Algebraicity modulo p of generalized hypergeometric series [formula omitted

Abstract

Let f(z)=nFn−1(α,β) be the hypergeometric series with parameters α=(α1,…,αn) and β=(β1,…,βn−1,1) in (Q∩(0,1])n, let dα,β be the least common multiple of the denominators of α1,…,αn, β1,…,βn−1 written in lowest form and let p be a prime number such that p does not divide dα,β and f(z)∈Z(p)[[z]]. Recently in [11], it was shown that if for all i,j∈{1,…,n}, αi−βj∉Z then the reduction of f(z) modulo p is algebraic over Fp(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>2dα,β then there is a nonzero polynomial Pp(Y)∈Fp(z)[Y] having degree at most p2nφ(dα,β) and height at most 5n(n+1)!p2nφ(dα,β) such that Pp(f(z)modp)=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 Pp(Y). We illustrate this construction by applying it to some explicit hypergeometric series.