#################################################################
##
##  Higher genus icosahedral Painleve curves
##  Philip Boalch (June 2005)
##
#################################################################
##
## Painleve VI solutions to accompany math.AG/0506407
##
## We will give the solutions functions y,t both as they appear in the article 
## and an equivalent solution at 'physical' values of the parameters theta, 
## where the corresponding Fuchsian system has finite (icosahedral) monodromy 
## group
##
## All the solutions have been checked symbolically
##
#################################################################


#################################################################
## 20 branch genus one (soln 45, soln 44):


#soln 45:
th:=[0,1/10,0,9/10];
y:=1/2-1/2*( 16 *s* (5 *s - 1)+ v*w)/(-1+s)/(1+3*s)/v;
t:=1/2-1/2*(27*s^5-315*s^4-370*s^3+170*s^2-25*s+1)/(-1+9*s)/(-1+s)^2/w;
v2:=(-1+9*s)*(-1+s);
w2:=(s^2-18*s+1)*(-1+9*s)*(-1+s);

#sibling, icosahedral soln 44:
th:=[0,3,0,7]/10;
y:=1/2-1/2*(64*(5*s-1)*s^2+(s-1)*v*w)/(3*s^3+75*s^2-15*s+1)/v;



#parameterisation of solution curve:
w12:=(s^2-18*s+1);
w:=v*w1;
w1:=-(-1+18*j-j^2)/(-18+2*j);
s:= (-1+j^2)/(-18+2*j);
v:=z/(j-9)/2;
z2:=(9*j^2-2*j+9)*(j^2-2*j+17);

simplify({w^2-w2},{z^2=z2});
simplify({v^2-v2},{z^2=z2});


####################
# 'physical' parameters:
# soln 45:

th:=[1/2, 3/5, 1/2, 3/5];
y:=1/2+1/2*(48*v*s*(5*s-1)*(91*s^3+43*s^2-7*s+1)*(s^2-18*s+1)-(-1+s)*(-1+9*s)*(
357*s^5-525*s^4+1730*s^3-490*s^2-55*s+7)*w)/(1+3*s)/(-1+9*s)/(s^2-18*s+1)/(357
*s^3+301*s^2-89*s+7)/(-1+s)^2;
t:=1/2-1/2*(27*s^5-315*s^4-370*s^3+170*s^2-25*s+1)/(-1+9*s)/(-1+s)^2/w;
v2:=(-1+9*s)*(-1+s);
w2:=(s^2-18*s+1)*(-1+9*s)*(-1+s);

#soln 44: as for 45 except:
th:=[1/2, 4/5, 1/2, 4/5];
y:=
1/2+1/2*(64*v*s^2*(5*s-1)*(57*s^3+261*s^2-69*s+7)*(s^2-18*s+1)-(-1+9*s)*(33*s^
5+2655*s^4-4710*s^3+1070*s^2-75*s+3)*(-1+s)^2*w)/(-1+s)/(-1+9*s)/(s^2-18*s+1)/
(3*s^3+75*s^2-15*s+1)/(33*s^3+49*s^2-21*s+3);




#################################################################
## 40 branch genus three (soln 51, soln 50):


#soln 51:
v2:=-2*(j+1)*(5*j^2-2*j+13);
w2:=2*(j-9)*(j^2-1);
s:= (-1+j^2)/(-18+2*j);
u:=w/2/(j-9);
y:=
1/2+1/8*(
(9*j^2-2*j+9)*(j^2-18*j+1)*(j^2-2*j+17)
+16*(j+3)*(j+1)*v*w)/(3*j-7)/(j-9)/(j-1)^2/v;
t:=1/2+1/256*(27*s^5-315*s^4-370*s^3+170*s^2-25*s+1)/(5*s-1)/s/u^3;
th:=[1,1,1,19]/20;

