This is a Maple worksheet demonstrating
the quartic transformation of Painleve VI functions
and pullback transformations of corresponding
isomonodromic 2x2 Fuchsian systems,
supplementing the articleQuadratic transformations of the sixth Painleve equation
in terms of Riemann-Hilbert correspondence
by Marta Mazzocco (Loughborough University, UK)
and Raimundas Vidunas (Kobe University, Japan)submitted to the "Journal of ...",
available at http://arxiv.org/abs/1011.xxxx
Version: 2.0.0
Date: November 15, 2010
<Text-field style="Heading 1" layout="Heading 1">Routines</Text-field>with(linalg);This routine computes an RS-pullback transformation of a 2x2 matrix Fuchsian equation.
The starting Fuchsian equation is assumed to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYxNiVRI2R6RidGNEY3LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSYmUHNpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInpGJy9GNVEldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3Ris= = 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,
where F0 is the first argument (a 2x2 matrix), and z is the second argument.
The pullback covering is assumed to be 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,
where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is the thrid argument, and x is the fourth argument.
The inverse Schlesinger transformation is given by the last argument S,
which is either a 2x2 matrix, or a list [SC, SM],
where SC is an algebraic function of x
(presumably, 1 divided by a square root of a polynomial),
and SM is a 2x2 matrix (with entries polynomial in x, presumably).
In the latter case, the inverse Schlesinger transformation is the product
of SC (an algebraic or function function) and SM (a polynomial matrix).
The meaning of some internal variables:
the list [SCi, SMi] is used in the same way as [SC, SM],
and defines the direct Schlesinger transformation
(as a product of algebraic function SCi and polynomial matrix SMi).
In particular, the product SC SCi SM SMi is the identity matrix.
In fact, SMi is the adjoint matrix of SM, so we have
det(SM) det(SMi) = 1 and SC SCi det(SM) = 1.RSpullback:= proc( F0, z, Phi, x, S )
local FM, SC, SM, iSC, iSM, dPx,
dSC, diM, dIM, QM, DM;
FM:= map(normal, subs(z=Phi,evalm(F0)) );
dPx:= normal( diff(Phi,x) );
if type(S, matrix) then
SC:= 1; SM:= S
else
SC:= S[1]; SM:= S[2];
fi;
iSC:= 1/det(SM)/SC;
iSM:= adjoint(SM);
diM:= map(normal, map(diff,iSM,x) );
dSC:= normal( diff(iSC,x)/iSC );
dIM:= map(normal, evalm(diM+dSC*iSM) );
QM:= map(normal, evalm(FM&*iSM*dPx-dIM) );
QM:= map(normal, evalm(SM&*QM) );
QM:= map(normal, evalm(SC*iSC*QM) );
### presenting the output, as a matrix divided by a denominator
DM:= primpart( denom(QM[1,1]) );
# map(collect, evalm(QM*DM), x)/collect(DM,x)
map(normal, evalm(QM*DM))/factor(DM)
end:
Given: a polynomial F in variable x, with coefficients dependant on parameter(s).
