##########################################################
##
## Some solutions to Painleve VI
##
## Philip Boalch 2004
##
##########################################################
##(see end for simple maple code to check these solutions)
##########################################################

#############################
#Tetrahedral solution 6
th:=[1/2, 1/3, 1/3, 1/2];

y:=-1/3*s*(s+1)*(s-3)^2/(s+3)/(s-1)^2;
t:=-(s+1)^3*(s-3)^3/(s+3)^3/(s-1)^3;

F:=
27*(1+t)*(y^6+t^3)-108*t*y*(y^4+t^2)-27*y^2*(t+t^2)*(y^2+t)
+(200*t^2+8*t^3+8*t)*y^3;



#############################
#Octahedral solution 7
th:=[1/2, 1/2, 1/4, 2/3];

F:=-125*t^5+702*t^4-1701*t^3+2916*t^2+(-654*t^4+3996*t^3-14094*t^2)*y+
(516*t^4-4821*t^3+30942*t^2+243*t)*y^2+
(-128*t^4+2080*t^3-36928*t^2-864*t)*y^3+(-48*t^3
+26304*t^2+624*t)*y^4-10752*t^2*y^5+(1728*t^2+128*t-64)*y^6;

#   {ga = 1/32, al = 1/18, de = 3/8, be = -1/8}

y:=9/4*(2*s^3-3*s+4)/(2*s^2+6*s+1)/(s+1)/(s-1)^2*s;
t:=27/4/(s^2-1)^3*s^2;

##############################
#Octahedral solution 8, 6 branches
th:=[1/3, 3/4, 1/3, 3/4];

F:=y^6+49*t^2-80*t^3+32*t^4-54*y*t^2+48*y*t^3+9*y^2*t+
54*y^2*t^2-48*y^2*t^3-4*y^3*t-48*y^3*t^2+32*y^3*t^3+
15*y^4*t-6*y^5*t;

# y:=-s*(5*s+3)*(s^2-2*s-1)/(3*s+1)/(s+1)/(3*s^2-1);
# t:=s^4*(5*s+3)^2/(s+1)^3/(3*s+1)/(3*s^2-1);

y:=1/2*(2*s^2-1)*(3*s-1)/(2*s^2+2*s-1)/(s-1)/s;
t:=-1/8*(3*s-1)^2/(2*s^2+2*s-1)/(s-1)/s^3;



#########################
#Octahedral solution 9, 8 branches   
th:=[1/3, 1/4, 1/2, 2/3];

F:=
-256*t^7-64*t^5+(1536*t^6+384*t^5)*y+(-32*t^6-4272*t^5-176*t^4)*y^2+(160*t^5+
4352*t^4-32*t^3)*y^3+(-9*t^5+1467*t^4-1467*t^3+9*t^2)*y^4+(32*t^4-4352*t^3-160
*t^2)*y^5+(176*t^3+4272*t^2+32*t)*y^6+(-384*t^2-1536*t)*y^7+(64*t^2+256)*y^8;

y:=s^3*(2*s^2-4*s+3)*(s^2-2*s+2)/(1-2*s+2*s^2)/(2-4*s+3*s^2);
t:=s^4*(2*s^2-4*s+3)^2/(2-4*s+3*s^2)^2;

#########################
#Octahedral solution 10, 8 branches  
th:=[1/2, 1/4, 1/2, 3/4];

F:=
-512*t^5+512*t^6+125*t^4+125*y^8-16*t*y^8-936*t*y^7-2048*y*t^6+1024*y*t^5+2048
*y^2*t^6+3072*y^2*t^5-1624*y^2*t^4-6144*y^3*t^5-1920*y^3*t^4+1096*y^3*t^3+8640
*y^4*t^4+320*y^4*t^3-210*y^4*t^2-7040*y^5*t^3+8*y^5*t^2+3592*y^6*t^2-92*y^6*t-\
64*y^7*t^2+16*y^8*t^2+24*y*t^4+4*y^2*t^3-32*y^3*t^2+32*y^5*t;

y:=32*s*(s+1)*(5*s^2+6*s-3)/(5+2*s+s^2)/(3*s^2+2*s+3)^2;
t:= 1024*s^3*(s+1)^2/(s^2+6*s+1)/(3*s^2+2*s+3)^3;