#sibling, soln 50:
th:=[3, 3, 3, 17]/20;
y:=
1/2+1/8*(
(9*j^2-2*j+9)*(j^2-18*j+1)*(j^2-2*j+17)^2
+8*(j-1)*(j^3+57*j^2-69*j+75)*v*w   )/(3*j^3-21*j^2-15*j-31)/w^2/v;

#put in terms of s,v,w:
s:='s':y:=subs(j=s,y);t:=subs(j=s,t);v2:=subs(j=s,v2);w2:=subs(j=s,w2);


#parameterisation:
g:=5*q^4+6*p^2*q^2+6*q^2+1+6*p^2+5*p^4;

#map in one direction:
p:=4*(j+1)/v;
q:=w/v;
#check well defined:     simplify(g,{w^2=w2,v^2=v2});

#inverse map:
p:='p': q:='q':
w := -200*q*p*(5*q^2+6*p^2+1)/(55*q^4+166*q^2-84*q^2*p^2+156*p^2+31); 
v := -(1000*p*q^2+1200*p^3+200*p)/(55*q^4+166*q^2-84*q^2*p^2+156*p^2+31);
j := -(28*v^2+800-4*w^2)/(800-3*v^2-15*w^2);
#check well-defined:  simplify(w^2-w2,{g}); simplify(v^2-v2,{g});


#check symmetries (slow):
t+subs(p=-p,t);
simplify(%,{g});
y+subs(p=-p,y);
simplify(%,{g});
	
t+subs(q=-q,t);
simplify(%,{g});
y-subs(q=-q,y*(t-1)/(t-y));
simplify(%,{g});

(y-t)/(y-1)-subs({p=q,q=p},y);
simplify(%,{g});
t-subs({p=q,q=p},t);
simplify(%,{g});

####################
# 'physical' parameters:
# soln 51:
v2:=-2*(j+1)*(5*j^2-2*j+13);
w2:=2*(j-9)*(j^2-1);
s:= (-1+j^2)/(-18+2*j);
u:=w/2/(j-9);
t:=1/2+1/256*(27*s^5-315*s^4-370*s^3+170*s^2-25*s+1)/(5*s-1)/s/u^3;
y:=1/2+1/4*(v*(j^2-18*j+1)*(9*j^2-2*j+9)*(105*j^9-1323*j^8+14724*j^7-43548*j^6+
98190*j^5-365850*j^4+693492*j^3-1227276*j^2+334737*j-420755)*(j^2-2*j+17)+8*(5
*j^2-2*j+13)*(957*j^10-14490*j^9-6975*j^8+1053960*j^7-5805270*j^6+16553700*j^5
-30587910*j^4+47354760*j^3-43708455*j^2+25673510*j-5270907)*(j+1)^2*w)/(3*j-7)
/(j-9)/(j+1)/(5*j^2-2*j+13)/(147*j^9-3087*j^8+90252*j^7-139116*j^6-588294*j^5-\
1306098*j^4-1839012*j^3-4673628*j^2-3285621*j-1100599)/(-1+j)^2;
th:=[1/2, 1/2, 1/2, 3/5];

#soln 50: as above, except:
y:=
1/2+1/8*(v*(9*j^2-2*j+9)*(j^2-18*j+1)*(15*j^7-189*j^6+2331*j^5-8225*j^4+3885*j
^3-18935*j^2+5481*j-8939)*(j^2-2*j+17)^2+4*(-1+j)*(j+1)*(5*j^2-2*j+13)*(633*j^
10-14130*j^9+84045*j^8-234520*j^7+109410*j^6-2745260*j^5+861490*j^4-7647640*j^
3-220955*j^2-5222770*j-698943)*w)/(-1+j)/(3*j-7)/(j-9)/(5*j^2-2*j+13)/(3*j^3-\
21*j^2-15*j-31)/(j^6-14*j^5+2023*j^4-1316*j^3+4991*j^2-3822*j+2233)/(j+1)^2;
th:=[1/2, 1/2, 1/2, 4/5];