Returns: an expression of the polynomial F, with coefficients to x factored.FactorCollect:= proc( F, x )
local F0;
F0:= subs(x=0,normal(F));
map(factor, collect( normal(F-F0), x)) + factor(F0)
end:
This routine gives comparison of two matrices or list-vectors,
by displaying entrywise quotients of the corresponding entries (in a matrix form).CompareMatrices:= proc( AA, BB )
local A, B, nr, nc, entratio;
### Routine for handling quotient of 2 expressions
entratio:= proc(X,Y)
if Y=0 then
if X=0 then `0/0`
else infinity fi
else normal(X/Y) fi
end;
### If the input is vectors, convert to matrices
if type(AA,list) then
A:= matrix([AA]);
B:= matrix([BB]);
else
A:= evalm(AA); B:= evalm(BB)
fi;
### Building the output matrix of quotients
nr:= rowdim(A); nc:= coldim(A);
if nr=rowdim(B) and nc=coldim(B) then
matrix( [ seq( [
seq( entratio(A[i,j],B[i,j]), j=1..nc) ],
i=1..nr ) ] )
else
error "Dimensions of the arguments must be compatable"
fi;
end:
<Text-field style="Heading 1" layout="Heading 1">Transformations of Painleve VI</Text-field>The quartic folding transformation, as in Section 3.2 of
T. Tsuda, K. Okamoto, H. Sakai, "Folding transformations of the Painleve equations",
Mathematische Annalen 331, 713-738 (2005).QQ:= (q^2-t)^2/(4*q*(q-1)*(q-t));
PP:= 4*q*(q-1)*(q-t)*(4*q*(q-1)*(q-t)*p-(2*e-1/2)*(q*(q-1)+q*(q-t)+(q-1)*(q-t)))/((q^2-t)*(q^2-2*q+t)*(q^2-2*q*t+t));Here is the inverse transformation.q = (sqrt(Q)+sqrt(Q-1))*(sqrt(Q)+sqrt(Q-t));
evala( subs(%,QQ) );Here is the transformation scheme of relevant Painleve VI equations, including the intermediate quartic transformation.
The upper row represents the folding quadratic transformations (and their quartic composition);
the vertical columns represent Okamoto transformations, and
the lower row represents Kitaev's quadratic transformations (and an Okamoto transformation).
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, 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 , 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 ; Okamoto Okamoto Okamoto to 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,
or 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 .We denote by 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 a solution of 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,
and we denote by 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 a solution of 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.SE:= e=a/2-1/4;
SAI:= a=theta[infinity];Now we relate the solution 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 of 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
with the solution 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 of 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
by recaling first their relation to 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 and 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.SY:= q=y+(e-1/2)/p;
YE:= factor(subs(SY, QQ+(1/2-2*e)/PP));
PE:= factor(subs(SY, PP));Here we relate to the article aexpressions.L1:= (y*p+(2*a-3)/4)^2-t*p^2;
L2:= (y*p+(2*a-3)/4)*(y*p-(2*a-1)/4)-t*p^2;
L3:= (y*p+(2*a-3)/4)*(y*p-p+(2*a-3)/4)*(y*p-t*p+(2*a-3)/4);
L4:= ((y*p-p+(2*a-3)/4)^2+(t-1)*p^2)*((y*p-p*t+(2*a-3)/4)^2-t*(t-1)*p^2);
L5:= (y*p-a/4)*(y*p+(2*a-3)/4)^2-p*(t+1)*(y*p-1/4)*(y*p+(2*a-3)/4)+t*p^2*(y*p+(a-2)/4);
L6:= y*(y*p+(2*a-3)/4)^4-(y*p+(2*a-1)/4)*(y*p+(2*a-3)/4)^3*(t+1)
+t*p*((4*a-1)/2)*(y*p+(2*a-3)/4)^2+t*(t+1)*p^2*(y*p-(2*a+1)/4)*(y*p+(2*a-3)/4)-p^3*t^2*(y*p-1/2);The solution 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 once more.YA:= L1*L2/4/p/L5;
PA:= 16*p*L3*L5/L1/L4;
normal(YA-subs(SE,YE)),
normal(PA-subs(SE,PE));
This is a solution 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 of 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.YY:= L1*L6/4/L3/L5;
factor( subs(SY,SE,QQ+(1/2+2*e)/PP) - % );
<Text-field style="Heading 1" layout="Heading 1">The traceless Fuchsian system</Text-field>This is a general form of the traceless normalization of the isomonodromic Fuchsian system.FS:= D(Phi)/Phi=1/2/z*matrix( [[u[0], w[0]*(theta[0]-u[0])], [(theta[0]+u[0])/w[0], -u[0]]])+
1/2/(z-1)*matrix( [[u[1], w[1]*(theta[1]-u[1])], [(theta[1]+u[1])/w[1], -u[1]]])+
1/2/(z-t)*matrix( [[u[t], w[t]*(theta[t]-u[t])], [(theta[t]+u[t])/w[t], -u[t]]]);
FM:= evalm(op(2,FS));The substitutions to organize the upper-right entry according to the traceless condition, and the diagonal entrieas at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRKCZpbmZpbjtGJ0Y0RjdGKw==.SW:= w[0]= k*y/t/(u[0]-theta[0]), w[1]= -k*(y-1)/(t-1)/(u[1]-theta[1]),
w[t]= k*(y-t)/t/(t-1)/(u[t]-theta[t]), u[t] = -theta[infinity]-u[0]-u[1];
factor( subs(SW, [FM[1,1], FM[1,2]] ) );The normalization condition on the [2,1]-entry gives a quadratic relation between 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.
