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

Abstract

Let $f(z)={}_nF_{n-1}(\mathbfα,\mathbfβ)$ be the hypergeometric series with parameters $\mathbfα = (α_1,\ldots,α_n)$ and $\mathbfβ = (β_1,\ldots,β_{n-1},1)$ in $(\mathbb{Q}\cap(0,1])^n$, let $d_{\mathbfα,\mathbfβ}$ be the least common multiple of the denominators of $α_1,\ldots,α_n$, $β_1,\ldots,β_{n-1}$ written in lowest form and let $p$ be a prime number such that $p$ does not divide $d_{\mathbfα,\mathbfβ}$ and $f(z)\in\mathbb{Z}_{(p)}[[z]]$. Recently in \cite{vmsff}, it was shown that if for all $i,j\in\{1,\ldots,n\}$, $α_i-β_j\notin\mathbb{Z}$ 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 $p>2d_{\mathbfα,\mathbfβ}$ then there is a nonzero polynomial $P_p(Y)\in\mathbb{F}_p(z)[Y]$ having degree at most $p^{2^nφ(d_{\mathbfα,\mathbfβ})}$ and height at most $5^n(n+1)!p^{2^{n}φ({d_{\mathbfα,\mathbfβ})}}$ such that $P_p(f(z)\bmod p)=0$, where $φ$ 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.