Arithmetic properties of the Taylor coefficients of differentially algebraic power series

Abstract

Let f = ∑ n = 0 ∞ f n x n ∈ ℚ ¯ 〚 x 〛 be a solution of an algebraic differential equation Q ( x , y ( x ) , ⋯ , y ( k ) ( x ) ) = 0 , where Q is a multivariate polynomial with coefficients in ℚ ¯ . The sequence ( f n ) n ≥ 0 satisfies a non-linear recurrence, whose expression involves a polynomial M of degree s . When the equation is linear, M is its indicial polynomial at the origin. We show that when M is split over ℚ , there exist two positive integers δ and ν such that the denominator of f n divides δ n + 1 ( ν n + ν ) ! 2 s for all n ≥ 0 , generalizing a well-known property when the equation is linear. This proves in this case a strong form of a conjecture of Mahler that Popken’s lower bound n - 𝒪 ( n log ( n ) ) for | f n | ≠ 0 is not optimal. This also enables us to make Sibuya–Sperber’s bound | f n | v ≤ e 𝒪 ( n )