On the $p$-integrality of $A$-hypergeometric series

TL;DR

This paper characterizes when certain $A$-hypergeometric series solutions have $p$-integral coefficients based on the $p$-integrality and rationality of the associated vector $v$, linking algebraic and number-theoretic properties.

Contribution

It provides a characterization of $p$-integrality of coefficients for $A$-hypergeometric series solutions in terms of the properties of the vector $v$.

Findings

Identifies conditions on $v$ for $p$-integral series coefficients

Links $p$-integrality to rationality and minimal negative support

Provides criteria for $p$-integrality within a specific interval

Abstract

Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $\beta = Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients.