiJW1VIiImIiY9JVEiIiciJyU+aiQiIigiKCcpUmMkIiIpIik1PS1PIiIqLUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNQ==

Asymptotics factor(discrim(algU,U1));

LCQqKCIiJSIiIilJInRHNiIiIiNGJSksKiomIiQ7I0YlKUYnIiIkRiVGJSomIiMiKUYlRiZGJSEiIiomIiM9RiVGJ0YlRiVGJUYyRi9GJUYy

The equation satisfied by the radius disc:=op(1,op(3,%));

LCoqJiIkOyMiIiIpSSJ0RzYiIiIkRiVGJSomIiMiKUYlKUYnIiIjRiUhIiIqJiIjPUYlRidGJUYlRiVGLg==

plot(disc,t=0..0.2);

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rhoest:=fsolve(disc); 1/%;

JCIrRnldYHchIzY=

JCIrPi9mMTghIik=

Nature of this singularity ? with(plots): implicitplot(algU,t=0..0.1,U1=1..2,numpoints=10000);

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The other branches at t=0 algeqtoseries(algU,t,U1,3);

NyUrK0kidEc2IiMhIiIiI0shIiMjIiImIiIpRicjISImIiIlIiIhLUkiT0clKnByb3RlY3RlZEc2IyIiIkY1KytGJCNGNSIiI0YnI0YnRjhGMEY5RjVGMUY4KytGJEY1RjBGOEY1IiM1RjhGMSIiJA==

implicitplot(algU,t=0..0.1,U1=-6..12,numpoints=10000);

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Two branches meet at the singular point. Let us look at them solve(disc);

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rhoex:=op(1,[%]);evalf(%);

LCgqJiMiIiIiI0NGJSksJiIjOEYlKiYiIztGJSkiIiMjRiVGLUYlRiUjRiUiIiRGJSEiIiomIyIiKEYmRiVGJ0YxRiUjRiUiIilGJQ==

JCIrQ3ldYHchIzY=

Expansion of the branches near rho: the first solution is the holomorphic one devens:=algeqtoseries(subs(t=rhoex*(1-epsilon),algU),epsilon,U1,2);

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devens:=algeqtoseries(subs(t=rhoex*(1-epsilon),algU),epsilon,U1,3): nops(devens);

IiIj

First solution: the regular branch, close to the value 6 coeff(op(1,devens),epsilon,0);evalf(%);

LCwqKCMiIiQiI2MiIiIpLCYiIzhGJyomIiM7RicpIiIjI0YnRi5GJ0YnI0YuRiVGJ0YtRichIiIqKCMiI0ZGJkYnKUYpI0YnRiVGJ0YtRidGJyomIyIjZCIkQyNGJ0YoRidGJyomIyIjXkY6RidGNUYnRjEjIiNURixGJw==

JCIrUTIlPSpmISIq

the two other branches meet at t=rho, and we have indeed a map-lie singularity with an exponent 3/2 op(2,devens);

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

The full system of Section 5.2.2. The letters a and b stand for 0 and 1. For instance, Ua is the bivariate series U0(t;x), and Ua1 is its specialization at x=1. eqTba:=t+t*Ua1: eqU:=1+2*Ua: eqUa:=2*t*x*U*Ua+t*x*U+t*x*Ua1+t*x^2*(Ubaa1+Uaba1+Ubba1)+t*x/(x-1)*(Ua-x^2*Ua1)+t*x^2*Tba: #eqUba:=x*Tba+Ubba+Uaba: eqUba:=x*Tba+2*t*x*U*Uba+t*x*Ua+t*x^2*(Uaba1+Ubba1)+t*x/(x-1)*(Uba-x^2*Uba1)+t*x^2*Tba; eqUaa:=2*t*x*U*Uaa+t*x*Ua+t*x^2*Ubaa1+t*x/(x-1)*(Uba-x^2*Uba1): eqUbaa:=2*t*x*U*Ubaa+t*x*Uba+t*x^2*Ubaa1+ t*x/(x-1)*(Ubaa-x^2*Ubaa1): eqUaba:=2*t*x*U*Uaba+t*x*(Uba+x*U*Tba)+t*x^2*Uaba1+ t*x/(x-1)*(Uaba-x^2*Uaba1): eqUbba:=2*t*x*U*Ubba+t*x*(Ubb+x*U*Tba)+t*x^2*Ubba1+ t*x/(x-1)*(Ubba-x^2*Ubba1):

LC4qJkkieEc2IiIiIkkkVGJhR0YlRiZGJiosIiIjRiZJInRHRiVGJkYkRiZJIlVHRiVGJkkkVWJhR0YlRiZGJiooRipGJkYkRiZJI1VhR0YlRiZGJiooRipGJilGJEYpRiYsJkkmVWFiYTFHRiVGJkkmVWJiYTFHRiVGJkYmRiYqKkYqRiZGJEYmLCZGJEYmRiYhIiJGNiwmKiZJJVViYTFHRiVGJkYwRiZGNkYsRiZGJkYmKihGKkYmRjBGJkYnRiZGJg==

Expressions of our four main series (Section 5.4.2). exU:=1+2*Ua; exUbis:=1+4*t*x*U*Ua+2*t*x*U+2*t*x*Ua1+2*t*x^2*(Uba1+Ubaa1)+2*t*x/(x-1)*(Ua-x^2*Ua1); exUba:=t*x*(1+Ua1)+2*t*x*U*Uba+t*x*Ua+t*x^2*Uba1+t*x/(x-1)*(Uba-x^2*Uba1); exUbaa:=2*t*x*U*Ubaa+t*x*Uba+t*x^2*Ubaa1+t*x/(x-1)*(Ubaa-x^2*Ubaa1);

LCYiIiJGIyomIiIjRiNJI1VhRzYiRiNGIw==

LC4iIiJGIyosIiIlRiNJInRHNiJGI0kieEdGJ0YjSSJVR0YnRiNJI1VhR0YnRiNGIyoqIiIjRiNGJkYjRihGI0YpRiNGIyoqRixGI0YmRiNGKEYjSSRVYTFHRidGI0YjKipGLEYjRiZGIylGKEYsRiMsJkklVWJhMUdGJ0YjSSZVYmFhMUdGJ0YjRiNGIyosRixGI0YmRiNGKEYjLCZGKEYjRiMhIiJGNiwmKiZGLkYjRjBGI0Y2RipGI0YjRiM=

LCwqKEkidEc2IiIiIkkieEdGJUYmLCZGJkYmSSRVYTFHRiVGJkYmRiYqLCIiI0YmRiRGJkYnRiZJIlVHRiVGJkkkVWJhR0YlRiZGJiooRiRGJkYnRiZJI1VhR0YlRiZGJiooRiRGJilGJ0YrRiZJJVViYTFHRiVGJkYmKipGJEYmRidGJiwmRidGJkYmISIiRjUsJiomRjJGJkYxRiZGNUYtRiZGJkYm

LCoqLCIiIyIiIkkidEc2IkYlSSJ4R0YnRiVJIlVHRidGJUklVWJhYUdGJ0YlRiUqKEYmRiVGKEYlSSRVYmFHRidGJUYlKihGJkYlKUYoRiRGJUkmVWJhYTFHRidGJUYlKipGJkYlRihGJSwmRihGJUYlISIiRjIsJiomRi9GJUYuRiVGMkYqRiVGJUYl

let us check that the two systems are consistent factor(exUbis-1-2*eqUa);

LCQqKiIiIyIiIkkidEc2IkYlKUkieEdGJ0YkRiUsKkklVWJhMUdGJyEiIkkmVWFiYTFHRidGJUkmVWJiYTFHRidGJUkkVGJhR0YnRiVGJUYs

We use a forward equation factor(subs(Uba1=Tba+Uaba1+Ubba1,%));

IiIh

factor(exUba-eqUba);

LCQqJkkieEc2IiIiIiwwKihJJFRiYUdGJUYmSSJ0R0YlRiZGJEYmRiYqKEkmVWFiYTFHRiVGJkYqRiZGJEYmRiYqKEklVWJhMUdGJUYmRipGJkYkRiYhIiIqKEkmVWJiYTFHRiVGJkYqRiZGJEYmRiYqJkkkVWExR0YlRiZGKkYmRi9GKUYmRipGL0YmRi8=

factor(subs(Uba1=Tba+Uaba1+Ubba1,%));

LCQqJkkieEc2IiIiIiwoKiZJJFVhMUdGJUYmSSJ0R0YlRiYhIiJJJFRiYUdGJUYmRipGK0YmRis=

factor(subs(Tba=eqTba,%));

IiIh

factor(exUbaa-eqUbaa);

IiIh

We now get rid of Ua, and obtain a system with three equations and three unknown series (and their specializations at x=1) syst:=normal(subs(Ua=(U-1)/2, Ua1=(U1-1)/2,[exUbis-U,exUba-Uba,exUbaa-Ubaa]));

NyUqJixCKioiIiMiIiIpSSJVRzYiRiZGJ0kidEdGKkYnKUkieEdGKkYmRidGJyooSSNVMUdGKkYnRitGJylGLSIiJEYnISIiKipGJkYnSSVVYmExR0YqRidGK0YnRjBGJ0YnKipGJkYnSSZVYmFhMUdGKkYnRitGJ0YwRidGJyoqRiZGJ0YoRidGK0YnRi1GJ0YyKihGL0YnRitGJ0YsRidGJyoqRiZGJ0Y0RidGK0YnRixGJ0YyKipGJkYnRjZGJ0YrRidGLEYnRjIqJkYrRidGMEYnRicqKEYpRidGK0YnRi1GJ0YnKihGL0YnRitGJ0YtRidGMiomRitGJ0YsRidGMiomRilGJ0YtRidGMkYpRidGLUYnRidGMkYnLCZGLUYnRidGMkYyLCQqKCNGJ0YmRicsNiosIiIlRidGKUYnSSRVYmFHRipGJ0YrRidGLEYnRicqLEZGRidGKUYnRkdGJ0YrRidGLUYnRjIqKEYpRidGK0YnRixGJ0YnRjhGJ0Y5RjJGPEYyRj1GMioqRiZGJ0ZHRidGK0YnRi1GJ0YnKihGJkYnRkdGJ0YtRidGMiomRiZGJ0ZHRidGJ0YnRkBGMkYnKiYsMiosRiZGJ0YpRidJJVViYWFHRipGJ0YrRidGLEYnRicqLEYmRidGKUYnRlBGJ0YrRidGLUYnRjIqKEZHRidGK0YnRixGJ0YnKihGNkYnRitGJ0YsRidGMiooRkdGJ0YrRidGLUYnRjIqKEZQRidGK0YnRi1GJ0YnKiZGUEYnRi1GJ0YyRlBGJ0YnRkBGMg==

indets(syst) minus {t,x};

PChJIlVHNiJJI1UxR0YkSSRVYmFHRiRJJVViYTFHRiRJJVViYWFHRiRJJlViYWExR0Yk

factor(subs(t=0,syst));

NyUsJkkiVUc2IiEiIiIiIkYnLCRJJFViYUdGJUYmLCRJJVViYWFHRiVGJg==

We can compute the first coefficients of all these series substitutions:=proc(n): {U=User(n-1), U1=subs(x=1,User(n-1)), Uba1=subs(x=1,Ubaser(n-1)), Uba=Ubaser(n-1), Ubaa=Ubaaser(n-1),Ubaa1=subs(x=1,Ubaaser(n-1))};end: User:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(substitutions(n),syst[1]+U),t,n+1)),polynom): fi:end: Ubaser:=proc(n) option remember: if n=0 then 0 else convert(normal(series(subs(substitutions(n),syst[2]+Uba),t,n+1)),polynom): fi:end: Ubaaser:=proc(n) option remember: if n=0 then 0 else convert(normal(series(subs(substitutions(n),syst[3]+Ubaa),t,n+1)),polynom): fi:end: User(1);

LCYqKCIiIyIiIkkidEc2IkYlSSJ4R0YnRiVGJUYlRiU=

User(8);

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

subs(x=1,%);

LDQqJiIoKXAqXCQiIiIpSSJ0RzYiIiIpRiVGJSomIidXeE5GJSlGJyIiKEYlRiUqJiImcXokRiUpRiciIidGJUYlKiYiJUtVRiUpRiciIiZGJUYlKiYiJC8mRiUpRiciIiVGJUYlKiYiI21GJSlGJyIiJEYlRiUqJiIjNUYlKUYnIiIjRiVGJSomRkFGJUYnRiVGJUYlRiU=

User(12);

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

subs(x=1,%);

LDwqJiIsIW9RbjZUIiIiKUkidEc2IiIjN0YlRiUqJiIrcyVScyRRRiUpRiciIzZGJUYlKiYiKjlFQWskRiUpRiciIzVGJUYlKiYiKXVERk5GJSlGJyIiKkYlRiUqJiIoKXAqXCRGJSlGJyIiKUYlRiUqJiInV3hORiUpRiciIihGJUYlKiYiJnF6JEYlKUYnIiInRiVGJSomIiVLVUYlKUYnIiImRiVGJSomIiQvJkYlKUYnIiIlRiVGJSomIiNtRiUpRiciIiRGJUYlKiZGMUYlKUYnIiIjRiVGJSomRlBGJUYnRiVGJUYlRiU=

ser:=subs(x=1,User(60)): with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],t,60): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,t=0.07..0.09);

JCIrJio0YmV3ISM2

1/%;

JCIrRit0MDghIik=

JSFH

We now attack the solution of the system eqs:=map(numer,syst);

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

indets(eqs);nops(eqs);

PCpJIlVHNiJJI1UxR0YkSSRVYmFHRiRJJVViYTFHRiRJJVViYWFHRiRJJlViYWExR0YkSSJ0R0YkSSJ4R0Yk

IiIk

The first equation only depends on U(x) and on series that do not depend on x indets(eqs[1]);degree(eqs[1],U);

PChJIlVHNiJJI1UxR0YkSSVVYmExR0YkSSZVYmFhMUdGJEkidEdGJEkieEdGJA==

IiIj

The quadratic method factor(discrim(eqs[1],U));

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

The discriminant with respect to x factors, and we must choose the correct factor res:=factor(discrim(%,x)): nops(res);

IiIl

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NiYiIiIiIiUiJC8jIiIj

NiY8IjwlSSNVMUc2IkklVWJhMUdGJkkmVWJhYTFHRiY8JkYlRidGKEkidEdGJjwjRio=

We need to choose the factor that actually vanishes n:=10: normal(series(subs(Ubaa1=subs(x=1,Ubaaser(n)), U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)), op(2,res)),t,n+1));

KzNJInRHNiIiIiMiIiQiI0EiIiUiJEEjIiImIiVXQSIiJyImV0kjIiIoIic5M0MiIikiKD8hZUQiIioiKTFLZEYiIzU=

op(3,res) is a very big thing which we do not want to see n:=20: normal(series(subs(Ubaa1=subs(x=1,Ubaaser(n)), U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)), op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

Alas, this big thing is the correct identity, our first identity between the three uknown series, specialized at 1 alg1:=op(3,res):nops(alg1);

IiQvIw==

In the second equation, there is one more series that depends on x, namely Uba indets(eqs[2]);

PChJIlVHNiJJI1UxR0YkSSRVYmFHRiRJJVViYTFHRiRJInRHRiRJInhHRiQ=

collect(eqs[2],[Uba,Ubaa],factor);

LCYqJiwsKioiIiUiIiJJIlVHNiJGJ0kidEdGKUYnKUkieEdGKSIiI0YnRicqKkYmRidGKEYnRipGJ0YsRichIiIqKEYtRidGKkYnRixGJ0YnKiZGLUYnRixGJ0YvRi1GJ0YnSSRVYmFHRilGJ0YnKihGKkYnRixGJywsKiZGKEYnRixGJ0YnKiZJI1UxR0YpRidGLEYnRicqKEYtRidJJVViYTFHRilGJ0YsRidGL0YoRi9GN0YvRidGJw==

The coefficient of the linear term (also called "kernel") ker:=factor(subs(x=X,U=UX,coeff(%,Uba)));

LCwqKiIiJSIiIkkjVVhHNiJGJSlJIlhHRiciIiNGJUkidEdGJ0YlRiUqKkYkRiVGJkYlRilGJUYrRiUhIiIqKEYqRiVGKUYlRitGJUYlKiZGKkYlRilGJUYtRipGJQ==

collect(factor(ker/2),UX,factor);

LCoqLCIiIyIiIkkiWEc2IkYlSSJ0R0YnRiUsJkYmRiVGJSEiIkYlSSNVWEdGJ0YlRiUqJkYmRiVGKEYlRiVGJkYqRiVGJQ==

We apply to this one the kernel method, by cancelling the coeff of Uba. This gives a second equation between X and U(X) RHS:=subs(Uba=0,x=X,U=UX,eqs[2]);

LCwqKEkjVTFHNiIiIiIpSSJYR0YlIiIjRiZJInRHRiVGJkYmKihJI1VYR0YlRiZGJ0YmRipGJkYmKipGKUYmSSVVYmExR0YlRiZGJ0YmRipGJiEiIiooRiRGJkYoRiZGKkYmRi8qKEYsRiZGKEYmRipGJkYv

An equation for X algX:=subs(x=X,op(5,factor(resultant(RHS,ker,UX))));

LC4qKiIiIyIiIkkjVTFHNiJGJSlJIlhHRidGJEYlSSJ0R0YnRiVGJSoqIiIlRiVJJVViYTFHRidGJUYoRiVGKkYlISIiKipGJEYlRiZGJUYpRiVGKkYlRi4qJkYpRiVGKkYlRi5GKUYlRiVGLg==

collect(algX,X,factor);

LCgiIiIhIiIqKiIiI0YjSSJ0RzYiRiMsJkkjVTFHRihGIyomRiZGI0klVWJhMUdGKEYjRiRGIylJIlhHRihGJkYjRiMqJiwoKihGJkYjRipGI0YnRiNGJEYnRiRGI0YjRiNGLkYjRiM=

UXsol:=factor(solve(ker,UX));UXsol:=factor(solve(RHS,UX));

LCQqLCMiIiIiIiNGJSwoKiZJIlhHNiJGJUkidEdGKkYlRiVGKSEiIkYlRiVGJUYpRixGK0YsLCZGKUYlRiVGLEYsRiw=

LCQqJiwoKiZJI1UxRzYiIiIiSSJYR0YnRihGKCooIiIjRihJJVViYTFHRidGKEYpRighIiJGJkYtRigsJkYpRihGKEYtRi1GLQ==

factor(%+U1);

LCQqKiIiIyIiIkkiWEc2IkYlSSVVYmExR0YnRiUsJkYmRiVGJSEiIkYqRiU=

We replace U(X) by this value in the first equation... alg2b:=(numer(factor(subs(U=UXsol,x=X,eqs[1]))));

LFgqKiIiIyIiIilJI1UxRzYiRiRGJSlJIlhHRigiIiRGJUkidEdGKEYlRiUqLCIiKUYlRidGJUklVWJhMUdGKEYlRilGJUYsRiUhIiIqKEYnRiUpRioiIiVGJUYsRiVGMCoqRi5GJSlGL0YkRiVGKUYlRixGJUYlKipGJEYlRi9GJUYyRiVGLEYlRiUqKkYkRiVJJlViYWExR0YoRiVGMkYlRixGJUYlKipGM0YlRiZGJSlGKkYkRiVGLEYlRjAqLEYuRiVGJ0YlRi9GJUY6RiVGLEYlRiUqKkYkRiVGJ0YlRilGJUYsRiVGJSoqRjNGJUYvRiVGKUYlRixGJUYwKipGM0YlRjhGJUYpRiVGLEYlRjAqJkYyRiVGLEYlRiUqKkYkRiVGJkYlRipGJUYsRiVGJSoqRitGJUYnRiVGOkYlRixGJUYwKipGM0YlRi9GJUY6RiVGLEYlRiUqKkYkRiVGOEYlRjpGJUYsRiVGJSooRiRGJUYpRiVGLEYlRjAqJkYnRiVGOkYlRiUqKkYkRiVGJ0YlRipGJUYsRiVGJSooRiRGJUYvRiVGOkYlRjAqJkY6RiVGLEYlRiUqKEYkRiVGJ0YlRipGJUYwKihGJEYlRi9GJUYqRiVGJSokRjpGJUYlRidGJSomRiRGJUYqRiVGMEYlRiU=

and eliminate X. This gives our second equation between the three unknown series alg2:=op(4,factor(resultant(algX,alg2b,X)));

LGBwKigiIiUiIiIpSSNVMUc2IkYkRiUpSSJ0R0YoIiIjRiVGJSoqIiIpRiUpRiciIiRGJUklVWJhMUdGKEYlRilGJSEiIiosIiM7RiUpRidGK0YlRjBGJUkmVWJhYTFHRihGJUYpRiVGMSooIiInRiVGLkYlRilGJUYlKipGM0YlRjRGJUYwRiVGKUYlRjEqKkYkRiVGNEYlRjVGJUYpRiVGMSoqRi1GJUYnRiUpRjBGK0YlRilGJUYxKiwiI0NGJUYnRiVGMEYlRjVGJUYpRiVGMSooRitGJUYuRiVGKkYlRiUqKiIjP0YlRjRGJUYwRiVGKkYlRjEqKkYkRiVGNEYlRjVGJUYqRiVGJSomRjRGJUYpRiVGJSoqIiNTRiVGJ0YlRjtGJUYqRiVGJSosRi1GJUYnRiVGMEYlRjVGJUYqRiVGMSoqIiM5RiVGJ0YlRjBGJUYpRiVGMSoqRjdGJUYnRiVGNUYlRilGJUYxKihGM0YlKUYwRi9GJUYqRiVGMSoqIiNLRiVGO0YlRjVGJUYqRiVGJSooRi1GJUY7RiVGKUYlRjEqKkYzRiVGMEYlKUY1RitGJUYqRiVGMSoqRi1GJUYwRiVGNUYlRilGJUYxKiZGNEYlRipGJUYxKipGJEYlRidGJUYwRiVGKkYlRjEqKEYrRiVGJ0YlRilGJUYxKigiI0dGJUY7RiVGKkYlRiUqKkYtRiVGMEYlRjVGJUYqRiVGJSooRjdGJUYwRiVGKUYlRjEqKEYkRiVGT0YlRipGJUYxKihGK0YlRjVGJUYpRiVGMSokRjRGJUYxKihGN0YlRidGJUYwRiVGJSooRitGJUYnRiVGNUYlRiUqKEYrRiVGJ0YlRipGJUYxKiZGLUYlRjtGJUYxKihGLUYlRjBGJUY1RiVGMSooIiM3RiVGMEYlRipGJUYlKiRGKUYlRjEqJkYrRiVGJ0YlRiUqJkY3RiVGMEYlRjEqJkYrRiVGNUYlRjFGKkYlRiVGMQ==

Checking n:=10: normal(series(subs(Ubaa1=subs(x=1,Ubaaser(n)), U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)),alg2),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

In the third equation of the system, there is one more series that depends on x, namely Ubaa indets(eqs[3]);

PChJIlVHNiJJJFViYUdGJEklVWJhYUdGJEkmVWJhYTFHRiRJInRHRiRJInhHRiQ=

collect(eqs[3],[Uba,Ubaa],factor);

