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.

```