#put in terms of s,v,w:
s:='s':y:=subs(j=s,y);t:=subs(j=s,t);v2:=subs(j=s,v2);w2:=subs(j=s,w2);



#################################################################
## 30 branch genus two (soln 47, soln 48):


##soln 47:
th:=[2, 7, 2, 23]/30;
u2:=3*(s+5)*(4*s^2+15*s+15);
v2:=(s+5)*s*(s+2)*(s-3);
w2:=s*(s+5)*(s+2)*(s+3);

y:=1/2+           
1/2*((s^2-5)*u2*v+s*(s-3)*(s+1)*w^3)/(s^3+s^2-9*s-15)/(s-3)/w^2/(s+5); 

t:=1/2+
1/4*(2*s^7+10*s^6-90*s^4-135*s^3+297*s^2+945*s+675)*(s+5)^2/w^3/s/(s^2-9);

#parameterisation:
z2:=(j^2+9)*(j+9)*(j+1)*(j^2+4*j+9);
x:=1/2*(-9+j^2)/j;
s:=1/2*( 9+j^2)/j;
v:=z/4*(j-3)/j^2;
w:=z/4*(j+3)/j^2;

#check:     simplify(v^2-v2,{z^2=z2}); simplify(w^2-w2,{z^2=z2});


##soln 48:
th:=[2/15, 1/30, 2/15, 29/30];
y:=1/2+
1/2*(3*(4*s^2+15*s+15)*(s^2-5)*v+
s*(s-3)*(s+2)*(s^2-6*s-15)*w)/s/(s+3)^2/v^2;

####################
# 'physical' parameters:
#soln 47:
th:=[1/2, 2/3, 1/2, 4/5];
u2:=3*(s+5)*(4*s^2+15*s+15);
v2:=(s+5)*s*(s+2)*(s-3);
w2:=s*(s+5)*(s+2)*(s+3);
y:=1/2+1/2*(-v*(4*s^2+15*s+15)*(17*s^6+29*s^5-190*s^4-270*s^3+1425*s^2+3825*s+
2700)*(s^2-5)+s^2*(s-3)*(s+5)*(s+3)*(s+2)*(3*s^6-85*s^4-180*s^3+255*s^2+1044*s
+795)*w)/s/(s-3)/(s+5)/(s+3)/(s+2)/(s^3+s^2-9*s-15)/(3*s^6+12*s^5+7*s^4-14*s^3
+19*s^2+90*s+75);
t:=1/2+
1/4*(2*s^7+10*s^6-90*s^4-135*s^3+297*s^2+945*s+675)*(s+5)^2/w^3/s/(s^2-9);

#soln 48: as above except:
th:=[1/2, 2/5, 1/2, 2/3];
y:=
1/2+1/2*(-9*v*(4*s^2+15*s+15)*(s^2-5)*(s^5+3*s^4-3*s^3-125*s^2-450*s-450)+s*(s
-3)*(s+5)*(s+2)*(19*s^6+90*s^5-45*s^4-540*s^3+675*s^2+4050*s+3375)*w)/s^2/(s-3
)/(s+5)/(s+2)/(19*s^5+35*s^4-90*s^3-450*s^2-1125*s-1125)/(s+3)^2;




#################################################################
## 36 branch genus three (soln 49):

th:=[0,1,0,5]/6;
u2:=(8*s^2-11*s+8)*s;
v2:=(s-2)*(2*s^2+s+2)*(2*s-1);
w12:=(s^2-7*s+1);
w2:=v2*w12;

y:=1/2-1/2*(9*s*(s-1)*u2+(s-2)*w*v)/(s^3+12*s^2-12*s+4)/(2*s-1)/v;
t:=1/2
-1/4*(s+1)*(32*s^8-320*s^7+1112*s^6-2420*s^5+3167*s^4-2420*s^3+1112*s^2-320*s+
32)/w/(2*s^2+s+2)/(2*s-1)^2/(s-2)^2;



