Algebraicity modulo p of generalized hypergeometric series [formula omitted].

Let f (z) = n F n − 1 (α , β) be the hypergeometric series with parameters α = (α 1 , ... , α n) and β = (β 1 , ... , β n − 1 , 1) in (Q ∩ (0 , 1 ]) n , let d α , β be the least common multiple of the denominators of α 1 , ... , α n , β 1 , ... , β n − 1 written in lowest form and let p be a prime number such that p does not divide d α , β and f (z) ∈ Z (p) [ [ z ] ]. Recently in [11] , it was shown that if for all i , j ∈ { 1 , ... , n } , α i − β j ∉ Z then the reduction of f (z) modulo p is algebraic over 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 > 2 d α , β then there is a nonzero polynomial P p (Y) ∈ F p (z) [ Y ] having degree at most p 2 n φ (d α , β) and height at most 5 n (n + 1) ! p 2 n φ (d α , β) such that P p (f (z) mod 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. [ABSTRACT FROM AUTHOR]