Algebraicity modulo p of generalized hypergeometric series $_nF_{n-1}$

TL;DR

This paper proves that generalized hypergeometric series modulo a prime p are algebraic with explicitly bounded degree and height, providing a constructive method to find the algebraic polynomial over finite fields.

Contribution

It establishes explicit bounds on the degree and height of algebraic polynomials satisfied by hypergeometric series modulo p, and offers a constructive approach for obtaining these polynomials.

Findings

For primes p > 2d_{α,β}, hypergeometric series modulo p satisfy a polynomial equation.

Explicit bounds on the degree and height of the polynomial are provided.

The method allows for explicit construction of the polynomial P_p(Y).

Abstract

Let $f(z)={}_nF_{n-1}(\mathbf{\alpha},\mathbf{\beta})$ be the hypergeometric series with parameters $\mathbf{\alpha} = (\alpha_1,\ldots,\alpha_n)$ and $\mathbf{\beta} = (\beta_1,\ldots,\beta_{n-1},1)$ in $(\mathbb{Q}\cap(0,1])^n$, let $d_{\mathbf{\alpha},\mathbf{\beta}}$ be the least common multiple of the denominators of $\alpha_1,\ldots,\alpha_n$, $\beta_1,\ldots,\beta_{n-1}$ written in lowest form and let $p$ be a prime number such that $p$ does not divide $d_{\mathbf{\alpha},\mathbf{\beta}}$ and $f(z)\in\mathbb{Z}_{(p)}[[z]]$. Recently in \cite{vmsff}, it was shown that if for all $i,j\in\{1,\ldots,n\}$, $\alpha_i-\beta_j\notin\mathbb{Z}$ then the reduction of $f(z)$ modulo $p$ is algebraic over $\mathbb{F}_p(z)$. A standard way to measure the complexity of an algebraic power series is to estimate its degree and its height. In this work, we prove that if $p>2d_{\mathbf{\alpha},\mathbf{\beta}}$ then there is a nonzero polynomial $P_p(Y)\in\mathbb{F}_p(z)[Y]$ having degree at most $p^{2^n\varphi(d_{\mathbf{\alpha},\mathbf{\beta}})}$ and height at most $5^n(n+1)!p^{2^{n}\varphi({d_{\mathbf{\alpha},\mathbf{\beta}})}}$ such that $P_p(f(z)\bmod p)=0$, where $\varphi$ is the Euler's totient function. Furthermore, our method of proof provides us a way to make an explicit construction of the polynomial $P_p(Y)$. We illustrate this construction by applying it to some explicit hypergeometric series.