#parameterisation:
z2:=(j^2-4*j+13)*(2*j^4+2*j^3-3*j^2-58*j+107)*(2*j^2-2*j+5);
w1:=-(-1+7*j-j^2)/(-7+2*j); 
s:= (-1+j^2)/(-7+2*j);
v:=z/(2*j-7)^2;
w:=w1*v;

#check:     simplify(v^2-v2,{z^2=z2}); simplify(w^2-w2,{z^2=z2});

####################
# 'physical' parameters:
# soln 49:
th:=  [1/2, 2/3, 1/2, 2/3];
v2:=(s-2)*(2*s^2+s+2)*(2*s-1);
w12:=(s^2-7*s+1);
w2:=v2*w12;

t:=1/2
-1/4*(s+1)*(32*s^8-320*s^7+1112*s^6-2420*s^5+3167*s^4-2420*s^3+1112*s^2-320*s+
32)/w/(2*s^2+s+2)/(2*s-1)^2/(s-2)^2;
y:=
1/2+1/4*(-w*(2*s-1)*(2*s^2+s+2)*(152*s^9+2592*s^8-17028*s^7+26952*s^6-40203*s^
5+58473*s^4-55092*s^3+28728*s^2-6552*s+728)*(s-2)^2+18*s^2*(s-1)*(8*s^2-11*s+8
)*(s^2-7*s+1)*(44*s^7-88*s^6+429*s^5+685*s^4-2960*s^3+3426*s^2-1762*s+476)*v)/
(s-2)/(2*s^2+s+2)/(s^2-7*s+1)/(76*s^7-332*s^6+696*s^5-220*s^4-1075*s^3+1344*s^
2-728*s+364)/(s^3+12*s^2-12*s+4)/(2*s-1)^2;




#################################################################
## 72 branch genus seven (soln 52):

th:=[1,1,1,11]/12;
v2:=-(j+1)*(j^2-2*j+6)*(4*j^2-13*j+19);
w2:= (j-1)*(2*j-7)*(j+1)*(4*j^2-13*j+19)*(2*j^2+j+17);
s:= (-1+j^2)/(-7+2*j);
u:=w/(-7+2*j)^2;

t:=1/2+
1/54*(s+1)*(32*s^8-320*s^7+1112*s^6-2420*s^5+3167*s^4-2420*s^3+1112*s^2-320*s+
32)*u/s^3/(s-1)/(8*s^2-11*s+8)^2;

y:=
1/2+1/6*(
2*(5+2*j^2-2*j)*(1-7*j+j^2)*(107-3*j^2+2*j^4-58*j+2*j^3)*(13+j^2-4*j)^2+
9*(j-1)*(j^3+27*j^2-57*j+79)*w*v
)/(j-1)/(2*j^2+j+17)/(j^3-3*j^2+3*j-11)/(j+1)/(-7+2*j)^2/v;


#parameterisation:

g:=
9*(q^6*p^2+q^2*p^6)+
18*q^4*p^4+
4*(q^6+p^6)+
26*(q^4*p^2+q^2*p^4)+
8*(q^4+p^4)+
57*q^2*p^2+
20*(q^2+p^2)+
16;

#map in one direction:
p:=1/3*w/(j-1)/v;
q:=1/3*v/(6+j^2-2*j);

#check well-defined:   simplify(g,{v^2=v2,w^2=w2});

#write down inverse map (v,w,j as functions of p,q):
u:='u': x:='x':
w:=u;
v:=x*(j+1)*(4*j^2-13*j+19)/w;

j := 1/4*(1420*u^6*x^2-64*x^8*u^2+191700*x^6*u^2+15*u^8+1165995*x^8+64*x^10+
15270*x^4*u^4+524880000*x^6+253125*u^6+7391250*u^4*x^2+78823125*x^4*u^2)/(70*u
^6*x^2+15210*x^6*u^2+5*u^8+30375*x^8+64*x^10+1060*x^4*u^4+127119375*x^6+50625*
u^6+658125*u^4*x^2+14124375*x^4*u^2);