LCgqKkkidEc2IiIiIkkieEdGJUYmLCZGJ0YmRiYhIiJGJkkkVWJhR0YlRiZGJiomLCwqKiIiI0YmSSJVR0YlRiZGJEYmKUYnRi5GJkYmKipGLkYmRi9GJkYkRiZGJ0YmRikqJkYkRiZGJ0YmRiZGJ0YpRiZGJkYmSSVVYmFhR0YlRiZGJiooRiRGJkYwRiZJJlViYWExR0YlRiZGKQ==

factor(subs(x=X,U=UX,coeff(%,Ubaa))/ker);

IyIiIiIiIw==

We apply to this one the kernel method, by cancelling the coeff of Ubaa rhs3:=factor(subs(x=X,Ubaa=0, Uba=UbaX,eqs[3]));

KihJIlhHNiIiIiJJInRHRiRGJSwoKiZJJVViYVhHRiRGJUYjRiVGJSomSSZVYmFhMUdGJEYlRiNGJSEiIkYpRixGJQ==

UbaXsol:=factor(solve(rhs3,UbaX));

KihJIlhHNiIiIiJJJlViYWExR0YkRiUsJkYjRiVGJSEiIkYo

We need another expression of UbaX : we differentiate the second equation collect(eqs[2],[Uba,Ubaa],factor);

LCYqJiwsKioiIiUiIiJJIlVHNiJGJ0kidEdGKUYnKUkieEdGKSIiI0YnRicqKkYmRidGKEYnRipGJ0YsRichIiIqKEYtRidGKkYnRixGJ0YnKiZGLUYnRixGJ0YvRi1GJ0YnSSRVYmFHRilGJ0YnKihGKkYnRixGJywsKiZGKEYnRixGJ0YnKiZJI1UxR0YpRidGLEYnRicqKEYtRidJJVViYTFHRilGJ0YsRidGL0YoRi9GN0YvRidGJw==

diff(subs(U=U(x),Uba=Uba(x),eqs[2]),x);indets(%);

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

PCpJI1UxRzYiSSVVYmExR0YkSSJ0R0YkSSJ4R0YkLUkiVUdGJDYjRictSSRVYmFHRiRGKi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKEYnLUYuNiRGK0Yn

op(3,numer(factor(subs(diff(U(x),x) =UpX,U(x)=UXsol,diff(Uba(x), x) =0,Uba(x)=UbaXsol,x=X,UX=UXsol,%%))));

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

indets(%);

PChJI1UxRzYiSSVVYmExR0YkSSZVYmFhMUdGJEkkVXBYR0YkSSJYR0YkSSJ0R0Yk

UpXsol:=factor(solve(%%,UpX));

LCQqLCIiIyIiIiw4KiwiIiVGJUkjVTFHNiJGJUkmVWJhYTFHRipGJSlJIlhHRipGJEYlSSJ0R0YqRiVGJSosIiIpRiVJJVViYTFHRipGJUYrRiVGLEYlRi5GJSEiIiosIiInRiVGKUYlRitGJUYtRiVGLkYlRjIqLEYoRiVGMUYlRitGJUYtRiVGLkYlRiUqKkYkRiVGKUYlRitGJUYuRiVGJSooRjFGJUYtRiVGLkYlRjIqKEYrRiVGLUYlRi5GJUYyKiZGMUYlRi5GJUYlKiZGK0YlRi1GJUYlKiZGK0YlRi5GJUYlRitGMkYlRi5GMiksJkYtRiVGJUYyRiRGMiwoKihGKEYlRitGJUYtRiVGJUYtRiVGJUYyRjJGJQ==

subs(U1=A1,Uba1=B1,Ubaa1=C1,X=X,UpXsol);

LCQqLCIiIyIiIiw4KiwiIiVGJUkjQTFHNiJGJUkjQzFHRipGJSlJIlhHRipGJEYlSSJ0R0YqRiVGJSosIiIpRiVJI0IxR0YqRiVGK0YlRixGJUYuRiUhIiIqLCIiJ0YlRilGJUYrRiVGLUYlRi5GJUYyKixGKEYlRjFGJUYrRiVGLUYlRi5GJUYlKipGJEYlRilGJUYrRiVGLkYlRiUqKEYxRiVGLUYlRi5GJUYyKihGK0YlRi1GJUYuRiVGMiomRjFGJUYuRiVGJSomRitGJUYtRiVGJSomRitGJUYuRiVGJUYrRjJGJUYuRjIpLCZGLUYlRiVGMkYkRjIsKCooRihGJUYrRiVGLUYlRiVGLUYlRiVGMkYyRiU=

collect(4*A1*C1*X^2*t-8*B1*C1*X^2*t-6*A1*C1*X*t+4*B1*C1*X*t+2*A1*C1*t-B1*X*t-C1*X*t+B1*t+C1*X+C1*t-C1,[A1,C1,B1],factor,distributed);

LCoqLiIiIyIiIkkidEc2IkYlLCYqJkYkRiVJIlhHRidGJUYlRiUhIiJGJSwmRipGJUYlRitGJUkjQTFHRidGJUkjQzFHRidGJUYlKi4iIiVGJUYqRiVGJkYlRihGJUkjQjFHRidGJUYuRiVGKyooRiZGJUYsRiVGMUYlRisqKCwmRiZGJUYlRitGJUYsRiVGLkYlRis=

We use in the first equation our expression of U'(X) alg3a:=(numer(factor(subs(diff(U(x), x)=UpXsol, U(x)=UXsol,x=X,diff(subs(U=U(x),eqs[1]),x)))));

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

indets(alg3a);

PCdJI1UxRzYiSSVVYmExR0YkSSZVYmFhMUdGJEkiWEdGJEkidEdGJA==

Elimination of X alg3:=op(4,factor(resultant(algX,alg3a,X))): We now have three equations in the three series that do not depend on x indets(alg3*alg2*alg1);

PCZJI1UxRzYiSSVVYmExR0YkSSZVYmFhMUdGJEkidEdGJA==

Checking the last one n:=10: normal(series(subs(Ubaa1=subs(x=1,Ubaaser(n)), U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)),alg3),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

degree(alg1,U1);degree(alg2,U1);degree(alg3,U1); degree(alg1,Uba1);degree(alg2,Uba1);degree(alg3,Uba1); degree(alg1,Ubaa1);degree(alg2,Ubaa1);degree(alg3,Ubaa1);

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res:=factor(resultant(alg1,alg2,Ubaa1)): nops(res);

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seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

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eq1:=op(3,res): n:=10: normal(series(subs(U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)),eq1),t,n+1));

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res:=factor(resultant(alg2,alg3,Ubaa1)): nops(res);

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seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

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Choosing the correct factor n:=10: normal(series(subs( U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)),op(3,res)),t,n+1));

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n:=10: normal(series(subs( U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)),op(4,res)),t,n+1));

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n:=10: normal(series(subs( U1=subs(x=1,User(n)), Uba1=subs(x=1,Ubaser(n)),op(5,res)),t,n+1));

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Careful here ! choose the correct factor eq2:=op(5,res):nops(%);

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indets(eq1);indets(eq2);

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degree(eq1,U1);degree(eq1,Uba1); degree(eq2,U1);degree(eq2,Uba1);

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Final step res:=factor(resultant(eq1,eq2,Uba1)): nops(res);

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seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

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n:=6: normal(series(subs(U1=subs(x=1,User(n)),op(4,res)),t,n+1)); normal(series(subs(U1=subs(x=1,User(n)),op(5,res)),t,n+1)); normal(series(subs(U1=subs(x=1,User(n)),op(1,op(6,res))),t,n+1)); normal(series(subs(U1=subs(x=1,User(n)),op(7,res)),t,n+1));

KytJInRHNiIhL1tXbSE0NHgpIiIlITFvbkx5bEZRPyIiJiIzU11AYyUpR3dWcSIiJy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIig=

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KzNJInRHNiIiIiIiIiFGJUYlIiM3IiIjIiN3IiIkIiRxJiIiJSIlT1oiIiYiJi1BJSIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiKA==

KzNJInRHNiIhIiIiIiEiIiQiIiJGJyIiIyIjPkYnIiRLIiIiJSIlUTUiIiYiJTEqKSIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIiKA==

Checking further n:=20:normal(series(subs(U1=subs(x=1,User(n)),op(5,res)),t,n+1));

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algU1:=collect(op(5,res),U1,factor): indets(%);

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degree(algU1,U1);

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save(algU1,'algU1classic');

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Three finite branches at t=0: with(gfun): algeqtoseries(algU1,t,U1,4,true);

NyUrLUkidEc2IiMhI0AiIiUiIiEjIiIqRigiIiIjIiVkKikiI10iIiMjISkoXGUiSCIlK10iIiQtSSJPRyUqcHJvdGVjdGVkRzYjRixGKCspRiRGLEYpIyIleTUiI0ZGLEY1RjAsKkYsRiwqJkYwRixGJEYsRiwqJi1JJ1Jvb3RPZkc2JEY3SShfc3lzbGliR0YlNiMsKComIiNERiwpSSNfWkdGQkYwRixGLComIiQlW0YsRklGLCEiIiIlU0JGLEYsKUYkRjBGLEYsLUY2NiMqJClGJCMiIihGNEYsRiw=

allvalues(%);

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factor(coeff(algU1,U1,27));

LCQqJiI2T2IlcDFKSHEpR3giIiIiKUkidEc2IiIjS0YlRiU=

disc:=factor(discrim(algU1,U1)): nops(disc);

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seq(nops(op(i,disc)),i=1..nops(disc));

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seq(indets(op(i,disc)),i=1..nops(disc));

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op(2,disc);

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There will be 5 candidates for the minimal polynomial of the radius, called d3, ..., d7 d3:=op(3,disc): d4:=op(1,op(4,disc)): d5:=op(1,op(5,disc)): d6:=op(1,op(6,disc)): d7:=op(1,op(7,disc)): Looking at the roots fsolve(d3,t=1/16..1/10); fsolve(d4,t=1/16..1/10); fsolve(d5,t=1/16..1/10);1/%;

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fsolve(d6,t=1/16..1/10); [fsolve(d7,t=1/16..1/10)];map (Z->1/Z,%);

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3 candidates for the radius. We need to look more closely at the branches with(plots): implicitplot(algU1,t=0..1/10,U1=0.9..1.4,numpoints=10000);

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separate the branches U1=1+2t+... res:=factor(subs(U1=1+2*t+t^2*S,algU1)): nops(res);

IiIj

op(1,res);

KiQpSSJ0RzYiIiImIiIi

algS:=op(2,res): with(gfun): algeqtoseries(algS,t,S,4,true);

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

Our branch is higher than the other near 0 allvalues(%);

NiQ3IystSSJ0RzYiIyIkTSMiI0QiIiEjIiZNcyQiJEQnIiIiIyIoQS5TIiIlREoiIiMjIitpeiYpZTkiJ0QxUiIiJC1JIk9HJSpwcm90ZWN0ZWRHNiNGLiIiJTcjKy1GJSIjNUYqIiNtRi4iJC8mRjIiJUtVRjZGN0Y7

A neater picture with(plots): implicitplot(algS,t=0..0.08,S=9..26,numpoints=10000);

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Here we see more clearly the 3 candidates and their respective roles implicitplot(algS,t=0.05..0.08,S=19..27,numpoints=10000);

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_YG6#$"#q!""-%-BOUNDS_WIDTHG6#$"%?M!""-%.BOUNDS_HEIGHTG6#$"%![$!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%SM!""-%.BOUNDS_HEIGHTG6#$"%?N!""-%)CHILDRENG6"Ig==

Even better implicitplot(algS,t=0.07..0.08,S=21..27,numpoints=10000);

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The radius is the second candidate 0.07658551141;

JCIrVDZiZXchIzY=

1/%;

JCIrLSt0MDghIik=

Nature of the singularity ? We compute estimates for the exponent ser:=subs(x=1,User(100)): with(gfun):liste:=seriestolist(series(ser,t,101)): plot([seq([1/n,n^2*(1-liste[n+1]*liste[n-1]/liste [n]^2)],n=5..nops(liste)-1)],x=0..0.10,y=-2.51..-2.45,style=point);

