# Algebraic Expressivity Certificates for Shallow Polynomial Neural Networks

arXiv: https://arxiv.org/abs/2609.28500 · HTML: https://mlsanity.com/paper/2609.28500/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 14 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 3 of 3)

- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)

## Abstract (arXiv)

We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry. Over mathbbC, a width-r network with activation zmapsto zd computes a sum of r d-th powers of linear forms, whose Zariski closure is a Veronese secant variety. Ideal elimination therefore yields polynomial certificates of nonrepresentability. We implement this construction as a generic architecture-to-certificate pipeline. For quadratics, we recover the exact symmetric determinantal description and explain its dimension through orthogonal symmetry. In higher degree, the implementation recovers classical catalecticant and secant equations and maps the practical reach of direct elimination across a finite architecture sweep. We also derive the exact population loss floor for a rank-two quadratic network on the sphere, illustrating how an algebraic obstruction induces irreducible approximation error.