#where

u := -135/4*q*p*(-932962265619*q^10*p^2-648699955578*q^6*p^2-333915984108*q^
10-40546964308*q^2-262521148808*q^6-132804826048*q^4-251532907008*q^12-\
346269732240*q^8-911364378120*q^8*p^2-14253839444*p^2-316388535716*q^4*p^2-\
97968680237*q^2*p^2-556095829086*p^4*q^10-384477479085*p^4*q^6-725922381672*p^
2*q^12-536954497305*p^4*q^8-192656197443*q^4*p^4-63107220939*q^2*p^4-\
9787840476*p^4-151119297648*q^14-72068049648*q^16-442319557356*q^14*p^2-\
438569757774*p^4*q^12-260399822058*p^4*q^14-216745093188*p^2*q^16-82317160035*
q^18*p^2-4537443258*q^22*p^2-125892664830*q^16*p^4-27633751020*q^18-1455439752
*q^22-8126402112*q^20-229582512*q^24-23700142836*q^20*p^2-11180987199*q^20*p^4
-47731373415*q^18*p^4+57395628*q^26-459165024*p^2*q^24-2912828121*p^4*q^22+
129140163*p^2*q^26+129140163*p^4*q^24-5600373968)/(-1257878745632*q^10*p^2-\
658540484196*q^6*p^2-574438524192*q^10-64549837872*q^2-365997568464*q^6-\
187289253248*q^4-494436505592*q^12-522124761172*q^8-1048653413117*q^8*p^2+
86093442*p^2*q^28-12655008727*p^2-296404162084*q^4*p^2-88673989916*q^2*p^2-\
358051152490*p^4*q^10-129824741992*p^4*q^6-1166874688726*p^2*q^12-255488924380
*p^4*q^8-45633345356*q^4*p^4-9466997554*q^2*p^4-746087880*p^4-340206237088*q^
14-193575403764*q^16-840740195304*q^14*p^2-370012084344*p^4*q^12-283998656736*
p^4*q^14-490571847441*p^2*q^16-231839516412*q^18*p^2-25185343284*q^22*p^2-\
165784024512*q^16*p^4-88411262832*q^18-9673853328*q^22-32401157328*q^20-\
1716436332*q^24-84347218728*q^20*p^2-28148658300*q^20*p^4-79306677990*q^18*p^4
-306110016*q^26-4512465531*p^2*q^24-6832205496*p^4*q^22+86093442*q^26*p^4-\
650483784*p^2*q^26-1511418204*p^4*q^24+38263752*q^28-12145248972);

