`loaded new kovacicsols`

Tools for the proofs of results in Hoeij & Weil, Issac'05restart: with(DEtools): _Envdiffopdomain:=[Dx,x]: with(diffalg):

Warning, the name singularities has been redefined

`loaded new kovacicsols`

## Change of variable in a differential operator. ## May also use the function Dchangevar pullback := proc(L, a) local i, f; global x, Dx; f := add( mult(subs(x = a, coeff(L, Dx, i)), Dx/diff(a, x) $ i) , i = 0 .. degree(L, Dx)); sort(collect(f/lcoeff(f, Dx), Dx, X->factor(simplify(X,symbolic))), Dx) end:## singularities of a differential operator singularites:=proc(L) local co, liste; co := lcoeff(collect(numer(normal(L)), Dx), Dx); liste := select(X -> has(X, x), factors(co)[2]); [seq(RootOf(i[1], x), i = liste), infinity] end proc:## local (generalized) exponents localex:=proc(L) seq(gen_exp(L,_t,x=p),p=singularites(L)) end proc:

Dn:=Dx^2+x*Dx/((x-1)*(x+1))-1/4/((x-1)*n^2*(x+1));

NiM+SSNEbkc2IiwoKiRJI0R4R0YlIiIjIiIiKipJInhHRiVGKkYoRiosJkYsRiohIiJGKkYuLCZGLEYqRipGKkYuRioqKEYtRi5JIm5HRiUhIiNGL0YuI0YuIiIl

## special case for quaternion group D2 Q:=eval(Dn,n=2); SQ:= symmetric_product(Q,Dx-coeff(Q,Dx,1)/2); r1:=4*x/(x-1)/(x+1); ## logarithmic derivative of the invariant of degree 6 D2I:=symmetric_product(SQ,Dx+r1/6);

NiM+SSJRRzYiLCgqJEkjRHhHRiUiIiMiIiIqKkkieEdGJUYqRihGKiwmRixGKiEiIkYqRi4sJkYsRipGKkYqRi5GKiomRi1GLkYvRi4jRi4iIzs=

NiM+SSNTUUc2IiwmKiRJI0R4R0YlIiIjIiIiKigsJiokSSJ4R0YlRilGKiIiJEYqRiosJkYuRiohIiJGKiEiIywmRi5GKkYqRipGMiNGLyIjOw==NiM+SSNyMUc2IiwkKihJInhHRiUiIiIsJkYoRikhIiJGKUYrLCZGKEYpRilGKUYrIiIlNiM+SSREMklHNiIsKCokSSNEeEdGJSIiIyIiIioqSSJ4R0YlRiosJkYsRiohIiJGKkYuLCZGLEYqRipGKkYuRihGKiMiIiUiIiQqKCwmKiRGLEYpRipGMkYqRipGLSEiI0YvRjYjISImIiRXIg==

coeff(symmetric_power(D2I,6),Dx,0);

NiMiIiE=

symmetric_product(SQ,Dx+r1/8);

NiMsKCokSSNEeEc2IiIiIyIiIioqSSJ4R0YmRigsJkYqRighIiJGKEYsLCZGKkYoRihGKEYsRiVGKEYoKiZGK0YsRi1GLCNGLCIjOw==

s:=(6*lambda-1)*(6*lambda+1)/144; St:=pullback( Dx^2+(7*x-4)/6/x/(x-1)*Dx-s/x/(x-1), 1/(x+1) ); St_A4:=subs(lambda=1/3,St); St_S4:=subs(lambda=1/4,St); St_A5:=subs(lambda=1/5,St); ##### former standard equations: A4S:=St_A4: S4S:=St_S4: A5S:=St_A5: A5I:=A5S:

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b6:=1/2/((x+1)*x): L:=symmetric_product(St_A4,Dx+b6/6); ## better choice 1 (gives nicer formula). A4I:=pullback(L,-2*x/(x+1)); verif:=coeff(symmetric_power(A4I,6),Dx,0);

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NiM+SSZ2ZXJpZkc2IiIiIQ==

## other choice 2: normalise the invariant of degree 8 to 1 so that ## the standard equation is a simple pullback of S4I. Gives nicer formula b8:=-1/3/(x+1); L:=symmetric_product(St_A4,Dx+b8/8); ## better choice (makes it a pullback of S4I) A4I8:=pullback(L,-2/(x+1)); verif:=coeff(symmetric_power(A4I8,8),Dx,0);

NiM+SSNiOEc2IiwkKiQsJkkieEdGJSIiIkYqRiohIiIjRisiIiQ=

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NiM+SSZ2ZXJpZkc2IiIiIQ==

b8:=-1/24/(x+1); L:=symmetric_product(St_S4,Dx+b8); ## better choice S4I:=pullback(L,x/(-x+1));

