ewan . Delanoy on Tue, 03 Feb 2009 08:08:05 +0100

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Re: System of two polynomial equations

    Bill Alombert wrote :

>>      if F and G are two polynomials in two variables x
>> and y, the system {F(x,y)=0,G(x,y)=0}, is usually
>> equivalent to a system of the form {P(x)=0,y=Q(x)}, where
>> P and Q are univariate polynomials.
>Could you provide an example and a more definite statement ?
>Obviously this imply that there are no pair of solution of the form
>{(x0,y0),(x0,y1)} with y0!=y1 because y0=Q(x0)=y1.

   You're right, a more accurate statement would be : the initial
system {F(x,y)=0,G(x,y)=0} is equivalent to a union of systems of the form
{P_i(x)=0, y=Q_i(x)}, where the P_i are the irreducible factors
of polresultant(F(x,y),G(x,y),y).

>I suggest you factorize P over K and then factorize G over (K[x]/P).
>In case where K=Q, you can use nffactor to factorize Q.

  Yes, nffactor certainly looks like exactly what I need. How is nffactor
implemented ? Does it perform in a purely algebraic way, or
does it use some "guesswork" as in lindep ?