Notes on symmetries and reductions of algebraic equations

TL;DR

This note analyzes how symmetry properties and invariants in algebraic equations can dramatically reduce problem complexity. By exploiting simple linear symmetries, reciprocal structures, and generalized reciprocal forms, it shows how to lower polynomial degree from $2n$ to $n$ or from $2n+1$ to $n$, often via substitutions like $z = x^2$ or $z = x + 1/x$, and provides concrete examples that yield reduced-degree equations such as $z^n + bz^m + c = 0$. It also demonstrates a two-step equation framework, where $P(P(x)) - x = 0$ factors into $(P(x) - x)Q(x)$, enabling solution through two simpler equations, with practical test cases for numerical solvers. The results offer systematic reduction techniques for symmetric or reciprocal polynomial forms and illustrate their applicability to constructing tractable test problems for numerical methods in algebraic equation solving.

Abstract

Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative examples are provided. The obtained results and solutions can be used as test problems for numerical methods of solving algebraic equations.