########################
#Octahedral solution 11: (12 branches g=0)
th:=[1/3, 1/2, 1/2, 2/3];

F:=
-8964*t^9+12614*t^8-8964*t^7+2401*t^6+2401*t^10+(-1920*t^9+4992*t^8+4992*t^7-\
1920*t^6)*y+(3210*t^9-10536*t^8-19140*t^7-10536*t^6+3210*t^5)*y^2+(-352*t^9+
672*t^8+56000*t^7+56000*t^6+672*t^5-352*t^4)*y^3+(-144*t^9+2127*t^8-46764*t^7-\
163878*t^6-46764*t^5+2127*t^4-144*t^3)*y^4+(-672*t^8+4320*t^7+199104*t^6+
199104*t^5+4320*t^4-672*t^3)*y^5+(-224*t^8-116*t^7-55376*t^6-361656*t^5-55376*
t^4-116*t^3-224*t^2)*y^6+(-672*t^7+4320*t^6+199104*t^5+199104*t^4+4320*t^3-672
*t^2)*y^7+(-144*t^7+2127*t^6-46764*t^5-163878*t^4-46764*t^3+2127*t^2-144*t)*y^
8+(-352*t^6+672*t^5+56000*t^4+56000*t^3+672*t^2-352*t)*y^9+(3210*t^5-10536*t^4
-19140*t^3-10536*t^2+3210*t)*y^10+(-1920*t^4+4992*t^3+4992*t^2-1920*t)*y^11+(
2401*t^4-8964*t^3+12614*t^2-8964*t+2401)*y^12;


y:=4*(s+1)*(3*s^2-4*s+2)*(7*s^4+16*s^3+4*s^2-4)/(-28+32*s-4*s^2+s^4)/s^
3/(s-2)/(s^2+4*s+6);
t:=16*(3*s^2-4*s+2)^2*(s+1)^4/s^4/(s^2+4*s+6)^2/(s-2)^4;

#####################################
#Octahedral solution 12, 12 branches, genus 1
th:=[1/2, 1/2, 1/2, 2/3];

#{de = 3/8, ga = 1/8, al = 1/18, be = -1/8} 

F:=
67108864*t^12-358612992*t^11+811745280*t^10-995639040*t^9+698779305*t^8-\
266389722*t^7+43046721*t^6+(-88080384*t^11+339148800*t^10-520128000*t^9+
383599800*t^8-131220000*t^7+16218792*t^6)*y+(147259392*t^10-495621120*t^9+
696460680*t^8-472260780*t^7+141105240*t^6-14407956*t^5)*y^2+(-4456448*t^10-\
138381312*t^9+317787520*t^8-332049240*t^7+177662160*t^6-29644056*t^5+629856*t^
4)*y^3+(3538944*t^10-18063360*t^9+284440896*t^8-441352080*t^7+296567190*t^6-\
117982332*t^5+12181590*t^4-314928*t^3)*y^4+(-1572864*t^10+9388032*t^9-\
105587712*t^8-50668320*t^7+144804360*t^6-42486768*t^5+16467624*t^4-769824*t^3)
*y^5+(524288*t^10-3997696*t^9+43289600*t^8+30594496*t^7+88555480*t^6-154575404
*t^5+33524928*t^4-2512620*t^3+93312*t^2)*y^6+(393216*t^9-5357568*t^8-29845632*
t^7-69990720*t^6+57945720*t^5+21785232*t^4-5791176*t^3+435456*t^2)*y^7+(897024
*t^8-5422080*t^7+76830336*t^6-62151600*t^5+21616185*t^4-14109642*t^3+1417905*t
^2-62208*t)*y^8+(407552*t^7-11596672*t^6-6614560*t^5+9587680*t^4-2084880*t^3+
2001424*t^2-152064*t)*y^9+(4086912*t^6-8188800*t^5+15207600*t^4-11036640*t^3+
2750760*t^2-284376*t)*y^10+(-1119744*t^6+2618880*t^5-2528640*t^4-349440*t^3+
1286400*t^2-368448*t)*y^11+(186624*t^6-559872*t^5+729920*t^4-526720*t^3+354560
*t^2-184512*t+38416)*y^12;
  
