Zeros of p-adic differential polynomials

Abstract

Let $$\mathbb{K}$$ be a complete algebraically closed p-adic field of characteristic zero. We consider a differential polynomial of the form F = a n f n f (k) + a n−1 f n−1 + ... + a 0 where the a j are small functions with respect to f and f is a meromorphic function in $$\mathbb{K}$$ or inside an open disk. Using p-adic methods, we can prove that when N(r, f) = S(r, f), then F must have infinitely many zeros, as in complex analysis.