NistJSdDVVJWRVNHNiQ3XHE3JCQiIiMhIiIkITFvRSpRJnBnKlsjISM6NyQkIjJtbW1tbW1tbSIhIzwkITFoWygpZSVva1kjISM6NyQkIjImRzlkRzlkRzkhIzwkITIkZUksOi9qYkMhIzs3JCQiJEQiISIkJCEyL3RLITMsdl5DISM7NyQkIjE2NjY2NjY2NiEjOyQhMXhHXSMpW05eQyEjOjckJCIiIiEiIiQhMkMneS9rUGtfQyEjOzckJCIxIjQ0NDQ0NDQqISM8JCEybmlcMmUmcGFDISM7NyQkIjFMTExMTExMJCkhIzwkITJYJFFuXypScVgjISM7NyQkIjEkcDJCcDJCcCghIzwkITJjPCU0a3FWZkMhIzs3JCQiMVVyJkc5ZEc5KCEjPCQhMkVuKG8vbndoQyEjOzckJCIxbm1tbW1tbW0hIzwkITFqViNRdnJSWSMhIzo3JCQiJEQnISIlJCEyWWhBPHRHZ1kjISM7NyQkIjIwWnc2JUhOIyllISM9JCEyblZmNTNLelkjISM7NyQkIjFiYmJiYmJiYiEjPCQhMiVceSRwZyZvcEMhIzs3JCQiMVVvdCUqeTpqXyEjPCQhMmE1Wz12KEhyQyEjOzckJCIiJiEiIyQhMVssX3UqeUZaIyEjOjckJCIyO3cvPncvPnclISM9JCExS20oXElTVFojISM6NyQkIjJjYWFhYWFhYSUhIz0kITJMPz5nYyNSdkMhIzs3JCQiMjtfY3AzRXlNJSEjPSQhMnYpbzF2ZmF3QyEjOzckJCIya21tbW1tbTslISM9JCEyOiQqZkknKjR3WiMhIzs3JCQiIiUhIiMkITI7O1RnNiRmeUMhIzs3JCQiMmslUTpZUTpZUSEjPSQhMiQpKltTJT0uJnpDISM7NyQkIjJOcS5QcS5QcSQhIz0kITJRL2B3N1ouWyMhIzs3JCQiMXImRzlkRzlkJCEjPCQhMjpBOmY8SjZbIyEjOzckJCIyYicqbz8nZUZbTSEjPSQhMlEqZVYoKjMnPVsjISM7NyQkIjFMTExMTExMTCEjPCQhMlktZ2ZFVERbIyEjOzckJCIxLkhoXmshZUEkISM8JCEyJHlUd0NuPCRbIyEjOzckJCIlREohIiYkITF5eHVYN3gkWyMhIzo3JCQiMi8uLi4uLi4uJCEjPSQhMV9bKEhRR1ZbIyEjOjckJCIyYEIpZXFrPFRIISM9JCEyd2E4MUteW1sjISM7NyQkIjFkRzlkRzlkRyEjPCQhMUVbXzpITSZbIyEjOjckJCIyd3h4eHh4eHgjISM9JCExd1IhXHQwZVsjISM6NyQkIjEuRnEtRnEtRiEjPCQhMiQqKjRXKjNVaVsjISM7NyQkIjFAJW90JSp5OmojISM8JCExJT13QzFhbVsjISM6NyQkIjFrRDVrRDVrRCEjPCQhMWInM2NgVnFbIyEjOjckJCIjRCEiJCQhMkZLIz02QVQoWyMhIzs3JCQiMkQhUkMhUkMhUkMhIz0kITJrKVEuSzt3KFsjISM7NyQkIjIzUV80UV80USMhIz0kITElUVkpKT4kNClbIyEjOjckJCIycyQpW2BSImVEQiEjPSQhMkYoPVQmPTMlKVsjISM7NyQkIjJHRkZGRkZGRiMhIz0kITIudHA/djIoKVsjISM7NyQkIjJCQUFBQUFBQSMhIz0kITImNE1iZEgqKilbIyEjOzckJCIyM0V5TS84UjwjISM9JCExeiRbJXBaRSpbIyEjOjckJCIxJjNvV2RmdzcjISM8JCExI1xQTTJDJipbIyEjOjckJCIyS0xMTExMTDMjISM9JCEyLnQnWyJvcigqWyMhIzs3JCQiMTcxYEVqIjMvIyEjPCQhMngwLyJSJDMrXCMhIzs3JCQiIiMhIiMkITI8XSd5SlpCIVwjISM7NyQkIjBcRFBKJXlnPiEjOyQhMlZSYjJcXi9cIyEjOzckJCIySyNwMkJwMkI+ISM9JCExR2wxKT5mMVwjISM6NyQkIjInKT1JR1gjeicpPSEjPSQhMmtpcTZSZTNcIyEjOzckJCIyPCY9Jj0mPSY9Jj0hIz0kITJLciopcGNcNVwjISM7NyQkIjE9PT09PT09PSEjPCQhMjxebWI9TDdcIyEjOzckJCIyY0c5ZEc5ZHkiISM9JCEyJnBtTHQnNDlcIyEjOzckJCIyMUc3XCdmUWE8ISM9JCExKkhuZ1V6OlwjISM6NyQkIjJGW001JHo4QzwhIz0kITJ1JikqXDZHdSJcIyEjOzckJCIxKUdQVUQ6XHAiISM8JCExUTF0ciwhPlwjISM6NyQkIjJtbW1tbW1tbSIhIz0kITF3IWVgIz0wI1wjISM6NyQkIjEjMyZIaVVNUjshIzwkITJQWkElcCEpPiNcIyEjOzckJCIyO1gxZUEuSGgiISM9JCExJXBnNj1SQlwjISM6NyQkIjJzZSx0ZSx0ZSIhIz0kITJMbSZmPmFaI1wjISM7NyQkIiZEYyIhIickITJkPHlwLTJFXCMhIzs3JCQiMiZROllROllROiEjPSQhMWtrNElVdCNcIyEjOjckJCIyX15eXl5eXl4iISM9JCExOXNZVHMmR1wjISM6NyQkIjJlJEdWOHRgI1wiISM9JCEyTDc/L0V3SFwjISM7NyQkIjJ3NiVITiMpZXE5ISM9JCExXm4zdTk0JFwjISM6NyQkIjIxJSk9Qk92I1w5ISM9JCEyKDRLSWVJPyRcIyEjOzckJCIyJkc5ZEc5ZEc5ISM9JCEyJXpeVnk2SiRcIyEjOzckJCIyQE5EVXFdJTM5ISM9JCEyWSJHOCEqZlQkXCMhIzs3JCQiMikpKSkpKSkpKSkpKSkpUSIhIz0kITJaR0ErazxOXCMhIzs3JCQiMGopcDhJJylwOCEjOyQhMXgqUmpFO09cIyEjOjckJCIyOU5eOE5eOE4iISM9JCEyVnY5JioqPnIkXCMhIzs3JCQiMk1MTExMTExMIiEjPSQhMSUzLkYnXCFRXCMhIzo3JCQiMjBAJW90JSp5OjghIz0kITIuUSJHc18qUVwjISM7NyQkIjIpKUgsKClILCgpSCIhIz0kITFNJSpIUUkpUlwjISM6NyQkIjEjR14/R14/RyIhIzwkITBWKSpbT29TXCMhIzk3JCQiMm03NVt5QWVFIiEjPSQhMjxwJ3BdODolXCMhIzs3JCQiJEQiISIlJCExKilmOSozS1VcIyEjOjckJCIyeWNNNyF6Y003ISM9JCEyPSc0JylvMUolXCMhIzs3JCQiMjgmPjcmPjcmPjchIz0kITJPKFIjUjwoUSVcIyEjOzckJCIyUVYzckY+Wz8iISM9JCExc3U7JW9oV1wjISM6NyQkIjIvPncvPncvPiIhIz0kITFlZFZ2VWAlXCMhIzo3JCQiMlRITiMpZXFrPCIhIz0kITFueSMpPl1nJVwjISM6NyQkIjInPVduKHAhemk2ISM9JCEyRTZCZilSbiVcIyEjOzckJCIyPUtjdEdEJVw2ISM9JCEyPXMhKipRN3UlXCMhIzs3JCQiMmtqampqamo4IiEjPSQhMk0iWz1UbyFbXCMhIzs3JCQiMnYoemgwYmZCNiEjPSQhMmQiZnNeMyhbXCMhIzs3JCQiMjc2NjY2NjY2IiEjPSQhMXArMkZMJFxcIyEjOjckJCIxKjQsKik0LCopNCIhIzwkITEnW201SyUqXFwjISM6NyQkIjIvOFI8X2NwMyIhIz0kITFlWTUmKVEwJlwjISM6NyQkIjI3SS9zIilvXzIiISM9JCExMDxEbz82JlwjISM6NyQkIjJEL01zeUhRMSIhIz0kITFCMlE8Km9eXCMhIzo3JCQiMiVvdCUqeTpqXzUhIz0kITEqeSFIeFdBJlwjISM6NyQkIjJtbW1tbW07LyIhIz0kITFgIz4zenlfXCMhIzo3JCQiMmthXl0keSM0LiIhIz0kITIyKW8lKikqPUwmXCMhIzs3JCQiMTFgRWoiMy8tIiEjPCQhMiVvaiozJVFRJlwjISM7NyQkIjItLCwsLCwsLCIhIz0kITJicUNVbE1hXCMhIzs3JCQiIiIhIiMkITAkKTRcUCVbJlwjISM5LSUmQ09MT1JHNiclJFJHQkckIilDKWVxJSEiKSQiIiEhIiIkIignPiFcJiEiKS0lK19BVFRSSUJVVEVHNiMvJSdzb3VyY2VHJSxtYXRoZGVmYXVsdEctJSVWSUVXRzYkOyQiIiEhIiIkIiIiISIiOyQhJF4jISIjJCEkWCMhIiMtJiUmX0FYSVNHNiMiIiI2Jy0lK19HUklETElORVNHNictJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJSlfVklTSUJMRUc2IyIiIS0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0mJSZfQVhJU0c2IyIiIzYnLSUrX0dSSURMSU5FU0c2Jy0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lKV9WSVNJQkxFRzYjIiIhLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSUrQVhFU0xBQkVMU0c2JC1JI21pRzYjL0krbW9kdWxlbmFtZUc2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The system of Section 6.2.1 (expressions of the various series) eqU:=-U+1+U*Up; eqUp:=-Up+2*t*x*U+t*U1*(2*x*U0p1+x/(x-1)*(Up-x*Up1)); eqU0p:=-U0p+t*x*U+t*U1*(x*U0p1+x/(x-1)*(U0p-x*U0p1));

LCgqJkkiVUc2IiIiIkkjVXBHRiVGJkYmRiQhIiJGJkYm

LChJI1VwRzYiISIiKioiIiMiIiJJInRHRiRGKEkieEdGJEYoSSJVR0YkRihGKCooRilGKEkjVTFHRiRGKCwmKihGJ0YoRipGKEklVTBwMUdGJEYoRigqKEYqRigsJkYqRihGKEYlRiUsJiomSSRVcDFHRiRGKEYqRihGJUYjRihGKEYoRihGKA==

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An equation in U only eqcat:=numer(factor(subs(U0p1=Up1/2,U=1/(1-Up),U1=1/(1-Up1),eqUp)));

LEAqKClJI1VwRzYiIiIjIiIiSSRVcDFHRiZGKEkieEdGJkYoISIiKihGJEYoSSJ0R0YmRihGKkYoRisqKkYlRihGKUYoRi1GKEYqRihGKCoqRidGKEYpRihGLUYoKUYqRidGKEYrKiZGJEYoRilGKEYoKiZGJEYoRipGKEYoKihGJUYoRilGKEYqRihGKCooRiVGKEYtRihGKkYoRigqKEYpRihGLUYoRipGKEYoKihGJ0YoRi1GKEYwRihGKCokRiRGKEYrKiZGJUYoRilGKEYrKiZGJUYoRipGKEYrKihGJ0YoRi1GKEYqRihGK0YlRig=

