Counting roots of truncated hypergeometric series over finite fields

TL;DR

This paper investigates the roots of truncated hypergeometric series over finite fields, establishing asymptotic bounds and exploring connections to elliptic curves and K3 surfaces, with computational illustrations.

Contribution

It provides new asymptotic upper bounds on the roots of truncated hypergeometric series over finite fields and links these to algebraic geometric structures.

Findings

Asymptotic upper bounds of O(p^{11/12}) on roots in _p

Connections to elliptic curves and K3 surfaces with sharp bounds in some cases

Computational evidence supporting theoretical results

Abstract

We consider natural polynomial truncations of hypergeometric power series defined over finite fields. For these truncations, we establish asymptotic upper bounds of order $O(p^{11/12})$ on the number of roots in the prime field $\mathbb{F}_p$. We discuss the correspondence to families of elliptic curves and K3 surfaces of certain such hypergeometric polynomials, for which sharp bounds are obtained in some cases. We include some computations to illustrate and supplement our results.