Here is an attractive parametrization of that condition.SU:= u[0]= ((s^2-2*theta[infinity]*t*s-theta[infinity]^2*t*y*(y-t-1))/(t*(y-1)*(y-t))
-theta[0]^2+theta[1]^2*(t-1)*y/(t*(y-1))-theta[t]^2*(t-1)*y/(y-t))/(2*theta[infinity]),
u[1]= ((s^2-2*theta[infinity]*y*s+theta[infinity]^2*t*y*(y-t+1))/((1-t)*y*(y-t))
-theta[1]^2+theta[0]^2*t*(y-1)/((t-1)*y)+theta[t]^2*t*(y-1)/(y-t))/(2*theta[infinity]),
u[t]= ((s^2-2*s*t*y*theta[infinity]+theta[infinity]^2*t*y*(y+t-1))/(t*(t-1)*y*(y-1))
-theta[t]^2-theta[0]^2*(y-t)/((t-1)*y)+theta[1]^2*(y-t)/(t*(y-1)))/(2*theta[infinity]);The Fuchsian system matrix in the the parameters 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.FMM:= map(normal, subs(SW,SU,evalm(FM)) ):
Checking the size and the degree of the [2,1]-entry.normal( FMM[2,1] ):
length(%), degree(numer(%),z);The inverse expressions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==.SS:= s= t*u[0]*(y-1)+y*u[1]*(t-1)+t*y*theta[infinity];
factor(subs(SS, SU, s));The relation of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUYyNi1RIixGJ0Y1RjgvRjxGVkY9Rj9GQUZDRkVGRy9GS1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJzRidGVEZXRjVGKw== to the Hamiltonian coordinates 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.S2P:= y=q, s=-2*p*q*(q-1)*(q-t)+theta[0]*(q-1)*(q-t)+theta[1]*q*(q-t)+(theta[t]-theta[infinity])*q*(q-1)+t*q*theta[infinity];
P2S:= q=y, p=-(s-t*y*theta[infinity])/2/y/(y-1)/(y-t)+theta[0]/2/y+theta[1]/2/(y-1)+(theta[t]-theta[infinity])/2/(y-t);
factor(subs(S2P, P2S, s)), factor(subs(P2S, S2P, p));
<Text-field style="Heading 2" layout="Heading 2">The Hamiltonian formulation (for comparison)</Text-field>The Hamiltonian for the isomonodromic 2x2 Fuchsian system determined by a Painleve VI solutionHAM:= (q*(q-1)*(q-t)*p^2-(theta[0]*(q-1)*(q-t)+theta[1]*q*(q-t)+(theta[t]-1)*q*(q-1))*p
+(1-theta[0]-theta[1]-theta[t]-theta[infinity])/2*(1-theta[0]-theta[1]-theta[t]+theta[infinity])/2*(q-t))/(t*(t-1));The relation of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==.solve(Dq = diff(HAM, p), p):
HP:= convert(%, parfrac, q);FSS:= 1/z*matrix([[a011,a012],[a021,-a011]])+
1/(z-1)*matrix([[a111,a112],[a121,-a111]])+1/(z-t)*matrix([[at11,at12],[at21,-at11]]);SA:= a011= q^2*(q-1)*(q-t)*(p^2+p*((theta[infinity]-theta[0])/q-theta[1]/(q-1)-theta[t]/(q-t))
+((theta[0]^2+theta[infinity]^2)*(q-1-t)+theta[1]^2*(q+1-t)+theta[t]^2*(q-1+t))/(4*q*(q-1)*(q-t))
+(theta[0]*theta[1]*(q-t)+theta[0]*theta[t]*(q-1)+theta[1]*theta[t]*q)/(2*q*(q-1)*(q-t))