We apply the quadratic method factor(discrim(eqcat,Up));

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factor(discrim(op(2,%),x));

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algUp1:=op(3,%);

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algu1:=collect(numer(factor(subs(Up1=1-1/U1,algUp1))),U1,factor);

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The system of Section 6.2.2 We denote T and U for general orientations, Tp and Up for the primitive ones. The letters a and b stand for 0 and 1. For instance, Ua is the bivariate series U0(t;x), and Ua1 is its specialization at x=1. eqTpba:=t+t*U1*Upa1; eqU:=1+U*Up; eqUa:=U*Upa; eqUba:=U*Upba; eqUaa:=U*Upaa; eqUp:=2*Upa; eqUpa:=t*x*U+t*U1*(x*Upa1+x^2*(Upbaa1+Upaba1+Upbba1)+x/(x-1)*(Upa-x^2*Upa1)+x^2*Tpba); #eqUpba:=x*Tpba+Upbba+Upaba; eqUpba:=x*Tpba+t*x*Ua+t*U1*(x^2*Upaba1+x^2*Upbba1+x/(x-1)*(Upba-x^2*Upba1)+x^2*Tpba); eqUpaa:=t*x*Ua+t*U1*(x^2*Upbaa1+x/(x-1)*(Upaa-x^2*Upaa1)); eqUpbaa:=t*x*Uab+t*U1*(x^2*Upbaa1+x/(x-1)*(Upbaa-x^2*Upbaa1)); eqUpaba:=t*x*Uab+t*U1*(x^2*Upaba1+x/(x-1)*(Upaba-x^2*Upaba1)); eqUpbba:=t*x*Uaa+t*U1*(x^2*Upbba1+x/(x-1)*(Upbba-x^2*Upbba1));

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LCgqJkkieEc2IiIiIkklVHBiYUdGJUYmRiYqKEkidEdGJUYmRiRGJkkjVWFHRiVGJkYmKihGKUYmSSNVMUdGJUYmLCoqJilGJCIiI0YmSSdVcGFiYTFHRiVGJkYmKiZGL0YmSSdVcGJiYTFHRiVGJkYmKihGJEYmLCZGJEYmRiYhIiJGNiwmKiZJJlVwYmExR0YlRiZGL0YmRjZJJVVwYmFHRiVGJkYmRiYqJkYvRiZGJ0YmRiZGJkYm

LCYqKEkidEc2IiIiIkkieEdGJUYmSSNVYUdGJUYmRiYqKEYkRiZJI1UxR0YlRiYsJiomKUYnIiIjRiZJJ1VwYmFhMUdGJUYmRiYqKEYnRiYsJkYnRiZGJiEiIkYyLCYqJkkmVXBhYTFHRiVGJkYtRiZGMkklVXBhYUdGJUYmRiZGJkYmRiY=

LCYqKEkidEc2IiIiIkkieEdGJUYmSSRVYWJHRiVGJkYmKihGJEYmSSNVMUdGJUYmLCYqJilGJyIiI0YmSSdVcGJhYTFHRiVGJkYmKihGJ0YmLCZGJ0YmRiYhIiJGMiwmKiZGL0YmRi1GJkYySSZVcGJhYUdGJUYmRiZGJkYmRiY=

LCYqKEkidEc2IiIiIkkieEdGJUYmSSRVYWJHRiVGJkYmKihGJEYmSSNVMUdGJUYmLCYqJilGJyIiI0YmSSdVcGFiYTFHRiVGJkYmKihGJ0YmLCZGJ0YmRiYhIiJGMiwmKiZGL0YmRi1GJkYySSZVcGFiYUdGJUYmRiZGJkYmRiY=

LCYqKEkidEc2IiIiIkkieEdGJUYmSSRVYWFHRiVGJkYmKihGJEYmSSNVMUdGJUYmLCYqJilGJyIiI0YmSSdVcGJiYTFHRiVGJkYmKihGJ0YmLCZGJ0YmRiYhIiJGMiwmKiZGL0YmRi1GJkYySSZVcGJiYUdGJUYmRiZGJkYmRiY=

Alternative expressions of four main series (section 6.4) exUp:=2*t*x*U +U1*t*(x*Up1+2*x^2*(Upba1+Upbaa1)+x/(x-1)*(Up-x^2*Up1)); exUpba:=t*x*(1+U1*Up1/2)+ t*x*(U-1)/2 +t*U1*(x^2*Upba1+x/(x-1)*(Upba-x^2*Upba1)); exUpbaa:=t*x*Uba +U1*t*(x^2*Upbaa1+x/(x-1)*(Upbaa-x^2*Upbaa1));

LCYqKiIiIyIiIkkidEc2IkYlSSJ4R0YnRiVJIlVHRidGJUYlKihJI1UxR0YnRiVGJkYlLCgqJkYoRiVJJFVwMUdGJ0YlRiUqKEYkRiUpRihGJEYlLCZJJlVwYmExR0YnRiVJJ1VwYmFhMUdGJ0YlRiVGJSooRihGJSwmRihGJUYlISIiRjYsJiomRi5GJUYwRiVGNkkjVXBHRidGJUYlRiVGJUYl

LCgqKEkidEc2IiIiIkkieEdGJUYmLCZGJkYmKigjRiYiIiNGJkkjVTFHRiVGJkkkVXAxR0YlRiZGJkYmRiYqKkYqRiZGJEYmRidGJiwmSSJVR0YlRiZGJiEiIkYmRiYqKEYkRiZGLEYmLCYqJilGJ0YrRiZJJlVwYmExR0YlRiZGJiooRidGJiwmRidGJkYmRjFGMSwmKiZGNkYmRjVGJkYxSSVVcGJhR0YlRiZGJkYmRiZGJg==

LCYqKEkidEc2IiIiIkkieEdGJUYmSSRVYmFHRiVGJkYmKihGJEYmSSNVMUdGJUYmLCYqJilGJyIiI0YmSSdVcGJhYTFHRiVGJkYmKihGJ0YmLCZGJ0YmRiYhIiJGMiwmKiZGL0YmRi1GJkYySSZVcGJhYUdGJUYmRiZGJkYmRiY=

Checking the agreement with the first system factor(exUp-2*eqUpa);

LCQqLEkjVTFHNiIiIiJJInRHRiVGJkkieEdGJUYmLEIqKCIiI0YmSSVUcGJhR0YlRiYpRihGK0YmRiYqJkkkVXAxR0YlRiZGLUYmRiYqKEYrRiZJJVVwYTFHRiVGJkYtRiYhIiIqKEYrRiZJJ1VwYWJhMUdGJUYmRi1GJkYmKihGK0YmSSZVcGJhMUdGJUYmRi1GJkYyKihGK0YmSSdVcGJiYTFHRiVGJkYtRiZGJiooRitGJkYsRiZGKEYmRjIqJkYvRiZGKEYmRjIqKEYrRiZGMUYmRihGJkYmKihGK0YmRjRGJkYoRiZGMiooRitGJkY2RiZGKEYmRiYqKEYrRiZGOEYmRihGJkYySSNVcEdGJUYyRi9GJiomRitGJkkkVXBhR0YlRiZGJiomRitGJkYxRiZGMkYmLCZGKEYmRiZGMkYyRjI=

factor(subs(Upba1=Tpba+Upaba1+Upbba1,Up=2*Upa,Up1=2*Upa1,%));

IiIh

collect(exUpba-eqUpba,U1,factor);

LCYqLCMiIiIiIiNGJSwsKihGJkYlSSVUcGJhRzYiRiVJInhHRipGJUYlKihGJkYlSSdVcGFiYTFHRipGJUYrRiVGJSooRiZGJUkmVXBiYTFHRipGJUYrRiUhIiIqKEYmRiVJJ1VwYmJhMUdGKkYlRitGJUYlSSRVcDFHRipGMEYlSSJ0R0YqRiVGK0YlSSNVMUdGKkYlRjAqKEYkRiVGK0YlLCoqJkkiVUdGKkYlRjRGJUYwKihGJkYlSSNVYUdGKkYlRjRGJUYlKiZGJkYlRilGJUYlRjRGMEYlRjA=

factor(subs(Upba1=Tpba+Upaba1+Upbba1,%));

LCQqKCMiIiIiIiNGJUkieEc2IkYlLCwqKEkjVTFHRihGJUkkVXAxR0YoRiVJInRHRihGJSEiIiomSSJVR0YoRiVGLUYlRi4qKEYmRiVJI1VhR0YoRiVGLUYlRiUqJkYmRiVJJVRwYmFHRihGJUYlRi1GLkYlRi4=

factor(subs(Tpba=eqTpba,U=1+2*Ua,Up1=2*Upa1,%));

IiIh

factor(exUpbaa-eqUpbaa);

LCQqKEkidEc2IiIiIkkieEdGJUYmLCZJJFVhYkdGJUYmSSRVYmFHRiUhIiJGJkYr

indets(exUp*exUpba*exUpbaa);

PC1JIlVHNiJJI1UxR0YkSSRVYmFHRiRJI1VwR0YkSSRVcDFHRiRJJVVwYmFHRiRJJlVwYmExR0YkSSZVcGJhYUdGJEknVXBiYWExR0YkSSJ0R0YkSSJ4R0Yk

syst:=subs(Uba=eqUba,U=1/(1-Up), U1=1/(1-Up1),[exUp-Up,exUpba-Upba,exUpbaa-Upbaa]);

NyUsKCoqIiIjIiIiSSJ0RzYiRiZJInhHRihGJiwmRiZGJkkjVXBHRighIiJGLEYmKigsJkYmRiZJJFVwMUdGKEYsRixGJ0YmLCgqJkYpRiZGL0YmRiYqKEYlRiYpRilGJUYmLCZJJlVwYmExR0YoRiZJJ1VwYmFhMUdGKEYmRiZGJiooRilGJiwmRilGJkYmRixGLCwmKiZGL0YmRjNGJkYsRitGJkYmRiZGJkYmRitGLCwqKihGJ0YmRilGJiwmRiZGJiooI0YmRiVGJkYuRixGL0YmRiZGJkYmKipGP0YmRidGJkYpRiYsJiokRipGLEYmRiZGLEYmRiYqKEYnRiZGLkYsLCYqJkYzRiZGNUYmRiYqKEYpRiZGOEYsLCYqJkY1RiZGM0YmRixJJVVwYmFHRihGJkYmRiZGJkYmRklGLCwoKipGJ0YmRilGJkYqRixGSUYmRiYqKEYnRiZGLkYsLCYqJkYzRiZGNkYmRiYqKEYpRiZGOEYsLCYqJkY2RiZGM0YmRixJJlVwYmFhR0YoRiZGJkYmRiZGJkZSRiw=

indets(syst) minus {t,x};

PChJI1VwRzYiSSRVcDFHRiRJJVVwYmFHRiRJJlVwYmExR0YkSSZVcGJhYUdGJEknVXBiYWExR0Yk

nops(syst);

IiIk

subs(t=0,syst);

NyUsJEkjVXBHNiIhIiIsJEklVXBiYUdGJUYmLCRJJlVwYmFhR0YlRiY=

We can compute the first coefficients of all these series substitutions:=proc(n): {Up=Upser(n-1), Up1=subs(x=1,Upser(n-1)), Upba1=subs(x=1,Upbaser(n-1)), Upba=Upbaser(n-1), Upbaa=Upbaaser(n-1),Upbaa1=subs(x=1,Upbaaser(n-1))};end: Upser:=proc(n) option remember: if n=0 then 0 else convert(normal(series(subs(substitutions(n),syst[1]+Up),t,n+1)),polynom): fi:end: Upbaser:=proc(n) option remember: if n=0 then 0 else convert(normal(series(subs(substitutions(n),syst[2]+Upba),t,n+1)),polynom): fi:end: Upbaaser:=proc(n) option remember: if n=0 then 0 else convert(normal(series(subs(substitutions(n),syst[3]+Upbaa),t,n+1)),polynom): fi:end: substitutions(1);

