Let $f,g\in \mathbb C[x_1,\ldots,x_n]_d$ be two given homogeneous polynomials of degree $d$. We say they are isomorphic if they differ by a linear coordinates change, i.e. a natural action by some element in $\mathrm{GL}_{n}(\mathbb C)$. I would like to know that, is there an effective way to tell whether two homogeneous polynomials are isomorphic?
I know the Torelli theorem says for curves we can just compute the Jacobian, and there is also a general Torelli theorem works for hypersurfaces. However, it only says the isomorphism classes are determined by some ring structures, and it is still not easy to tell whether two ring structures are isomophic. I would like to know if there are some numbers we can really compute.