-theta[infinity]*(theta[0]/q+theta[1]/(q-1)+theta[t]/(q-t))/(2*q))/(theta[infinity]*t),
a111 = q*(q-1)^2*(q-t)*(p^2+p*((theta[infinity]-theta[1])/(q-1)-theta[0]/q-theta[t]/(q-t))
+(theta[0]^2*(q-1-t)+(theta[1]^2+theta[infinity]^2)*(q+1-t)+theta[t]^2*(q-1+t))/(4*q*(q-1)*(q-t))
+(theta[0]*theta[1]*(q-t)+theta[0]*theta[t]*(q-1)+theta[1]*theta[t]*q)/(2*q*(q-1)*(q-t))
-theta[infinity]*(theta[0]/q+theta[1]/(q-1)+theta[t]/(q-t))/(2*(q-1)))/(theta[infinity]*(1-t)),
at11 = q*(q-1)*(q-t)^2*(p^2+p*((theta[infinity]-theta[t])/(q-t)-theta[0]/q-theta[1]/(q-1))
+(theta[0]^2*(q-1-t)+theta[1]^2*(q+1-t)+(theta[t]^2+theta[infinity]^2)*(q-1+t))/(4*q*(q-1)*(q-t))
+(theta[0]*theta[1]*(q-t)+theta[0]*theta[t]*(q-1)+theta[1]*theta[t]*q)/(2*q*(q-1)*(q-t))
-theta[infinity]*(theta[0]/q+theta[1]/(q-1)+theta[t]/(q-t))/(2*(q-t)))/(theta[infinity]*t*(t-1)),
a012 = -k*q/2/t, a112 = k*(q-1)/2/(t-1), at12 = -k*(q-t)/2/t/(t-1);SAA:= a021= (theta[0]^2/4-a011^2)/a012, a121= (theta[1]^2/4-a111^2)/a112, at21= (theta[t]^2/4-at11^2)/at12;map(normal, subs(SAA,SA,P2S,evalm(FSS)) ):
CompareMatrices( FMM, %);
<Text-field style="Heading 1" layout="Heading 1">Pullback transformations</Text-field>The pullback covering is a Belyi covering of degree 4,
with ramified fibers 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.FX:= (x^2-t)^2/4/x/(x-1)/(x-t);
factor(FX-1), factor(FX-t);
ST:= theta[0]=-1/2, theta[1]=-1/2, theta[t]=-1/2, theta[infinity]=a;
<Text-field style="Heading 2" layout="Heading 2">Pullback to P6(a,a,a,a) </Text-field>The shape of the gauge transformation is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjUkRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYjUSFGJy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNictSSRtdGRHRiQ2KC1GMTYlUSRQMTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSlyb3dhbGlnbkdGMy8lLGNvbHVtbmFsaWduR0YzLyUrZ3JvdXBhbGlnbkdGMy8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZOLUY7NigtRjE2JVEkUDEyRidGQEZDRkZGSEZKRkxGT0ZGRkhGSi1GODYnLUY7NigtRjE2JVEkUDIxRidGQEZDRkZGSEZKRkxGTy1GOzYoLUYxNiVRJFAyMkYnRkBGQ0ZGRkhGSkZMRk9GRkZIRkovJSZhbGlnbkdRJWF4aXNGJy9GR1EpYmFzZWxpbmVGJy9GSVEnY2VudGVyRicvRktRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0ZCLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGaW8vJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGZHAvJSZmcmFtZUdGZHAvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR1EmZmFsc2VGJy8lLWVxdWFsY29sdW1uc0dGXnEvJS1kaXNwbGF5c3R5bGVHRl5xLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmFwRjAvRkRRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYn.
