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

TL;DR

The paper develops a framework for exploiting nonclassical and hidden symmetries in polynomial equations by introducing an auxiliary variable to convert problems into symmetric systems. It formalizes methods for handling symmetric and mixed-type systems, and for reducing nonclassical systems to simpler subsystems, often via invariants like $\sigma_1=x+y$ and $\sigma_2=xy$. It provides concrete high-degree, parameterized examples (degrees six and nine) that are solvable in radicals and proposes test problems to benchmark computer algebra systems. The experimental analysis with Maple and Mathematica reveals current limitations in obtaining explicit radical solutions for parameterized problems, while numerical solutions remain robust, underscoring opportunities to enhance CAS capabilities and to guide future algorithmic development. The work thereby contributes both theoretical reduction techniques and practical benchmarks to improve symbolic solving of parametric polynomial equations.

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.

Figures (3)

• Figure 1: Figure 1. Scheme for solving classical symmetric systems of polynomial equations.
• Figure 2: Figure 2. Scheme for solving nonclassical symmetric systems of polynomial equations.
• Figure 3: Figure 3. Scheme for solving systems of polynomial equations of the form \ref{['eee01']} under condition \ref{['eee02']}.