Reduction of Algebraic Parametric Systems by Rectification of their Affine Expanded Lie Symmetries

TL;DR

This paper introduces a method to reduce algebraic parametric systems by identifying affine symmetries and rewriting the system in invariant coordinates, leading to a significant simplification of the problem.

Contribution

It presents a novel algorithm that finds affine Lie symmetries to reduce the number of parameters in algebraic systems, with quasi-polynomial complexity.

Findings

The algorithm effectively reduces parameters in algebraic systems.

Affine symmetries can be used to simplify complex algebraic problems.

The method has a quasi-polynomial arithmetic complexity.

Abstract

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine derivations} that induces \emph{expanded} Lie point symmetries of considered system. By rewriting original problem in an invariant coordinates set for these symmetries, we \emph{reduce} the number of involved parameters. We present an algorithm based on this standpoint whose arithmetic complexity is \emph{quasi-polynomial} in input's size.