#Homework 10 by Ben Miles for Experimental math assigned 2/20 OK to post
#Background
GuessAlg:=proc(L,P,x, MaxDegP) local g,DegP:
for DegP from 1 to MaxDegP do
 g:=GuessAlg1(L,P,x,DegP):
 if g<>FAIL then
     RETURN(g):
 fi:
od:
FAIL:
end:
GuessAlg1:=proc(L,P,x, DegP) local g,Degx:
for Degx from 0 while (1+Degx)*(1+DegP)<nops(L)-6 do
g:=GuessAlg11(L,P,x, DegP,Degx):
if g<>FAIL then
 RETURN(g):
fi:
od:
FAIL:
end:
GuessAlg11:=proc(L,P,x, DegP, Degx) local g, F,a, i,j,
var,P1,F1,eq, var1,v:
F:=add(add(a[i,j]*x^i*P^j,i=0..Degx),j=0..DegP):
var:={seq(seq(a[i,j],i=0..Degx),j=0..DegP)}:
P1:=add(L[i]*x^(i-1),i=1..nops(L)):
F1:=expand(subs(P=P1,F)):
eq:={seq( coeff(F1,x,i)=0, i=0..nops(L)-1)}:
var1:=solve(eq,var):
F:=subs(var1,F):
if F=0 then
 RETURN(FAIL):
else
F:=subs({seq(v=1, v  in var)},F):
F:=add(factor(coeff(F,P,i))*P^i,i=0..degree(F,P)): 
RETURN(F):
fi:
end:



#Homework
#Number 2

DiagToAlg:=proc(R,x,y,P,t,MaxDegP) local L,Q:

Q:=mtaylor(R,{x,y}, 2*MaxDegP+22):
L:=[subs({x=0,y=0},R),seq(coeff(coeff(Q,x^i),y^i),i=1..MaxDegP+10)]:
GuessAlg(L,P,t, MaxDegP);
end:

#DiagToAlg(1/((1-x)*(1-y)),x,y,P,t,6)=1+(-1+t)*P
#DiagToAlg(1/(1-x-y),x,y,P,t,6)=-1+(1-4*t)*P^2
#DiagToAlg(1/(1-x-y+3*x*y),x,y,P,t,6)=-1/2+(1/2+t+(9/2)*t^2)*P^2
#DiagToAlg(1/(1-x-x^2-y-y^2),x,y,P,t,10)=FAIL



#Number 5. I am not really sure how to do this one.

LIF:=proc(F,z,P,n) :






end: