The Alexander-Hirschowitz theorem for neurovarieties

arXiv:2511.19703v2 Announce Type: replace-cross Abstract: We study the dimension and identifiability of neurovarieties associated to polynomial neural networks.

We give an independent geometric proof that the linear bounds $d_i\geq 2n_i-1$ on the activation degrees imply non defectiveness for any number of outputs, a dimension statement previously obtained from finite identifiability.

The proof is based on a direct analysis of the differential of the parameterization. We also investigate secant and Grassmann-secant obstructions outside this range and prove global identifiability for multi-output architectures under the same degree bounds.