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

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.