Integer and Rational Solutions to Polynomial Equations

TL;DR

This paper introduces a formalism for counting integer and rational solutions to polynomial equations, especially elliptic curves, by relating them through mappings between elliptic curves with different coefficients.

Contribution

It presents a novel formalism that uses elliptic curves and their mappings to systematically count solutions to polynomial equations with rational coefficients.

Findings

A new method to count solutions using elliptic curve mappings

Reduction of the counting problem to relations between elliptic curves

Framework applicable to polynomials parameterized by three integers

Abstract

A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational solutions to $y^2=P(x)$ is facilitated by another elliptic curve with integral coefficients. The problem of counting is described by two elliptic curves and a map between them.