Bill Allombert on Tue, 03 Feb 2009 13:41:19 +0100 |
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Re: System of two polynomial equations |
On Tue, Feb 03, 2009 at 08:06:11AM +0100, ewan.Delanoy@math.unicaen.fr wrote: > 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 ? nffactor use a totally rigorous algorithm. However what it does is more like a padic variant of lindep (it use LLL) than something purely algebraic. Cheers, Bill.