#elliptic parameterisation:
u2:=-(3*s^2+2*s+1)*(-1+4*s);
#so elliptic curve is:  u^2=u2

y:=
-1/4*(-1+4*s)*
((-1+4*s)*(3*s^2+2*s+1)*(22*s^5-30*s^4-65*s^3-25*s^2-15*s+5)
+u*(s-1)*(2*s+1)*(42*s^5+50*s^4+25*s^3+15*s^2+25*s+5))
/s/(2+s)/(49*s^4+10*s^3-10*s+5)/(1+2*s^2)^2;

t:=1/2*(3*s^2+2*s+1)*(-1+4*s)^2/(1+2*s^2)^3;

#or alternatively:

y:=1/2+1/4/(5*s^2+1)/(s+1)^2*(45*s^6+20*s^5+95*s^4+92*s^3+39*s^2-3)/u;
t:=1/2+(2*s+1)^2*(27*s^4+28*s^3+26*s^2+12*s+3)*s/u^3/(s+1)^3;
u2:=(2*s+1)*(9*s^2+2*s+1);

#############################
#Octahedral solution 13, g=0, 16 branches
th:=[1/2,1/2,1/2,3/4];

F:=
1173445494025*t^12-135962558464*t^9+(1130364928*t^14-8883634176*t^13+
36909994816*t^12-83311654348*t^11+108338295060*t^10-83311654348*t^9+
36909994816*t^8-8883634176*t^7+1130364928*t^6)*y^4+(-2218786816*t^14+
9041223680*t^13-14332013184*t^12+7505201320*t^11+7505201320*t^10-14332013184*t
^9+9041223680*t^8-2218786816*t^7)*y^3+(8748269568*t^14-50554318848*t^13+
123787299720*t^12-163960625880*t^11+123787299720*t^10-50554318848*t^9+
8748269568*t^8)*y^2+(-2952790016*t^15+14502854656*t^14-25819146240*t^13+
14268956600*t^12+14268956600*t^11-25819146240*t^10+14502854656*t^9-2952790016*
t^8)*y+(1032480*t^8-4230600*t^7+12412560*t^6-16553880*t^5+12412560*t^4-4230600
*t^3+1032480*t^2)*y^14+(781400*t^9-8556160*t^8+6564600*t^7-3164840*t^6-3164840
*t^5+6564600*t^4-8556160*t^3+781400*t^2)*y^13+(1528720*t^10-12722700*t^9+
95764300*t^8-155878360*t^7+171053580*t^6-155878360*t^5+95764300*t^4-12722700*t
^3+1528720*t^2)*y^12+(1081344*t^11-15119712*t^10+18036312*t^9-207800184*t^8+
169677240*t^7+169677240*t^6-207800184*t^5+18036312*t^4-15119712*t^3+1081344*t^
2)*y^11+(524288*t^12-9093120*t^11+104225352*t^10-119828456*t^9+945725696*t^8-\
1717982520*t^7+945725696*t^6-119828456*t^5+104225352*t^4-9093120*t^3+524288*t^
2)*y^10+(-2621440*t^12+41853440*t^11-417180280*t^10-287303880*t^9+575877160*t^
8+575877160*t^7-287303880*t^6-417180280*t^5+41853440*t^4-2621440*t^3)*y^9+(
22425600*t^12-207509760*t^11+2552630550*t^10-4376401020*t^9+4218803010*t^8-\
4376401020*t^7+2552630550*t^6-207509760*t^5+22425600*t^4)*y^8+(13107200*t^13-\
159170560*t^12-702519520*t^11+176737320*t^10+582470560*t^9+582470560*t^8+
176737320*t^7-702519520*t^6-159170560*t^5+13107200*t^4)*y^7+(8388608*t^14-\
104595456*t^13+1332452352*t^12-1131120368*t^11-487389376*t^10+889653480*t^9-\
487389376*t^8-1131120368*t^7+1332452352*t^6-104595456*t^5+8388608*t^4)*y^6+(-\
25165824*t^14+610467840*t^13-5677888512*t^12+11506770600*t^11-6448309104*t^10-\
6448309104*t^9+11506770600*t^8-5677888512*t^7+610467840*t^6-25165824*t^5)*y^5+
(15625*t^8-62500*t^7+123322*t^6-151216*t^5+165163*t^4-151216*t^3+123322*t^2-\
62500*t+15625)*y^16+(-125000*t^8+513424*t^7-1033560*t^6+520136*t^5+520136*t^4-\
1033560*t^3+513424*t^2-125000*t)*y^15+17179869184*t^8+17179869184*t^16-\
135962558464*t^15-940183290880*t^11+472243240960*t^14-940183290880*t^13+
472243240960*t^10;