See the Derivation section for defining conditions and computation.RD:= 1/sqrt((x^2-t)*(x^2-2*x+t)*(x^2-2*x*t+t));
P11:= (w[0]-w[1])*(w[0]-w[t])*(w[1]-w[t])*x^3
+(w[t]*(w[0]-w[1])*(w[0]+w[1]-2*w[t])*t-w[1]*(w[0]-w[t])*(w[0]+w[t]-2*w[1]))*x^2
-(w[0]^2+w[0]*w[1]+w[0]*w[t]-3*w[1]*w[t])*(w[1]-w[t])*t*x
-(w[t]*(w[0]-w[1])^2*t-w[1]*(w[0]-w[t])^2)*t;
P12:= (w[t]*(w[0]-w[1])*(w[0]*w[t]+w[1]*w[t]-2*w[0]*w[1])*t
-w[1]*(w[0]-w[t])*(w[0]*w[1]+w[1]*w[t]-2*w[0]*w[t]))*x^2
+2*t*w[0]*(w[1]-w[t])*(w[0]*w[t]+w[0]*w[1]-2*w[1]*w[t])*x
+(w[t]^2*(w[0]-w[1])^2*t-w[1]^2*(w[0]-w[t])^2)*t;
P21:= ((w[0]-w[1])*(w[0]+w[1]-2*w[t])*t-(w[0]-w[t])*(w[0]+w[t]-2*w[1]))*x^2
-2*t*(w[1]-w[t])*(2*w[0]-w[1]-w[t])*x-t*((w[0]-w[1])^2*t-(w[0]-w[t])^2);
P22:= (w[0]-w[1])*(w[0]-w[t])*(w[1]-w[t])*x^3
+((w[0]-w[1])*(w[0]*w[t]+w[1]*w[t]-2*w[0]*w[1])*t
-(w[0]-w[t])*(w[0]*w[1]+w[1]*w[t]-2*w[0]*w[t]))*x^2
+t*(w[1]-w[t])*(3*w[0]^2-w[0]*w[t]-w[0]*w[1]-w[1]*w[t])*x
+t*(w[t]*(w[0]-w[1])^2*t-w[1]*(w[0]-w[t])^2);The gauge matrix, with all substitutions.subs(SW, SU, ST, matrix( [[P11,P12],[P21,P22]] )):
GM1:= map(normal,%):
This is the heavy computation of the tpullback transformation.
It takes about 560s, 56.8MB on Maple 9.5,
or about 184s, 69.5MB on Maple 13.map(normal, subs(ST,evalm(FMM)) ):
MM:= RSpullback( %, z, FX, x, [RD,GM1] ):
map(length,GM1);SU;Substitutions only explode the gauge expressions.FactorCollect( subs(SW,ST,P12), x):
FactorCollect( subs(SW,SU,ST,P21), x):
length(%%), length(%);length(%);
<Text-field style="Heading 2" layout="Heading 2">Derivation of gauge matrices</Text-field>Recall, the shape of the gauge transformation is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjUkRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYjUSFGJy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNictSSRtdGRHRiQ2KC1GMTYlUSRQMTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSlyb3dhbGlnbkdGMy8lLGNvbHVtbmFsaWduR0YzLyUrZ3JvdXBhbGlnbkdGMy8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZOLUY7NigtRjE2JVEkUDEyRidGQEZDRkZGSEZKRkxGT0ZGRkhGSi1GODYnLUY7NigtRjE2JVEkUDIxRidGQEZDRkZGSEZKRkxGTy1GOzYoLUYxNiVRJFAyMkYnRkBGQ0ZGRkhGSkZMRk9GRkZIRkovJSZhbGlnbkdRJWF4aXNGJy9GR1EpYmFzZWxpbmVGJy9GSVEnY2VudGVyRicvRktRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0ZCLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGaW8vJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGZHAvJSZmcmFtZUdGZHAvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR1EmZmFsc2VGJy8lLWVxdWFsY29sdW1uc0dGXnEvJS1kaXNwbGF5c3R5bGVHRl5xLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmFwRjAvRkRRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYn.
