Tanguy Rivoal

Arithmetic properties of the Taylor coefficients of differentially algebraic power series

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 i...

A new transcendence measure for the values of the exponential function at algebraic arguments

Let P ∈ ℤ [ X ] ∖ { 0 } be of degree δ ≥ 1 and usual height H ≥ 1 , and let α ∈ ℚ ¯ * be of degree d ≥ 2 . Mahler proved in 1931 the following transcendence measure for e α : for any ε > 0 , there exists c > 0 such that | P ( e α ) | > c / H μ ( d , δ ) + ε where the exponent μ (...