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

• To: mathgroup at smc.vnet.net
• Subject: [mg105121] Re: [mg105080] Help with algorithm to find rational roots of a bivariate equation?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 21 Nov 2009 03:36:26 -0500 (EST)
• References: <200911201139.GAA03433@smc.vnet.net>

```Efficient? I think the only general algorithm for finding rational
points is that due to Poincare, who showed hoe to find all rational
points on a curve of genus 1. Since the genus is (n-1)(n-2)/2 - d, where
n is the degree and d the number of double points, your curve must have
2 double points. In fact, if I remember correctly, even the Poincare
method is not really an algorithm but a collection of special techniques
(but I may be wrong).
Anyway, I am sure that not only there is no such Mathematica program but
that there is no known algorithm for solving this type of problem at
all.

Andrzej Kozlowski

On 20 Nov 2009, at 20:39, TPiezas wrote:

> 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
>

```

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