Christian Krattenthaler and Tanguy Rivoal

Arithmetic properties of the Taylor coefficients of differentially algebraic power series

(25 pages)

Abstract. Let f = n=0fnxn [[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 . The sequence (fn)n0 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 , there exist two positive integers δ and ν such that the denominator of fn divides δn+1(νn + ν)!2s 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 Pólya–Popken’s upper bound nO(n log(n)) for the denominator of fn is not optimal. This also enables us to make Sibuya and Sperber’s bound |fn|v eO(n), for all finite places v of , explicit in this case. Our method is completely effective and rests upon a detailed p-adic analysis of the above mentioned non-linear recurrences. Finally, we present various examples of differentially algebraic functions for which the associated polynomial M is split over , among which are Weierstraß’ elliptic function, solutions of Painlevé equations, and Lagrange’s solution to Kepler’s equation.


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