Arithmetic properties of the Taylor coefficients of differentially algebraic power series
(25 pages)
Abstract.
Let f= ∑n=0∞fnxn∈ℚ[[x] be a solution of an algebraic differential
equation Q(x,y(x),…,y(k)(x)) = 0, where Qis a multivariate polynomial
with coefficients in ℚ. The sequence (fn)n≥0 satisfies a non-linear
recurrence, whose expression involves a polynomial Mof degree s. When
the equation is linear, Mis its indicial polynomial at the origin. We show
that when Mis split over ℚ, 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(nlog(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 vof ℚ, 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 ℚ, among which are Weierstraß’ elliptic ℘ function,
solutions of Painlevé equations, and Lagrange’s solution to Kepler’s
equation.