Nonclassical symmetries of polynomial equations and test problems with parameters for computer algebra systems

TL;DR

This paper explores nonclassical symmetries of polynomial equations, introduces methods for simplifying complex systems, and presents test problems to evaluate and improve computer algebra systems like Maple and Mathematica.

Contribution

It introduces new techniques for reducing and solving nonclassical symmetric polynomial systems and provides benchmark problems to assess computer algebra systems' capabilities.

Findings

Symmetric systems can be simplified via variable introduction.

Current systems struggle with analytical solutions for parametric equations.

Numerical solutions are feasible for fixed parameters.

Abstract

Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of polynomial equations by introducing a new additional variable. It has been established that symmetric systems of polynomial equations of mixed type, consisting of symmetric and anti-symmetric polynomials, can be transformed into simpler systems. A method is presented for solving nonclassical symmetric systems of two polynomial equations that change places when the unknowns are permuted. We study polynomial equations containing the second iteration of a given polynomial, which are reduced to nonclassical symmetric systems of equations. New higher-degree polynomial equations containing free parameters that admit solutions in radicals are found. Three such equations of the sixth and ninth degrees are further used as test problems with parameters for analyzing the capabilities of two leading computer algebra systems. It is shown that currently, the Maple and Mathematica systems do not allow us to efficiently find analytical solutions (in radicals) of polynomial equations with free parameters, but they allow us to obtain numerical solutions of equations for fixed numerical values of the parameters. The results of this work and the proposed test problems with parameters can be used to further improve existing computer algebra systems.