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sol_DnI:=hypergeomsols(diffop2de(Dn,y(x)),y(x)); RiemannPsols(diffop2de(Dn,y(x)),y(x));

NiM+SShzb2xfRG5JRzYiNyQtSSpoeXBlcmdlb21HSShfc3lzbGliR0YlNiU3JCwkKiRJIm5HRiUhIiIjIiIiIiIjLCRGLSNGL0YyNyNGMCwmRjBGMUkieEdGJUYwKiYsJkYyRjFGN0YyRjAtRig2JTckLCQqJiwmRi9GMUYuRjFGMUYuRi9GMCwkKiYsJkYxRjFGLkYxRjFGLkYvRjA3IyMiIiRGMkY2RjE=

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sol_D2S:=hypergeomsols(diffop2de(Q,y(x)),y(x)); RiemannPsols(diffop2de(Q,y(x)),y(x));

NiM+SShzb2xfRDJTRzYiNyQtSSpoeXBlcmdlb21HSShfc3lzbGliR0YlNiU3JCMhIiIiIiUjIiIiRi43IyNGMCIiIywmRjJGMEkieEdGJUYyKiYsJkYzRjBGNUYzRjItRig2JTckRi8jIiIkRi43IyNGPEYzRjRGMA==

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hypergeomsols(diffop2de(D2I,y(x)),y(x)); sol_D2I:=RiemannPsols(diffop2de(D2I,y(x)),y(x)) ;

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lprint(sol_S4I);

sol_S4I

eq:=diffop2de(A4I,y(x)): hypergeomsols(eq,y(x)); sol_A4I:=RiemannPsols(eq,y(x)); dsolve(eq,[hyper3]);

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latex(sol_A4I);

[{\it hypergeom} \left( [-1/12,1/4],[1/2],2\,{\frac {x}{x-1}} \right) \left( {\frac {x}{x-1}} \right) ^{-1/12},\sqrt {2} \left( {\frac {x}{ x-1}} \right) ^{{\frac {5}{12}}}{\it hypergeom} \left( [{\frac {5}{12} },3/4],[3/2],2\,{\frac {x}{x-1}} \right) ]

sol_S4I:=hypergeomsols(diffop2de(S4I,y(x)),y(x)); sol_S4I:=RiemannPsols(diffop2de(S4I,y(x)),y(x));

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hypergeomsols(diffop2de(A5I,y(x)),y(x)); sol_A5I:=RiemannPsols(diffop2de(A5I,y(x)),y(x));

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sol_A4S:=hypergeomsols(diffop2de(A4S,y(x)),y(x)); RiemannPsols(diffop2de(A4S,y(x)),y(x));

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sol_S4S:=hypergeomsols(diffop2de(S4S,y(x)),y(x)); RiemannPsols(diffop2de(S4S,y(x)),y(x));

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L:=S4I; PL:=pullback(L,F(x)): A0,A1:=coeff(PL,Dx,0),coeff(PL,Dx,1); sys:=map(numer,{A1-a1(x),A0-a0(x)}): R := differential_ring(ranking=[F,a1,a0], derivations=[x], notation=diff): GE:= Rosenfeld_Groebner(sys, R); EQ:=rewrite_rules(GE[1]); ## simplification: may set G=2a1+diff(a0)/a0 to simplify the formulas. the_f:=normal(subs(diff(a0(x),x)=G(x)*a0(x)-2*a1(x)*a0(x) , rhs(rewrite_rules(GE[1])[1]) )): pullback_S4I:=collect(the_f,G(x));

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L:=A4I; PL:=pullback(L,F(x)): A0,A1:=coeff(PL,Dx,0),coeff(PL,Dx,1); sys:=map(numer,{A1-a1(x),A0-a0(x)}): R := differential_ring(ranking=[F,a1,a0], derivations=[x], notation=diff): GE:= Rosenfeld_Groebner(sys, R); rewrite_rules(GE[1]); ## simplification: may set G=2a1+diff(a0)/a0 to simplify the formulas. the_pol:=collect(normal(subs(diff(a0(x),x)=G(x)*a0(x)-2*a1(x)*a0(x) , rhs(rewrite_rules(GE[1])[1]) )),F(x),X->collect(X,G(x)) ); PF:=F(x)^2-the_pol; pullback_A4I:=sqrt(the_pol);