We keep the leading terms of the local solutions at the infinity,
hence deg(P11) = deg(P22) = 3, deg(P12) = deg(P21) = 2.RD;The solutions at with the local exponent difference 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 at 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
have the asymptotics LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYjUSFGJy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNiYtSSRtdGRHRiQ2KC1JJW1zdWJHRiQ2JS1GMTYlUSJ3RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSNtbkdGJDYkUSIwRicvRkdRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSlyb3dhbGlnbkdGMy8lLGNvbHVtbmFsaWduR0YzLyUrZ3JvdXBhbGlnbkdGMy8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZmbkZURlZGWC1GODYmLUY7NigtRkw2JFEiMUYnRk9GVEZWRlhGWkZnbkZURlZGWC8lJmFsaWduR1ElYXhpc0YnL0ZVUSliYXNlbGluZUYnL0ZXUSdjZW50ZXJGJy9GWVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRkUvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZdcC8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZocC8lJmZyYW1lR0ZocC8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHUSZmYWxzZUYnLyUtZXF1YWxjb2x1bW5zR0ZicS8lLWRpc3BsYXlzdHlsZUdGYnEvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGZXBGMEZPLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIrRidGOi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZSLUYjNiUtRjE2JVEiT0YnLyUnaXRhbGljR0ZDRjotRj42LVEwJkFwcGx5RnVuY3Rpb247RidGOkZBRkRGRkZIRkpGTEZOL0ZRUSYwLjBlbUYnL0ZURmpuLUkobWZlbmNlZEdGJDYkLUYjNiMtRjE2JVEiekYnL0ZlblEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0Y6RjBGMC1GIzYjLUkmbXNxcnRHRiRGYG8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmFwLyUpYmV2ZWxsZWRHRkM=, 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, 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.
In the transformed solutions 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, 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, 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
the local exponent 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 has to be shifted to 0, therefore
the vector 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 must be divisible by 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,
the vector 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 must be divisible by 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,
the vector 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 must be divisible by 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.
Here are cubic and quadratic polynomials with undetermined coefficients.TP:= c[0]*x^3+c[1]*x^2+c[2]*x+c[3];
TQ:= c[4]*x^2+c[5]*x+c[6];Computing the upper entries of the (inverse) Schlesinger transformation
from the divisibility conditions.rem( TP*w[0]+TQ, x^2-t, x), rem( TP*w[1]+TQ, x^2-2*x+t, x),
rem( TP*w[t]+TQ, x^2-2*t*x+t, x);
So1:= solve( map(coeffs,{%},x), {c[1],c[2],c[3],c[4],c[5],c[6]});Computing the lower entries of the (inverse) Schlesinger transformation
from the divisibility conditions.rem( TQ*w[0]+TP, x^2-t, x), rem( TQ*w[1]+TP, x^2-2*x+t, x),
rem( TQ*w[t]+TP, x^2-2*t*x+t, x);
So2:= solve( map(coeffs,{%},x), {c[1],c[2],c[3],c[4],c[5],c[6]});The gauge rows are determined up to a constant multiple.
Therefore we can pick one undetermined so to avoid denominators.Sc0:= c[0]=(w[0]-w[1])*(w[0]-w[t])*(w[1]-w[t]);We check that the solutions are the same.subs(So1,TP)-P11, subs(So1,TQ)-P12, subs(So2,TQ)-P21, subs(So2,TP)-P22:
normal( subs(Sc0, [%]) );