Rational invariants of even degree polynomials under the orthogonal group
Henri Breloer
TL;DR
The work addresses constructing a generating set for the field of rational invariants of the $O(n)$-action on the space $\mathbb{R}[x_1,\dots,x_n]_{2d}$ of even-degree homogeneous polynomials. It harnesses the Slice Lemma to reduce to a finite-group problem on a slice $\Lambda_{2d}^n$ with stabilizer $B(n)$, and then builds an equivariant basis via harmonic decomposition to decompose the problem into tractable pieces $W_1$ and $W_2$. The authors explicitly produce invariants: a set $\{q_j\}$ from $W_1$ and additional invariants $r_i$, $r_\mu$ (together with $p_l$, $q_l$, and $z$ on $W_1$ in the complex setting) that generate the field $\mathbb{R}(\Lambda_{2d}^n)^{B(n)}$, yielding a generating set for $\mathbb{R}(V_{2d}^n)^{O(n)}$ of size $\binom{n+2d-1}{2d}-\binom{n-1}{2}$. While not minimal, this construction leverages the finite-group step to circumvent the graph-isomorphism barrier and provides a practical pathway toward minimal generators and odd-degree extensions in related work.
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.