Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$.

- Is there a simple way to tell if $f$ and $g$ differ only by rotation of variables?
- How one can classify orbits or polynomials (i.e. parametrize the representatives)?

Some additional remarks:

There is an easy solution for $n=2$ - it suffices to factorize $$f(z_1,z_2) = c \prod_{i=1}^k (\alpha_i z_1 +\beta_i z_2 ),$$ where $\alpha_i>0$ and $|\alpha_i|^2+|\beta_i|^2=1$. Then one can produce the Gram matrix out of vectors $\psi_i=(\alpha_1,\beta_i)^\dagger$, which is clearly invariant under rotations.

$A\in GL(n)$, instead of $U\in SU(n)$, is also of my interest.

An alternative formulations: what are the orbits in the symmetric subspace of $H^{\otimes k}$ ($H$ - Hilbert space of the dimension $n$) under the action of elements of the form $U^{\otimes k}$, where $U\in SU(n)$.

The questions arises from a problem in quantum optics (states of fixed number of photons that can be reached with so-called passive optics). Numerical and partial answers are welcome as well.

One obvious invariant is the Bombieri norm (it's physical meaning is the conservations of the number of particles), but it gives only the normalization.

One naive approach is to calculate rotation-invariant integrals $$\int_{|z_1|^2+\cdots+|z_n|^2=1} h[p,f(z_1,\cdots,z_n)] dz_1\cdots dz_n, $$ for $h[p,x]$ depending on some parameter, e.g. $h[p,x]=|x|^p$. However, I hardly see how to prove that a set of the same integrals is a sufficient condition for $f(\vec{z}) = g(U \vec{z})$.