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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 $U\in SU(n)$.

  1. Is there a simple way to tell if $f$ and $g$ differ only by rotation of variables?
  2. 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 $f(\vec{z}) = g(U \vec{z})$.

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Let us consider the case of $SL(n)$ acting on complex homogeneous polynomials of degree $k$ (the case of $GL(n)$ seems to follow easily). The space of such polynomials is canonically identified with $Sym^k(\mathbf C^{n*})$ and I guess undestanding this representation is a classical subject, although I'm non an expert on this subject. For instance, for non-singular cubics in three variables there is the Weierstrass normal form $y^2 z+x^3+pxz^2 +qz^3$ where $p$, $q\in\mathbf C$. This is a linear variety that intersects each orbit of maximal dimension at exactly one point. Since there is such section, the algebra of invariants is free. According to my references, this algebra is generated by two polynomials respectively of degrees $4$ and $6$, and the stabilizer in general position is $(\mathbf Z_3)^2$. The singular cubics are described in detail in Chapter I, section 7 of H. Kraft, Geometrische Methoden in der Invariantetheorie, Vieweg.
Check also the very nice survey Invariant theory by V. L. Popov and E. B. Vinberg in Algebraic Geometry IV, Encyclopedia of Mathematical Sciences volume 55, Springer-Verlag for more results and references.

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