Rational Solutions of Underdetermined Polynomial Equations

TL;DR

This paper presents a computer algebra method for efficiently computing rational solutions to large, heavily underdetermined polynomial equations, facilitating the generation of mathematical puzzles used in contests.

Contribution

It introduces new modules for equation splitting in the CRACK system, enabling the solution of large, underdetermined polynomial equations.

Findings

Successfully solved large polynomial equations with up to 10^5 terms

Enhanced the CRACK system with new splitting modules

Enabled automated generation of mathematical puzzles

Abstract

In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our work provides a method to compute rational solutions of single polynomial equations that are typically large with $10^2 \ldots 10^5$ terms and that are heavily underdetermined. This functionality was obtained by adding modules for a new type of splitting of equations to the existing package CRACK that is normally used to solve polynomial algebraic and differential systems.