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#St:=Dx^2+1/6*(8*x+3)/(x+1)/x*Dx+_s/(x+1)^2/x ; L:=pullback(Dx^2+(7*x-4)/6/x/(x-1)*Dx-_s/x/(x-1), 1/(x+1) ); PSt:=pullback(L,F(x)): A0,A1:=coeff(PSt,Dx,0) , collect(coeff(PSt,Dx,1),diff(F(x),x$2),factor); K := field_extension(transcendental_elements=[_s]): R := differential_ring(ranking=[F,a1,a0], derivations=[x],field_of_constants=K, notation=diff): sys:=map(numer,{A1-a1(x),A0-a0(x)}): GE:= Rosenfeld_Groebner(sys, R); rewrite_rules(GE[1]); ## result letting G=A0'/A0+2*A1; pullback_formula:=F(x)=factor(normal( subs(a1(x)=1/2*(G(x)-diff(a0(x),x)/a0(x) ) , rhs(rewrite_rules(GE[1])[1]))));

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NiM+SSNHRUc2IjcmSTBjaGFyYWN0ZXJpemFibGVHRiVGJ0YnRic=NiM3JC8tSSJGRzYiNiNJInhHRicsJCooSSNfc0dJKnByb3RlY3RlZEdGLSIiIiwoKiYtSSNhMUdGJ0YoIiIjLUkjYTBHRidGKEYzIiIlKiQtSSVkaWZmR0YtNiRGNEYpRjNGLiooRjhGLkYxRi5GNEYuRjZGLkY0ISIkIiIqLy1GOTYkRjFGKSwkKigsKiokRjRGM0YuKihGMUYzRixGLkY0Ri4iI08qKEY4Ri5GLEYuRjFGLiIjSSomLUY5NiRGNC1JIiRHRi02JEYpRjNGLkYsRi4iIidGLkYsISIiRjRGUCNGUCIjNw==NiM+STFwdWxsYmFja19mb3JtdWxhRzYiLy1JIkZHRiU2I0kieEdGJSwkKihJI19zR0kqcHJvdGVjdGVkR0YuIiIiLUkjYTBHRiVGKSEiIi1JIkdHRiVGKSIiIyIiKg==

pullback_A4S:=rhs(subs(_s=s,lambda=1/3,pullback_formula)); pullback_S4S:=rhs(subs(_s=s,lambda=1/4,pullback_formula)); pullback_A5S:=rhs(subs(_s=s,lambda=1/5,pullback_formula)); pullback_A5I:=pullback_A5S:

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solution_A4I:=eval(sol_A4I,x=pullback_A4I);

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solution_S4I:=eval(sol_S4I,x=pullback_S4I);

NiM+SS1zb2x1dGlvbl9TNElHNiI3JComLUkqaHlwZXJnZW9tR0koX3N5c2xpYkdGJTYlNyQjISIiIiNDIyIiJkYvNyMjIiIiIiIjLCQqJi1JIkdHRiU2I0kieEdGJUY1LUkjYTBHRiVGOkYuIyEiKCIkVyJGNCwmRjdGPkYuRjRGLSooRjZGMy1GKTYlNyQjIiM2Ri8jIiM8Ri83IyMiIiRGNUY2RjRGQUYt

solution_A5I:=eval(sol_A5I,x=pullback_A5I);

NiM+SS1zb2x1dGlvbl9BNUlHNiI3JComLUkqaHlwZXJnZW9tR0koX3N5c2xpYkdGJTYlNyQjISIiIiNnIyIjPkYvNyMjIiIiIiIjLCQqJi1JIkdHRiU2I0kieEdGJUY1LUkjYTBHRiVGOkYuIyEjNiIkKyVGNCwmRjcjIiM2RkBGNEY0Ri0qKEY2RjMtRik2JTckIyIjSEYvIyIjXEYvNyMjIiIkRjVGNkY0RkFGLQ==

solution_A4S:=eval(sol_A4S,x=pullback_A4S);

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solution_S4S:=eval(sol_S4S,x=pullback_S4S);

NiM+SS1zb2x1dGlvbl9TNFNHNiI3JCwkKiwsJiomLUkiR0dGJTYjSSJ4R0YlIiIjLUkjYTBHRiVGLSEiIiMiIiYiI2siIiJGNiNGNCIjQ0Y0I0Y2Ri9GNUY5RipGOS1JKmh5cGVyZ2VvbUdJKF9zeXNsaWJHRiU2JTckIyIjPEY4IyIjREY4NyMjIiIkRi8sJEYqIyEiJkY1RjYjRjZGNSomRilGNy1GOzYlNyRGNyMiIzhGODcjRjlGRkY2