##parameterisation (from Mark van Hoeij):

t:=  1/32*(s-1)^2*(s+1)^2*(s^4+6*s^2+1)^3/s^2/(s^4+1)^3;
y:=(-1/4-1/4*I)*(s^8-2*s^7+2
*I*s^7-6*s^6-2*I*s^6+10*s^5+2*I*s^5+4*I*s^4+10*s^3-2*I*s^3+6*s^2-2*I*s^2-2*s
-2*I*s-1)*(s^2+2*I*s+1)*(s+1)*(s-1)*(s^2-2*I*s+1)^2/s/(s^2+I)/(s^6-3*s^5-3*I*s
^5+3*I*s^4+4*s^3-4*I*s^3+3*s^2+3*s+3*I*s+I)/(s^2+s+I*s-I)/(s^2-I)^2;


#Note F=0 is birational to the conic
#  u^2 + 1-2*x+3*x^2 = 0
#via the following map, defined over Q:
t := 
1/16*(72*x^9+72*x^8*u-312*x^8+584*x^7-96*x^7*u+32*x^6*u-632*x^6+440*x^5+80*x^5
*u-116*x^4*u-200*x^4+56*x^3+80*x^3*u-32*x^2*u-8*x^2+8*x*u-u)/x^2/(3*x^2-2*x+1)
^2/(x-1)^3;
y :=
1/2*(72*x^8*u+36*x^8-156*x^7*u-132*x^7+210*x^6+216*x^6*u-222*x^5*u-195*x^5+181
*x^4*u+116*x^4-105*x^3*u-44*x^3+10*x^2+40*x^2*u-9*x*u-x+u)/(12*x^3-12*x^2+6*x-\
1)/x/(3*x^2-2*x+1)/(x-1)^2;


#############################
#237 solution, genus one, 18 branches
th:=[2/7, 2/7, 2/7, 1/3];

u2:=(s^2+s+7)*s;
y:=
1/2-1/4*(3*s^8-2*s^7-4*s^6-204*s^5-536*s^4-1738*s^3-5064*s^2-4808*s-3199)/(s^6
+196*s^3+189*s^2+756*s+154)/(s^2+s+7)/(s+1)*u;
t:=
1/2-1/432*(s^9-84*s^6-378*s^5-1512*s^4-5208*s^3-7236*s^2-8127*s-784)/s/(s+1)^2
/(s^2+s+7)^2*u;


F:=
7398655171875*t^17+148206064500*t^16+19150282563282*t^15+148206064500*t^14+
7398655171875*t^13+(-14797310343750*t^17-119193616104750*t^16-174205118875788*
t^15-174205118875788*t^14-119193616104750*t^13-14797310343750*t^12)*y+(
262272331913625*t^16+1523462955401628*t^15+1667862195882390*t^14+
1523462955401628*t^13+262272331913625*t^12)*y^2+(-36543753774000*t^16-\
3375884975416560*t^15-10559125325510496*t^14-10559125325510496*t^13-\
3375884975416560*t^12-36543753774000*t^11)*y^3+(23379979887000*t^16+
1879622008160220*t^15+24381595835814456*t^14+52217459762534568*t^13+
24381595835814456*t^12+1879622008160220*t^11+23379979887000*t^10)*y^4+(-\
6848226000000*t^16-1151935832250504*t^15-20294095830649704*t^14-\
125248437685460880*t^13-125248437685460880*t^12-20294095830649704*t^11-\
1151935832250504*t^10-6848226000000*t^9)*y^5+(3480246000000*t^16+
371649738980016*t^15+14387154978990388*t^14+140547176465298176*t^13+
325086786630360888*t^12+140547176465298176*t^11+14387154978990388*t^10+
371649738980016*t^9+3480246000000*t^8)*y^6+(-1026432000000*t^16-\
134738240796576*t^15-6282202563271536*t^14-128059694064775152*t^13-\
410412946832497920*t^12-410412946832497920*t^11-128059694064775152*t^10-\
6282202563271536*t^9-134738240796576*t^8-1026432000000*t^7)*y^7+(256608000000*
t^16+21383144640000*t^15+2862276345300840*t^14+82971785546524938*t^13+
428565125897695536*t^12+469607517282365628*t^11+428565125897695536*t^10+
82971785546524938*t^9+2862276345300840*t^8+21383144640000*t^7+256608000000*t^6
)*y^8+(-7240337280000*t^15-991484631806336*t^14-46287895378500868*t^13-\
336874615686463476*t^12-448310526391887240*t^11-448310526391887240*t^10-\
336874615686463476*t^9-46287895378500868*t^8-991484631806336*t^7-7240337280000
*t^6)*y^9+(1219401216000*t^15+465330532949376*t^14+17684786413194144*t^13+
241265088714763806*t^12+266089870836016872*t^11+447436580570407860*t^10+
266089870836016872*t^9+241265088714763806*t^8+17684786413194144*t^7+
465330532949376*t^6+1219401216000*t^5)*y^10+(-221709312000*t^15-\
167460763987968*t^14-5360117987008704*t^13-136076799330833808*t^12-\
150557853627476688*t^11-252728154714722016*t^10-252728154714722016*t^9-\
150557853627476688*t^8-136076799330833808*t^7-5360117987008704*t^6-\
167460763987968*t^5-221709312000*t^4)*y^11+(48524215742464*t^14+
1301038919151104*t^13+69551430270896616*t^12+26807516850723084*t^11+
225938746267906728*t^10-11588803559941944*t^9+225938746267906728*t^8+
26807516850723084*t^7+69551430270896616*t^6+1301038919151104*t^5+
48524215742464*t^4)*y^12+(-9386285432832*t^14-400838873938944*t^13-\
33335590161232128*t^12+26331280164507192*t^11-138987204138618024*t^10-\
299578279646352*t^9-299578279646352*t^8-138987204138618024*t^7+
26331280164507192*t^6-33335590161232128*t^5-400838873938944*t^4-9386285432832*
t^3)*y^13+(1340897918976*t^14+82382081937408*t^13+15440782735493376*t^12-\
31301371764970320*t^11+74374424464699140*t^10-17686092769029696*t^9+
22963724118160152*t^8-17686092769029696*t^7+74374424464699140*t^6-\
31301371764970320*t^5+15440782735493376*t^4+82382081937408*t^3+1340897918976*t
^2)*y^14+(-12235782316032*t^13-5654355450974208*t^12+15001060138007392*t^11-\
25190987918726992*t^10+4279227499180464*t^9-2394262539871680*t^8-\
2394262539871680*t^7+4279227499180464*t^6-25190987918726992*t^5+
15001060138007392*t^4-5654355450974208*t^3-12235782316032*t^2)*y^15+(
1589958132922368*t^12-5503911731463168*t^11+11315640618091032*t^10-\
15437455816608789*t^9+25955998481173884*t^8-30601126597717758*t^7+
25955998481173884*t^6-15437455816608789*t^5+11315640618091032*t^4-\
5503911731463168*t^3+1589958132922368*t^2)*y^16+(-308196045324288*t^12+
1374371911231488*t^11-3435929778078720*t^10+5676742872485274*t^9-\
6722439283219662*t^8+3107254277581620*t^7+3107254277581620*t^6-\
6722439283219662*t^5+5676742872485274*t^4-3435929778078720*t^3+
1374371911231488*t^2-308196045324288*t)*y^17+(34244005036032*t^12-\
205464030216192*t^11+671928195778560*t^10-1476220701911040*t^9+
2375372767218961*t^8-2904271189787716*t^7+3043065912798822*t^6-\
2904271189787716*t^5+2375372767218961*t^4-1476220701911040*t^3+671928195778560
*t^2-205464030216192*t+34244005036032)*y^18;





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#####simple maple code to check parameterised rational solutions:
# first set up the theta values th[1],th[2],th[3],th[4], and y(s),t(s)

yp:=diff(y,s);tp:=diff(t,s);dydt:=yp/tp;
ypp:=diff(yp,s);tpp:=diff(tp,s);dypdt:=ypp/tp;dtpdt:=tpp/tp;
d2ydt2:=dypdt/tp-yp*dtpdt/tp^2;

f0:=th[1]; f2:=th[2]; f1:=th[3]; f9:=th[4];
al:=(f9-1)^2/2;be:=-f0^2/2;ga:=f1^2/2;de:=(1-f2^2)/2;

f:=-d2ydt2 +(1/y+1/(y-1)+1/(y-t))*(dydt)^2/2
-(1/t+1/(t-1)+1/(y-t))*dydt +((y*(y-1)*(y-t))/(t^2*((t-1)^2)))*
(al+be*t/(y^2)+ga*(t-1)/((y-1)^2)+de*t*(t-1)/((y-t)^2)); 
simplify(f);


#####simple maple code to check implicit solutions F(y,t)=0
# first set up the theta values th[1],th[2],th[3],th[4], and F(y,t) 

dg:=degree(F,y):  
a:='a': ap:='ap': app:='app':
for i from 0 to dg do
  a[i]:=coeff(y*F,y^(i+1));
  ap[i]:=diff(a[i],t);
  app[i]:=diff(ap[i],t);
od;
yp:=-sum('ap[i]*y^i','i'=0..dg)/sum('i*a[i]*y^(i-1)','i'=1..dg);
ypp:=-(
sum('app[i]*y^i','i'=0..dg)+
2*yp*sum('i*ap[i]*y^(i-1)','i'=1..dg)+
yp*yp*sum('i*(i-1)*a[i]*y^(i-2)','i'=2..dg))/sum('i*a[i]*y^(i-1)','i'=1..dg);

f0:=th[1]; f2:=th[2]; f1:=th[3]; f9:=th[4];
al:=(f9-1)^2/2;be:=-f0^2/2;ga:=f1^2/2;de:=(1-f2^2)/2;

f:=-ypp +(1/y+1/(y-1)+1/(y-t))*(yp)^2/2
-(1/t+1/(t-1)+1/(y-t))*yp +((y*(y-1)*(y-t))/(t^2*((t-1)^2)))*
(al+be*t/(y^2)+ga*(t-1)/((y-1)^2)+de*t*(t-1)/((y-t)^2)); 

simplify(f,{F=0});