PCgvSSNVcEc2IiIiIS9JJFVwMUdGJUYmL0klVXBiYUdGJUYmL0kmVXBiYTFHRiVGJi9JJlVwYmFhR0YlRiYvSSdVcGJhYTFHRiVGJg==

Upser(1);

LCQqKCIiIyIiIkkidEc2IkYlSSJ4R0YnRiVGJQ==

Upser(8);

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

subs(x=1,%);

LDIqJiIoUSxqIiIiIilJInRHNiIiIilGJUYlKiYiJyVbbSJGJSlGJyIiKEYlRiUqJiImQXgiRiUpRiciIidGJUYlKiYiJScqPkYlKUYnIiImRiVGJSomIiRXI0YlKUYnIiIlRiVGJSomIiNNRiUpRiciIiRGJUYlKiZGMUYlKUYnIiIjRiVGJSomRkBGJUYnRiVGJQ==

Upser(10);

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

subs(x=1,%);

LDYqJiIqRToiMzwiIiIpSSJ0RzYiIiM1RiVGJSomIilJdVo7RiUpRiciIipGJUYlKiYiKFEsaiJGJSlGJyIiKUYlRiUqJiInJVttIkYlKUYnIiIoRiVGJSomIiZBeCJGJSlGJyIiJ0YlRiUqJiIlJyo+RiUpRiciIiZGJUYlKiYiJFcjRiUpRiciIiVGJUYlKiYiI01GJSlGJyIiJEYlRiUqJkY5RiUpRiciIiNGJUYlKiZGSEYlRidGJUYl

Back to general orientations series(1/(1-%),t,10);

KzlJInRHNiIiIiIiIiEiIiNGJSIjNUYnIiNtIiIkIiQvJiIiJSIlR1UiIiYiJnl5JCIiJyInX2lOIiIoIihpJnlNIiIpIiktQipcJCIiKi1JIk9HJSpwcm90ZWN0ZWRHNiNGJUYo

ser:=subs(x=1,Upser(60)): with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],t,60): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,t=0.07..0.09);

JCIrUGIza3chIzY=

1/%;

JCIrM3J5LzghIik=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

We now attack the solution of the system eqs:=map(numer,syst);

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

indets(eqs);nops(eqs);

PCpJI1VwRzYiSSRVcDFHRiRJJVVwYmFHRiRJJlVwYmExR0YkSSZVcGJhYUdGJEknVXBiYWExR0YkSSJ0R0YkSSJ4R0Yk

IiIk

The first equation only depends on Up(x) and on series that do not depend on x indets(eqs[1]);

PChJI1VwRzYiSSRVcDFHRiRJJlVwYmExR0YkSSdVcGJhYTFHRiRJInRHRiRJInhHRiQ=

The quadratic method factor(discrim(eqs[1],Up));

LGpyKigpSSRVcDFHNiIiIiMiIiIpSSJ0R0YmRidGKClJInhHRiYiIidGKEYoKiwiIiVGKEYlRihJJlVwYmExR0YmRihGKUYoRitGKCEiIiosRi9GKEYlRihJJ1VwYmFhMUdGJkYoRilGKEYrRihGMSoqRi9GKClGMEYnRihGKUYoRitGKEYoKiwiIilGKEYwRihGM0YoRilGKEYrRihGKCoqRi9GKClGM0YnRihGKUYoRitGKEYoKipGJ0YoRiRGKEYpRigpRiwiIiZGKEYxKixGN0YoRiVGKEYwRihGKUYoRjtGKEYoKixGN0YoRiVGKEYzRihGKUYoRjtGKEYoKipGN0YoRjVGKEYpRihGO0YoRjEqLCIjO0YoRjBGKEYzRihGKUYoRjtGKEYxKipGN0YoRjlGKEYpRihGO0YoRjEqKiIiJEYoRiRGKEYpRigpRixGL0YoRigqLEY3RihGJUYoRjBGKEYpRihGRUYoRjEqLEY3RihGJUYoRjNGKEYpRihGRUYoRjEqKkYvRihGNUYoRilGKEZFRihGKCosRjdGKEYwRihGM0YoRilGKEZFRihGKCoqRi9GKEY5RihGKUYoRkVGKEYoKipGJ0YoRiRGKEYpRigpRixGREYoRjEqKkYnRihGJEYoRipGKEZFRihGMSosRi9GKEYlRihGMEYoRilGKEZMRihGKCosRi9GKEYlRihGMEYoRipGKEZFRihGKCosRi9GKEYlRihGM0YoRilGKEZMRihGKCosRi9GKEYlRihGM0YoRipGKEZFRihGKCoqRidGKEYlRihGKUYoRkVGKEYxKipGL0YoRjBGKEYpRihGRUYoRigqKkYvRihGM0YoRilGKEZFRihGKCooRiRGKEYpRigpRixGJ0YoRigqKkYvRihGJEYoRipGKEZMRihGMSosRjdGKEYlRihGMEYoRipGKEZMRihGMSosRjdGKEYlRihGM0YoRipGKEZMRihGMSoqRi1GKEYlRihGKUYoRkxGKEYxKipGJ0YoRiVGKEYqRihGRUYoRigqKkYvRihGMEYoRilGKEZMRihGMSoqRi9GKEYwRihGKkYoRkVGKEYxKipGL0YoRjNGKEYpRihGTEYoRjEqKkYvRihGM0YoRipGKEZFRihGMSoqIiM3RihGJEYoRipGKEZWRihGKCosRi9GKEYlRihGMEYoRipGKEZWRihGKCosRi9GKEYlRihGM0YoRipGKEZWRihGKCoqRi1GKEYlRihGKUYoRlZGKEYoKipGW29GKEYlRihGKkYoRkxGKEYoKipGN0YoRjBGKEYqRihGTEYoRigqKkY3RihGM0YoRipGKEZMRihGKCooRjdGKEYpRihGTEYoRigqKkYtRihGJEYoRipGKEYsRihGMSomRiRGKEZWRihGKCoqIiNFRihGJUYoRipGKEZWRihGMSoqRi9GKEYwRihGKkYoRlZGKEYxKipGL0YoRjNGKEYqRihGVkYoRjEqKCIiKEYoRilGKEZWRihGMSooRjdGKEYqRihGTEYoRjEqKEYnRihGJEYoRixGKEYxKipGW29GKEYlRihGKkYoRixGKEYoKihGJ0YoRiVGKEZWRihGMSooIiM5RihGKkYoRlZGKEYoKiRGJEYoRigqKEYvRihGJUYoRixGKEYoKihGLUYoRipGKEYsRihGMSokRlZGKEYoKiZGJ0YoRiVGKEYxKiZGJ0YoRixGKEYxRihGKA==

res:=factor(discrim(%,x)): nops(res);

IiIm

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NiciIiIiIiNGJCIkNihGJA==

Nic8IjwjSSRVcDFHNiI8JUYlSSZVcGJhMUdGJkknVXBiYWExR0YmPCZGJUYoRilJInRHRiY8I0Yr

op(3,res);op(5,res);

KiQpLChJJFVwMUc2IiIiIiomIiIjRidJJlVwYmExR0YmRichIiIqJkYpRidJJ1VwYmFhMUdGJkYnRisiIiVGJw==

KiQpSSJ0RzYiIiM1IiIi

We need to choose the factor that actually vanishes n:=10: normal(series(subs(Upbaa1=subs(x=1,Upbaaser(n)), Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)), op(1,op(3,res))),t,n+1));

KzNJInRHNiIiIiMiIiQiIz0iIiUiJG0iIiImIiUrOyIiJyImd2YiIiIoIidBUzsiIikiKGNAcyIiIioiKSdlPSU9IiM1

op(4,res) is a very big thing which we do not want to see n:=20: normal(series(subs(Upbaa1=subs(x=1,Upbaaser(n)), Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)), op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

Alas, this big thing is the correct identity, our first identity between the three uknown series, specialized at 1 alg1:=op(4,res):nops(alg1);

IiQ2KA==

In the second equation, there is one more series that depends on x, namely Upba indets(eqs[2]);

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collect(eqs[2],[Upba,Upbaa],factor);

LCYqKiIiIyIiIiwsKiZJJFVwMUc2IkYlSSJ4R0YpRiVGJSomSSJ0R0YpRiVGKkYlRiVGKCEiIkYqRi1GJUYlRiUsJkYlRi1JI1VwR0YpRiVGJUklVXBiYUdGKUYlRi0qKEYsRiVGKkYlLDIqKkYkRiVGL0YlSSZVcGJhMUdGKUYlRipGJUYlKiZGL0YlRipGJUYtRidGLSooRiRGJUY0RiVGKkYlRi1GL0YlRihGJSomRiRGJUYqRiVGJUYkRi1GJUYl

The coefficient of the linear term (also called "kernel"). We keep the factor that cancels for some series X(t) ker:=op(2,factor(subs(x=X,Up=UpX,coeff(%,Upba))));

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We apply to this one the kernel method, by cancelling the coeff of Upba RHS:=subs(Upba=0,x=X,Up=UpX,eqs[2]);

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An equation for X Xsol:=factor(solve(ker,X));

KiYsJiIiIiEiIkkkVXAxRzYiRiRGJCwoRiZGJEkidEdGJ0YkRiRGJUYl

factor(%-1);

LCQqJkkidEc2IiIiIiwoSSRVcDFHRiVGJkYkRiZGJiEiIkYpRik=

UpXsol:=factor(subs(X=Xsol,solve(RHS,UpX)));

KiYsKiooIiIjIiIiSSRVcDFHNiJGJkkmVXBiYTFHRihGJkYmKiZGJ0YmSSJ0R0YoRiYhIiIqJkYlRiZGKUYmRiwqJkYlRiZGK0YmRiZGJiwoRiRGJkYtRixGK0YmRiw=

factor(%-1);

LCQqKEkidEc2IiIiIiwmRiYhIiJJJFVwMUdGJUYmRiYsKCooIiIjRiZGKUYmSSZVcGJhMUdGJUYmRiYqJkYsRiZGLUYmRihGJEYmRihGKA==

We replace X and Up(X) by their values in the first equation... alg2:=op(3,numer(factor(subs(Up=UpXsol,x=Xsol,eqs[1]))));

LEAqJClJJFVwMUc2IiIiJCIiIkYoKigiIiVGKClGJSIiI0YoSSZVcGJhMUdGJkYoISIiKiZGK0YoSSJ0R0YmRihGKCoqRixGKEYlRihGLUYoRjBGKEYuKipGLEYoRiVGKEknVXBiYWExR0YmRihGMEYoRigqJkYlRigpRjBGLEYoRigqJkYsRihGK0YoRi4qKCIiKUYoRiVGKEYtRihGKCooRidGKEYlRihGMEYoRi4qKEYsRihGLUYoRjBGKEYoKihGLEYoRjNGKEYwRihGLiomRixGKEY1RihGLkYlRigqJkYqRihGLUYoRi4qJkYsRihGMEYoRig=

Checking n:=10: normal(series(subs(Upbaa1=subs(x=1,Upbaaser(n)), Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),alg2),t,n+1));

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In the third equation of the system, there is one more series that depends on x, namely Upbaa indets(eqs[3]);

PClJI1VwRzYiSSRVcDFHRiRJJVVwYmFHRiRJJlVwYmFhR0YkSSdVcGJhYTFHRiRJInRHRiRJInhHRiQ=

collect(eqs[3],[Upba,Upbaa],factor);

