On the complexity of invariant polynomials under the action of finite reflection groups

TL;DR

This paper investigates the computational complexity of finding invariant polynomials under finite reflection groups, providing an algorithm to transform polynomials into a canonical form and analyzing its efficiency.

Contribution

It introduces an algorithm for computing invariant polynomials under reflection group actions and analyzes its arithmetic complexity.

Findings

Algorithm successfully computes invariant polynomials

Complexity analysis provides bounds on computational resources

Applicable to multivariate polynomial rings over fields

Abstract

Let $\mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$. Let $(u_1, \dots, u_n)$ be a sequence of $n$ algebraically independent elements in $\mathbb{K}[x_1, \dots, x_n]$. Given a polynomial $f$ in $\mathbb{K}[u_1, \dots, u_n]$, a subring of $\mathbb{K}[x_1, \dots, x_n]$ generated by the $u_i$'s, we are interested infinding the unique polynomial $f_{\rm new}$ in $\mathbb{K}[e_1,\dots, e_n]$, where $e_1, \dots, e_n$ are new variables, such that $f_{\mathrm{new}}(u_1, \dots, u_n) = f(x_1, \dots, x_n)$. We provide an algorithm and analyze its arithmetic complexity to compute $f_{\mathrm{new}}$ knowing $f$ and $(u_1, \dots, u_n)$.