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 ? Ewan