# Homework 10 - Alex Gentul OK to share

info1:=proc(); print(`guesAlg(sequence L,symbol P,variable x,integer deg)`); end:
info1();


	# 2 - diagToAlg(function R,variable x,variable y,symbol P,t,deg) that inputs a rational function R of the two variables x and y, and outputs a guessed algebaric equation F(P,t) such that if you do mtaylor to R (do the two-dimensional Taylor expansion of R about the origin) and call its Taylor coefficients ai,j, then if 




diagToAlg:=proc(R,x,y,P,t,deg) local A,B:

A:=mtaylor(R,{x,y}, 3*deg):
B:=[subs({x=0,y=0},R),seq(coeff(coeff(A,x^i),y^i),i=1..2*deg)]:
GuessAlg(L,P,t,deg);

end:






















	##### from c10.txt:

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: