Optimalité pour un problème de Bézivin

Abstract

Let q be an integer such that |q|≥2 and s be a positive integer. In this article, we show that an entire function f such that lim ¯ r→+∞ ln|f| r ln 3 r<4s 27ln 2 |q| and taking Gaussian integer values on {q m +iq n ∣m,n∈ℕ}, as well as its s-1 first derivatives, is a polynomial. Moreover, the bound 4s 27ln 2 |q| is optimal. This generalizes and improves a result obtained by Bézivin in [1].