This is the naive version, showing all steps of the computation. Of course, it is much easier to compute the pullback from D2I to A4I directly !Dn:=Dx^2+x*Dx/((x-1)*(x+1))-1/4/((x-1)*n^2*(x+1)): Q:=eval(Dn,n=2); localex(A4S); ## check local exponents to see which transform to apply b1:=1/12/(x+1)-1/12/x; L1:=symmetric_product(A4S,Dx-b1); localex(L1); ## check local exponents ## arrange exponents are 0,1/3 at zero and infinity so they will become ordinary: L2:=pullback(L1,x-1); ## L3:=pullback(L2,x^3); localex(L3); ## shows that this is a standard equations with group D_2. alias(J=RootOf(_Z^2+_Z+1)): ## rearrange the singular points to get closer to desired equation. p3:=(x+1+2*J)/(x-1-2*J); ## sends to 1,1,inf ; inverse: (2*J+1)*(1+y)/(-1+y) L4:=pullback(L3,p3): L4:=collect(L4,Dx,X->factor(evala(expand(X))));; localex(L4); b4:=1/12/(x-1)+1/12/(x+1); L5:=symmetric_product(L4,Dx-b4); ## this is the standard equation for D2. ## but the inversion of the above steps gives an ugly formula so let's not use it

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NiU3JDckIyEiIiIjNy9JI190RzYiLCZJInhHRioiIiJGLUYtNyQjRi0iIiVGKDckNyQiIiEvRilGLDckI0YtIiIjRjQ3JDckRjMvRikqJEYsRiY3JCNGLSIiJEY6NiM+SSNiMUc2IiwmKiQsJkkieEdGJSIiIkYqRiohIiIjRioiIzcqJEYpRisjRitGLQ==NiM+SSNMMUc2IiwoKiRJI0R4R0YlIiIjIiIiKiosJkkieEdGJUYpRipGKkYqLCZGLUYqRipGKiEiIkYtRi9GKEYqI0YpIiIkKiZGLkYvRi0hIiMjISImIiRXIg==

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NiM+SSNMNEc2IiwoKiRJI0R4R0YlIiIjIiIiKipJInhHRiVGKiwmRixGKiEiIkYqRi4sJkYsRipGKkYqRi5GKEYqIyIiJSIiJCooLCYqJEYsRilGKkYyRipGKkYtISIjRi9GNiMhIiYiJFci

NiU3JDckIyIiJiIjNy9JI190RzYiLCZJInhHRioiIiIhIiJGLTckI0YuRidGKDckNyRGJS9GKSwmRixGLUYtRi03JEYwRjM3JDckRiUvRikqJEYsRi43JEYwRjg=NiM+SSNiNEc2IiwmKiQsJkkieEdGJSIiIiEiIkYqRisjRioiIzcqJCwmRilGKkYqRipGK0Ys

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The pullback formula from D_2^i to A_4^ip:=eval(3*sqrt(-3)*(F^2-1)/(F^3-9*F),F=x);

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verif:=collect(D2I-pullback(A4I,p),Dx,normal);

NiM+SSZ2ZXJpZkc2IiIiIQ==

latex(3*sqrt(-3)*(F^2-1)/(F^3-9*F));

{\frac {3\,i\sqrt {3} \left( {F}^{2}-1 \right) }{{F}^{3}-9\,F}}

D2I;

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## automorphisms of D2I (the other one is x->-x). p1:=(x+3)/(1-x); p2:=normal(eval(p1,x=p1)); p3:=normal(eval(p2,x=p1));

NiM+SSNwMUc2IiomLCZJInhHRiUiIiIiIiRGKUYpLCZGKUYpRighIiJGLA==NiM+SSNwMkc2IiomLCZJInhHRiUiIiIhIiRGKUYpLCZGKEYpRilGKSEiIg==NiM+SSNwM0c2IkkieEdGJQ==

verif1:=collect(pullback(D2I,p1)-D2I,Dx,normal); verif2:=collect(pullback(D2I,-x)-D2I,Dx,normal);

NiM+SSd2ZXJpZjFHNiIiIiE=NiM+SSd2ZXJpZjJHNiIiIiE=

P:=numer(p)-y*denom(p): P:=collect(P/lcoeff(P,x),x,factor); ## we may check that automorphisms of D2I preserve the pullback, of course: P1:=numer(normal(subs(x=p1,P))): verif1:= collect(P1/lcoeff(P1,x) - P,x,factor); P2:=numer(normal(subs(x=p2,P))): verif1:= collect(P2/lcoeff(P2,x) - P,x,factor);

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NiM+SSd2ZXJpZjFHNiIiIiE=

NiM+SSd2ZXJpZjFHNiIiIiE=