Asymptotic expected-dimension conjecture for polynomial neural network neurovarieties

Let $d53d=(d_0,d55fd53\dots,d_L)$ be a fixed vector of widths, with $d_i>1$ for $i=1,d55fd53\dots,L-1$, and let $r$ denote the activation degree. The neurovariety $d4b1_{d53d,r}$ is the image variety associated with this architecture, and its expected dimension is the dimension predicted by the parameter count and ambient dimension.

Asymptotic expected-dimension conjecture. There exists a threshold $d53fdd53f(d53d)$ such that, whenever $r>d53f$, the neurovariety $d4b1_{d53d,r}$ attains the expected dimension.

The claim concerns eventual non-defectivity as the activation degree grows. It was previously asserted as a theorem, but the proof contained a mistake; the statement has subsequently been proved in the cited work on activation functions.