Rational invariants of even degree polynomials under the orthogonal group

TL;DR

This paper constructs a generating set of rational invariants for even degree polynomials under the orthogonal group, revealing a surprising link to the graph isomorphism problem and generalizing previous work from three variables.

Contribution

It extends the understanding of invariants for orthogonal group actions on even degree polynomials to higher dimensions, connecting to graph isomorphism.

Findings

Established a generating set of invariants for general n

Linked the problem to the graph isomorphism problem

Generalized previous results from n=3 to arbitrary n

Abstract

In this article, we construct a generating set of rational invariants for the action of the orthogonal group $\text{O}(n)$ on the space $\mathbb{R}[x_1,\dots,x_n]_{2d}$ of real homogeneous polynomials of even degree $2d$. This generalizes a paper which addressed the case $n=3$. The main difficult with the generalization lies in a surprising connection to the graph isomorphism problem, a classical problem of computer science.