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Help with algorithm to find rational roots of a bivariate equation?

• To: mathgroup at smc.vnet.net
• Subject: [mg105080] Help with algorithm to find rational roots of a bivariate equation?
• From: TPiezas <tpiezas at gmail.com>
• Date: Fri, 20 Nov 2009 06:39:05 -0500 (EST)

Hello all,

Does anyone know an efficient algorithm using Mathematica that can
find _rational_ roots of a non-homogenous eqn in two variables with
deg > 4?  For ex, you want to find rational {x,y} such that,

F(x,y) = x^n + (P_1)x^(n-1) + (P_2)x^(n-2) + .... = 0

where the P_i are polynomials in y.

I _always_ come across this situation in the course of experimental
mathematics, and it would be great if Mathematica had a built-in
feature that solves _bivariate_ eqns in the rationals.  Right now, I
have a 22-deg eqn in x with coefficients in y.  I know three non-
trivial rational pairs {x,y} such that F(x,y) = 0, but I want to know
if there are others.  If there are, a certain family of identities
would have more members.

Any help will be appreciated.

- Titus

• Follow-Ups:
  • Re: Help with algorithm to find rational roots of a bivariate
    • From: Daniel Lichtblau <danl@wolfram.com>
  • Re: Help with algorithm to find rational roots of a bivariate equation?
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>