In this talk I will introduce the main topics about the new field of research called Neuroalgebraic Geometry. In particular I will focus on the definitions, results (and conjectures) contained in the papers

- J. Kileel, M. Tranger, J. Bruna, On the Expressive Power of Deep Polynomial Neural Networks, (2019).

- K. Kubjas, J. Li, M. Wiesmann, Geometry of polynomial neural networks, (2024).

- G. L. Marchetti, V. Shahverdi, S. Mereta, M. Trager, K. Kohn, An Invitation to Neuroalgebraic Geometry, (2025).

and I will talk also about my recent results on it and a list of suggested problems.

We generalize an induction-specialization technique due to Brambilla and Ottaviani into a construction we call the \emph{inductant}. With it, we obtain a computer-assisted proof that Segre-Veronese varieties $\mathbb P^m \times \mathbb P^n$ embedded by $O(1,2)$ are non-defective when $n\gg m^3$, $m\geq3$. This was already proved in the subabundant case by Abo and Brambilla, we provide a complementary result in the superabundant case.

Based on joint work with Nikhil Ken.

We describe the irreducible components of the Hilbert scheme of d points on affine space for d=9, 10. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a specific Hilbert function. As the main consequence, we establish the equality of cactus Grassmann and the secant Grassmann variety in the corresponding cases. This is a joint project with Hanieh Keneshlou and Klemen Sivic.

A very classical result, the Castelnuovo Lemma, states that, given d+3 points in general position in P^d, there is a unique rational normal curve of degree d passing through them. 

An easy argument shows that asking for a rational normal curve to pass through a point, imposes, in the parameter spaces of rational normal curves, the same number of conditions imposed by asking for a rational normal curve to be (d-1)-secant to a codimension 2 projective subspace of P^d. 

For this reason, it is quite natural to pose the following problem: given p points in P^d and l codimension 2 subspaces in P^d with p+l=d+3, is there a rational normal curve passing through the points and being (d-1)-secant to the l projective subspaces? 

A paper of Carlini and Catalisano (2007) solves all the cases except for (p,l)=(0,d+3). In this talk we show how to solve this missing case by using symmetric polynomials.