Expressivity of Shallow Neural Networks Over Finite Fields

Maksym Zubkov; Carol Wu; Shiwei Yang; Param Mody; Yifei Chen

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arXiv · 2607.17090

Expressivity of Shallow Neural Networks Over Finite Fields

Maksym Zubkov·Carol Wu·Shiwei Yang·Param Mody·Yifei Chen

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Abstract

We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.

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Maksym Zubkov, Carol Wu, Shiwei Yang, Param Mody, Yifei Chen. 2026-07-19. Expressivity of Shallow Neural Networks Over Finite Fields. https://arxiv.org/abs/2607.17090

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