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.