LCgqLEkidEc2IiIiIkkieEdGJUYmLCZGJ0YmRiYhIiJGJiwmRiZGKUkkVXAxR0YlRiZGJkklVXBiYUdGJUYmRikqKCwsKiZGK0YmRidGJkYmKiZGJEYmRidGJkYmRitGKUYnRilGJkYmRiYsJkYmRilJI1VwR0YlRiZGJkkmVXBiYWFHRiVGJkYpKipJJ1VwYmFhMUdGJUYmRiRGJilGJyIiI0YmRjFGJkYm

Same kernel as before factor(subs(x=X,Up=UpX,coeff(%,Upbaa))/ker);

LCYiIiJGI0kkVXBYRzYiISIi

We apply to this one the kernel method, by cancelling the coeff of Upbaa rhs3:=factor(subs(x=Xsol,Upbaa=0, Up=UpX,Upba=UpbaX,eqs[3]));

KipJInRHNiIiIiIpLCZGJSEiIkkkVXAxR0YkRiUiIiNGJSwoKiZJJFVwWEdGJEYlSSdVcGJhYTFHRiRGJUYlKiZJJlVwYmFYR0YkRiVGI0YlRiVGLkYoRiUpLChGKUYlRiNGJUYlRihGKkYo

UpbaXsol:=factor(solve(rhs3,UpbaX));

LCQqKEkidEc2IiEiIkknVXBiYWExR0YlIiIiLCZGKEYmSSRVcFhHRiVGKEYoRiY=

We need another expression of UpbaX : we differentiate the second equation collect(eqs[2],[Upba,Upbaa],factor);

LCYqKiIiIyIiIiwsKiZJJFVwMUc2IkYlSSJ4R0YpRiVGJSomSSJ0R0YpRiVGKkYlRiVGKCEiIkYqRi1GJUYlRiUsJkYlRi1JI1VwR0YpRiVGJUklVXBiYUdGKUYlRi0qKEYsRiVGKkYlLDIqKkYkRiVGL0YlSSZVcGJhMUdGKUYlRipGJUYlKiZGL0YlRipGJUYtRidGLSooRiRGJUY0RiVGKkYlRi1GL0YlRihGJSomRiRGJUYqRiVGJUYkRi1GJUYl

diff(subs(Up=Up(x),Upba=Upba(x),eqs[2]),x);indets(%);

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

PCpJJFVwMUc2IkkmVXBiYTFHRiRJInRHRiRJInhHRiQtSSNVcEdGJDYjRictSSVVcGJhR0YkRiotSSVkaWZmRyUqcHJvdGVjdGVkRzYkRihGJy1GLjYkRitGJw==

op(4,numer(factor(subs(diff(Up(x), x) =UppX,Up(x)=UpXsol,diff(Upba(x), x) =0,Upba(x)=UpbaXsol,x=Xsol,UpX=UpXsol,%%))));

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

indets(%);

PCdJJFVwMUc2IkkmVXBiYTFHRiRJJ1VwYmFhMUdGJEklVXBwWEdGJEkidEdGJA==

UppXsol:=factor(solve(%%,UppX));

LCQqLCIiIyIiIiwmRiUhIiJJJFVwMUc2IkYlRiUpLChGKEYlSSJ0R0YpRiVGJUYnRiRGJSwuKihGJEYlRihGJSlJJlVwYmExR0YpRiRGJUYlKiZGKEYlSSdVcGJhYTFHRilGJUYnKiZGJEYlRi9GJUYnKiZGMEYlRixGJUYlKiZGMkYlRixGJUYnRjJGJUYlKSwoKihGJEYlRihGJUYwRiVGJSomRiRGJUYwRiVGJ0YsRiUiIiRGJ0Yl

subs(Up1=A1,Upba1=B1,Upbaa1=C1,X=X,UppXsol);

LCQqLCIiIyIiIiwmRiUhIiJJI0ExRzYiRiVGJSksKEYoRiVJInRHRilGJUYlRidGJEYlLC4qKEYkRiVGKEYlKUkjQjFHRilGJEYlRiUqJkYoRiVJI0MxR0YpRiVGJyomRiRGJUYvRiVGJyomRjBGJUYsRiVGJSomRjJGJUYsRiVGJ0YyRiVGJSksKCooRiRGJUYoRiVGMEYlRiUqJkYkRiVGMEYlRidGLEYlIiIkRidGJQ==

collect(2*A1*B1^2-A1*C1-2*B1^2+B1*t-C1*t+C1,[C1,B1],factor);

LCgqJiwoSSNBMUc2IiEiIkkidEdGJkYnIiIiRilGKUkjQzFHRiZGKUYpKiYsJiomIiIjRilGJUYpRilGLkYnRikpSSNCMUdGJkYuRilGKSomRjBGKUYoRilGKQ==

We use in the first equation our expression of U'(X) alg3:=op(2,numer(factor(subs(diff(Up(x), x)=UppXsol, Up(x)=UpXsol,x=Xsol,diff(subs(Up=Up(x),eqs[1]),x)))));

LGp5KigiIiUiIiIpSSRVcDFHNiIiIidGJSlJJlVwYmExR0YoIiIjRiVGJSooIiM7RiUpRiciIiZGJSlGKyIiJEYlISIiKipGLkYlKUYnRiRGJUYxRiVJInRHRihGJUYlKixGLkYlRjVGJUYqRiVJJ1VwYmFhMUdGKEYlRjZGJUYlKioiIilGJUY1RiVGKkYlKUY2RixGJUYlKipGOkYlKUYnRjJGJUYxRiVGO0YlRiUqLEY6RiVGPUYlRipGJUY4RiVGO0YlRjMqKEYsRiVGJkYlRjhGJUYzKihGLkYlRi9GJUYqRiVGMyoqRilGJUYvRiVGOEYlRjZGJUYzKigiIyEpRiVGNUYlRjFGJUYlKioiIzdGJUY1RiVGKkYlRjZGJUYzKixGJEYlRjVGJUYrRiVGOEYlRjZGJUYzKioiIzVGJUY1RiVGK0YlRjtGJUYzKipGJEYlRjVGJSlGOEYsRiVGNkYlRjMqKkY6RiVGNUYlRjhGJUY7RiVGMyoqIiNrRiVGPUYlRjFGJUY2RiVGMyosRk1GJUY9RiVGKkYlRjhGJUY2RiVGMyosRiRGJUY9RiVGK0YlRjhGJUY7RiVGJSoqRixGJUY9RiVGK0YlKUY2RjJGJUYzKipGOkYlRj1GJUZKRiVGO0YlRjMqKkYpRiVGPUYlRjhGJUZRRiVGMyoqIiNDRiUpRidGLEYlRjFGJUY7RiVGMyosRlVGJUZWRiVGKkYlRjhGJUY7RiVGJSoqRi5GJUZWRiVGKkYlRlFGJUYlKixGLkYlRlZGJUYrRiVGOEYlRlFGJUYzKipGJEYlRlZGJUYrRiUpRjZGJEYlRjMqKkYkRiVGVkYlRkpGJUZRRiVGMyoqRixGJUZWRiVGOEYlRmVuRiVGMyooRkhGJUYvRiVGOEYlRiUqKCIjP0YlRjVGJUYqRiVGJSoqRilGJUY1RiVGK0YlRjZGJUYlKipGVUYlRjVGJUY4RiVGNkYlRiUqJkY1RiVGO0YlRjMqKCIkZyJGJUY9RiVGMUYlRjMqKiIjW0YlRj1GJUYqRiVGNkYlRiUqLEYuRiVGPUYlRitGJUY4RiVGNkYlRiUqKiIjT0YlRj1GJUYrRiVGO0YlRiUqKkYuRiVGPUYlRkpGJUY2RiVGJSoqRlVGJUY9RiVGOEYlRjtGJUYlKihGMEYlRj1GJUZRRiVGMyoqIiMnKkYlRlZGJUYxRiVGNkYlRiUqLEZpb0YlRlZGJUYqRiVGOEYlRjZGJUYlKipGYW9GJUZWRiVGKkYlRjtGJUYzKixGRUYlRlZGJUYrRiVGOEYlRjtGJUYzKipGVUYlRlZGJUYrRiVGUUYlRiUqKkZVRiVGVkYlRkpGJUY7RiVGJSoqIiM5RiVGVkYlRjhGJUZRRiVGJSooRjJGJUZWRiVGZW5GJUYzKipGVUYlRidGJUYxRiVGO0YlRiUqLEZVRiVGJ0YlRipGJUY4RiVGO0YlRjMqKiIjS0YlRidGJUYqRiVGUUYlRjMqLEZlcEYlRidGJUYrRiVGOEYlRlFGJUYlKioiIz1GJUYnRiVGK0YlRmVuRiVGJSoqRjpGJUYnRiVGSkYlRlFGJUYlKipGLEYlRidGJUY4RiVGZW5GJUYzKihGLEYlRidGJSlGNkYwRiVGMyooRmpuRiVGNUYlRjhGJUYzKipGVUYlRj1GJUYrRiVGNkYlRjMqKkZkb0YlRj1GJUY4RiVGNkYlRjMqKEYwRiVGPUYlRjtGJUYlKihGX29GJUZWRiVGMUYlRiUqKiIjc0YlRlZGJUYqRiVGNkYlRjMqLEZVRiVGVkYlRitGJUY4RiVGNkYlRjMqKkZhb0YlRlZGJUYrRiVGO0YlRjMqKkZVRiVGVkYlRkpGJUY2RiVGMyoqRlVGJUZWRiVGOEYlRjtGJUYzKihGYHBGJUZWRiVGUUYlRiUqKkZNRiVGJ0YlRjFGJUY2RiVGMyosRk1GJUYnRiVGKkYlRjhGJUY2RiVGMyoqRk1GJUYnRiVGKkYlRjtGJUYlKixGRUYlRidGJUYrRiVGOEYlRjtGJUYlKioiI1VGJUYnRiVGK0YlRlFGJUYzKipGVUYlRidGJUZKRiVGO0YlRjMqKkZIRiVGJ0YlRjhGJUZRRiVGMyooIiIqRiVGJ0YlRmVuRiVGJSooRjpGJUYxRiVGO0YlRjMqKkY6RiVGKkYlRjhGJUY7RiVGJSooRi5GJUYqRiVGUUYlRiUqKkYuRiVGK0YlRjhGJUZRRiVGMyooRmBwRiVGK0YlRmVuRiVGMyooRiRGJUZKRiVGUUYlRjMqKEYkRiVGOEYlRmVuRiVGJSomRiRGJUZccUYlRiUqKEZqbkYlRj1GJUY4RiVGJSooRmpuRiVGVkYlRipGJUYzKipGZG9GJUZWRiVGK0YlRjZGJUYlKipGVUYlRlZGJUY4RiVGNkYlRiUqKEZickYlRlZGJUY7RiVGMyooRkNGJUYnRiVGMUYlRjMqKkZhb0YlRidGJUYqRiVGNkYlRiUqLEYuRiVGJ0YlRitGJUY4RiVGNkYlRiUqKiIjR0YlRidGJUYrRiVGO0YlRiUqKkYuRiVGJ0YlRkpGJUY2RiVGJSoqRjpGJUYnRiVGOEYlRjtGJUYlKigiIzhGJUYnRiVGUUYlRjMqKEYuRiVGMUYlRjZGJUYlKipGLkYlRipGJUY4RiVGNkYlRiUqKEZVRiVGKkYlRjtGJUYzKipGJEYlRitGJUY4RiVGO0YlRjMqKEZqbkYlRitGJUZRRiVGJSooRjpGJUZKRiVGO0YlRiUqKEYsRiVGOEYlRlFGJUYlKiZGKUYlRmVuRiVGMyooRkhGJUZWRiVGOEYlRjMqKEYuRiVGJ0YlRipGJUYlKipGVUYlRidGJUYrRiVGNkYlRjMqKkYpRiVGJ0YlRjhGJUY2RiVGMyooIiIoRiVGJ0YlRjtGJUYlKiZGLkYlRjFGJUYlKihGRUYlRipGJUY2RiVGMyoqRiRGJUYrRiVGOEYlRjZGJUYzKihGKUYlRitGJUY7RiVGMyooRiRGJUZKRiVGNkYlRjMqJkYkRiVGUUYlRiUqKEYsRiVGJ0YlRjhGJUYlKiZGJEYlRipGJUYzKihGKUYlRitGJUY2RiVGJSomRixGJUY7RiVGMw==

indets(alg3);