x := -45/4*p
*(-932962265619*q^10*p^2-648699955578*q^6*p^2-333915984108*q^10-40546964308*q^
2-262521148808*q^6-132804826048*q^4-251532907008*q^12-346269732240*q^8-\
911364378120*q^8*p^2-14253839444*p^2-316388535716*q^4*p^2-97968680237*q^2*p^2-\
556095829086*p^4*q^10-384477479085*p^4*q^6-725922381672*p^2*q^12-536954497305*
p^4*q^8-192656197443*q^4*p^4-63107220939*q^2*p^4-9787840476*p^4-151119297648*q
^14-72068049648*q^16-442319557356*q^14*p^2-438569757774*p^4*q^12-260399822058*
p^4*q^14-216745093188*p^2*q^16-82317160035*q^18*p^2-4537443258*q^22*p^2-\
125892664830*q^16*p^4-27633751020*q^18-1455439752*q^22-8126402112*q^20-\
229582512*q^24-23700142836*q^20*p^2-11180987199*q^20*p^4-47731373415*q^18*p^4+
57395628*q^26-459165024*p^2*q^24-2912828121*p^4*q^22+129140163*p^2*q^26+
129140163*p^4*q^24-5600373968)/(-1257878745632*q^10*p^2-658540484196*q^6*p^2-\
574438524192*q^10-64549837872*q^2-365997568464*q^6-187289253248*q^4-\
494436505592*q^12-522124761172*q^8-1048653413117*q^8*p^2+86093442*p^2*q^28-\
12655008727*p^2-296404162084*q^4*p^2-88673989916*q^2*p^2-358051152490*p^4*q^10
-129824741992*p^4*q^6-1166874688726*p^2*q^12-255488924380*p^4*q^8-45633345356*
q^4*p^4-9466997554*q^2*p^4-746087880*p^4-340206237088*q^14-193575403764*q^16-\
840740195304*q^14*p^2-370012084344*p^4*q^12-283998656736*p^4*q^14-490571847441
*p^2*q^16-231839516412*q^18*p^2-25185343284*q^22*p^2-165784024512*q^16*p^4-\
88411262832*q^18-9673853328*q^22-32401157328*q^20-1716436332*q^24-84347218728*
q^20*p^2-28148658300*q^20*p^4-79306677990*q^18*p^4-306110016*q^26-4512465531*p
^2*q^24-6832205496*p^4*q^22+86093442*q^26*p^4-650483784*p^2*q^26-1511418204*p^
4*q^24+38263752*q^28-12145248972);

#check well-defined (slow): 
w:= simplify(w,{g}); 
v:= simplify(v,{g}); 
j:= simplify(j,{g}); 
 simplify(w^2/w2,{g});
 simplify(v^2/v2,{g});


####################
# 'physical' parameters:
# soln 52:
th:=[1/2, 1/2, 1/2, 2/3];
v2:=-(j+1)*(j^2-2*j+6)*(4*j^2-13*j+19);
w2:= (j-1)*(2*j-7)*(j+1)*(4*j^2-13*j+19)*(2*j^2+j+17);
s:= (-1+j^2)/(-7+2*j);
u:=w/(-7+2*j)^2;

t:=1/2+
1/54*(s+1)*(32*s^8-320*s^7+1112*s^6-2420*s^5+3167*s^4-2420*s^3+1112*s^2-320*s+
32)*u/s^3/(s-1)/(8*s^2-11*s+8)^2;
y:=
1/2+1/2*(v*(j^2-7*j+1)*(5+2*j^2-2*j)*(107-3*j^2+2*j^4-58*j+2*j^3)*(64*j^15-760
*j^14+4520*j^13-19720*j^12+129860*j^11-808342*j^10+3653920*j^9-12452190*j^8+
33322665*j^7-71627570*j^6+126805367*j^5-184436785*j^4+214743545*j^3-191317570*
j^2+112206335*j-43328339)*(13+j^2-4*j)^2+3*(j-1)*(j+1)*(j^2-2*j+6)*(1804*j^18-\
31752*j^17+263772*j^16-1480944*j^15+6414390*j^14-22205322*j^13+61089846*j^12-\
132726276*j^11+201923397*j^10-96501655*j^9-557646768*j^8+2180994939*j^7-\
4577345139*j^6+6510542913*j^5-6619276485*j^4+4101059661*j^3-1665964908*j^2+
1404983268*j-2348782241)*(4*j^2-13*j+19)*w)/(j-1)/(2*j^2+j+17)/(j^2-2*j+6)/(4*
j^2-13*j+19)/(j^3-3*j^2+3*j-11)/(4*j^15-60*j^14+420*j^13-2020*j^12+129360*j^11
-1159212*j^10+5886520*j^9-20329740*j^8+53261640*j^7-111409070*j^6+196954662*j^
5-297445710*j^4+367066445*j^3-330956745*j^2+181194585*j-50847329)/(j+1)^2/(2*j
-7)^2;


#put in terms of s,v,w:
s:='s':y:=subs(j=s,y);t:=subs(j=s,t);v2:=subs(j=s,v2);w2:=subs(j=s,w2);