Rational invariants of even degree polynomials under the orthogonal group

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.