PCZJJFVwMUc2IkkmVXBiYTFHRiRJJ1VwYmFhMUdGJEkidEdGJA==

We now have three equations in the three series that do not depend on x indets(alg3*alg2*alg1);

PCZJJFVwMUc2IkkmVXBiYTFHRiRJJ1VwYmFhMUdGJEkidEdGJA==

Checking the last one n:=10: normal(series(subs(Upbaa1=subs(x=1,Upbaaser(n)), Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),alg3),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

degree(alg1,Up1);degree(alg2,Up1);degree(alg3,Up1); degree(alg1,Upba1);degree(alg2,Upba1);degree(alg3,Upba1); degree(alg1,Upbaa1);degree(alg2,Upbaa1);degree(alg3,Upbaa1);

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res:=factor(resultant(alg1,alg2,Upbaa1)): nops(res);

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seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NiciIiIiIiNGJCIkSSdGJA==

Nic8IjwjSSJ0RzYiPCRJJFVwMUdGJkYlPCVGKEkmVXBiYTFHRiZGJTwjRig=

n:=10: normal(series(subs( Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),op(3,res)),t,n+1));

KztJInRHNiIiIiIiIiEhIidGJSEiJCIiIyEjSyIiJCEkWyMiIiUhJT9AIiImISYlUT4iIichJyM0Jz0iIighKFdNJj0iIikhKSEzKCoqPSIiKiEqX05DKj4iIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzY=

n:=10: normal(series(subs( Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

eq1:=op(4,res): res:=factor(resultant(alg2,alg3,Upbaa1)): nops(res);

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seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

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Nig8IjwjSSJ0RzYiPCNJJFVwMUdGJjwkRihGJTwlRihJJlVwYmExR0YmRiVGKg==

Choosing the correct factor n:=10: normal(series(subs( Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),op(4,res)),t,n+1));

KzlJInRHNiIhIiIiIiEiIiQiIiIiIiciIiMiI01GJyIkVyMiIiUiJScqPiIiJiImQXgiRikiJyVbbSIiIigiKFEsaiIiIikiKUl1WjsiIioiKkU6IjM8IiM1

n:=10: normal(series(subs( Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),op(5,res)),t,n+1));

KzdJInRHNiIhIiIiIiIhIiMiIiQhIz8iIiUhJCU+IiImISVDPiIiJyEmIWY+IiIoISdrVD8iIikhKFEsPCMiIiohKXdWWEIiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiYiIzY=

n:=10: normal(series(subs( Up1=subs(x=1,Upser(n)), Upba1=subs(x=1,Upbaser(n)),op(6,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

eq2:=op(6,res):nops(%);

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indets(eq1);indets(eq2);

PCVJJFVwMUc2IkkmVXBiYTFHRiRJInRHRiQ=

PCVJJFVwMUc2IkkmVXBiYTFHRiRJInRHRiQ=

degree(eq1,Up1);degree(eq1,Upba1); degree(eq2,Up1);degree(eq2,Upba1);

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IiIm

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Final step res:=factor(resultant(eq1,eq2,Upba1)): nops(res);

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seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NiciIiJGIyIiIyIkKXBGJA==

Nic8IjwjSSJ0RzYiPCNJJFVwMUdGJjwkRihGJUYp

n:=6: normal(series(subs(Up1=subs(x=1,Upser(n)),op(4,res)),t,n+1)); normal(series(subs(Up1=subs(x=1,Upser(n)),op(5,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

KzNJInRHNiIiIiIiIiEhIzdGJSIjSSIiIyEjRyIiJCEkLiIiIiUhJXM1IiImISY7MyIiIictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIig=

n:=20:normal(series(subs(Up1=subs(x=1,Upser(n)),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

algUp1:=collect(op(4,res),Up1,factor): indets(%);

PCRJJFVwMUc2IkkidEdGJA==

degree(algUp1,Up1);

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save(algUp1,'algU1prime');

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Back to general orientations algU1:=collect(factor(numer(subs(Up1=1-1/U1,algUp1))),U1,factor):

factor(coeff(algU1,U1,28));

LCQqJiIlIz4pIiIiKUkidEc2IiIjRUYlRiU=

disc:=factor(discrim(algU1,U1)): nops(disc);

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seq(nops(op(i,disc)),i=1..nops(disc));

NikiIiIiIiMiI2BGJEYkRiRGJA==

seq(indets(op(i,disc)),i=1..nops(disc));

Nik8IjwjSSJ0RzYiRiRGJEYkRiRGJA==

There will be 6 candidates for the minimal polynomial of the radius, called d2, ..., d6 op(1,disc);

IUwvXlx3LjRpIm8mSFJRJnliRWoqSCc9dmI=

op(2,disc);

KiQpSSJ0RzYiIiRtJyIiIg==

d3:=op(3,disc): d4:=op(1,op(4,disc)): d5:=op(1,op(5,disc)): d6:=op(1,op(6,disc)): op(7,disc);

KiQpLCgqJClJInRHNiIiIiMiIiJGKiomIiInRipGJ0YqRipGKkYqRixGKg==

Looking at the roots fsolve(d3,t=1/16..1/10); fsolve(d4,t=1/16..1/10); fsolve(d5,t=1/16..1/10); fsolve(d6,t=1/16..1/10);1/%;

JCIrJ3AmM2t3ISM2

JCIrITMoeS84ISIp

Only one candidate for the radius. We need to look more closely at the branches (just for the fun) with(plots): implicitplot(algU1,t=0..1/10,U1=1..1.4,numpoints=10000);

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implicitplot(algU1,t=0.06..1/10,U1=1..1.35,numpoints=100000);

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Nature of the singularity ? We compute estimates for the exponent User:=proc(n) series(1/(1-Upser(n)),t,n+1): end: subs(x=1,User(10));

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ser:=subs(x=1,series(User(100),t,101)): liste:=seriestolist(ser): plot([seq([1/n,n^2*(1-liste[n+1]*liste[n-1]/liste [n]^2)],n=5..nops(liste)-1)],x=0..0.10,y=-2.51..-2.45,style=point);

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

JSFH

read('syst_sup_prime_3'): subs(x=1,User(10));

KztJInRHNiIiIiIiIiEiIiNGJSIjNUYnIiNtIiIkIiQvJiIiJSIlO1UiIiYiJj93JCIiJyInVUFOIiIoIihVTFUkIiIpIikzWEZNIiIqIiphL2ZeJEYoLUkiT0clKnByb3RlY3RlZEc2I0YlIiM2

ser:=subs(x=1,User(100)): with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],t,99): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,t=0.07..0.09);

JCIrXiJcT24oISM2

1/%;

JCIraDQ7LjghIik=

plot([seq([1/n,coeff(ser,t,n+1)/coeff(ser,t,n)],n=3..98)],x=0..0.3,y=9..13,style=point);

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NiI=

Comparing the estimates of the radius for i from 1 to 10 do res:=hermite_pade( [1,ser,ser^2],t,10*i-1):dis5:=factor(op(2,res)^2-4*op(1,res)*op(3,res)):fsolve(dis5,t=1/14..1/12): print(1/%); od:

JCIrYFdPNDchIik=

JCIrUWlUbTchIik=

JCIrI1FQT0kiISIp

JCIrUyhvSUkiISIp

JCIrKVxdSkkiISIp

JCIrImZmSkkiISIp

JCIrZjM7LjghIik=

JCIrJCo0Oy44ISIp

JCIrZTQ7LjghIik=

JCIraDQ7LjghIik=

Nature of the singularity ? We compute estimates for the exponent ser:=series(ser,t,101): liste:=seriestolist(ser): plot([seq([1/n,n^2*(1-liste[n+1]*liste[n-1]/liste [n]^2)],n=5..nops(liste)-1)],x=0..0.10,style=point);

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

JSFH JSFH

read('syst_sup_prime_4'): subs(x=1,User(12));

Kz9JInRHNiIiIiIiIiEiIiNGJSIjNUYnIiNtIiIkIiQvJiIiJSIlO1UiIiYiJlt2JCIiJyInUTBOIiIoIigpKVtSJCIiKSIpZSwnUSQiIioiKjlqJmZNRigiK10uZDFPIiM2IixZVlFYI1EiIzctSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzg=

ser:=subs(x=1,User(100)): with(numapprox):with(gfun): res:=hermite_pade( [1,ser,ser^2],t,99): dis:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis,t=0.07..0.09);

JCIrKFE+PW8oISM2

1/%;

JCIrV1x4LDghIik=

plot([seq([1/n,coeff(ser,t,n+1)/coeff(ser,t,n)],n=3..98)],x=0..0.3,y=9..13,style=point);

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NiI=

for i from 2 to 10 do res:=hermite_pade( [1,ser,ser^2],t,10*i-1):dis5:=factor(op(2,res)^2-4*op(1,res)*op(3,res)):fsolve(dis5,t=1/13.5..1/12.5): print(1/%); od:

JCIrKnkyNUkiISIp

JCIrUV1XJ0giISIp

JCIrRWlDLDghIik=

JCIrXEkhPUkiISIp

JCIrUGt5LDghIik=

JCIrVT94LDghIik=

JCIrInB1PEkiISIp

JCIrI0d2PEkiISIp

JCIrV1x4LDghIik=

Nature of the singularity ? We compute estimates for the exponent ser:=series(ser,t,101): liste:=seriestolist(ser): plot([seq([1/n,n^2*(1-liste[n+1]*liste[n-1]/liste [n]^2)],n=5..nops(liste)-1)],x=0..0.10,style=point);

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NiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

read('syst_sup_prime_5'): subs(x=1,User(12));

Kz9JInRHNiIiIiIiIiEiIiNGJSIjNUYnIiNtIiIkIiQvJiIiJSIlO1UiIiYiJlt2JCIiJyInITRdJCIiKCIoK01RJCIiKSIpSWlsTCIiKiIqTSRSR01GKCIrNUdbaU4iIzYiLDdmU1x3JCIjNy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjOA==

ser5:=subs(x=1,User(100)): with(numapprox): res:=hermite_pade( [1,ser5,ser5^2],t,99): dis5:=factor(op(2,res)^2-4*op(1,res)*op(3,res)): fsolve(dis5,t=1/14..1/12);

JCIrbUsiKSlvKCEjNg==

1/%;

JCIrRTNmKzghIik=

plot([seq([1/n,coeff(ser5,t,n+1)/coeff(ser5,t,n)],n=3..98)],x=0..0.2,y=10..13,style=point);

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NiI=

for i from 2 to 10 do res:=hermite_pade( [1,ser5,ser5^2],t,10*i-1):dis5:=factor(op(2,res)^2-4*op(1,res)*op(3,res)):fsolve(dis5,t=1/14..1/12): print(map(Z->1/Z,[%])); od:

NyMkIitzMFQmRyIhIik=

NyMkIitqailwSCIhIik=

NyMkIis9czEsOCEiKQ==

NyMkIisxM04rOCEiKQ==

NyMkIitDd2IrOCEiKQ==

NyMkIit2LmcrOCEiKQ==

NyMkIispKXBlKzghIik=

NyUkIismPSRmKzghIikkIitFQnckSCJGJUYm

NyMkIitFM2YrOCEiKQ==

Nature of the singularity ? We compute estimates for the exponent ser:=series(ser,t,101): liste:=seriestolist(ser): plot([seq([1/n,n^2*(1-liste[n+1]*liste[n-1]/liste [n]^2)],n=5..nops(liste)-1)],x=0..0.10,y=-2.65..-2.45,style=point);

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Ig==

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