Zeros of the derivative of a p-adic meromorphic function and applications

Abstract

Let $K$ be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. We show that if the Wronskian of two entire functions in $K$ is a polynomial, then both functions are polynomials. As a consequence, if a meromorphic function $f$ on all $K$ is transcendental and has finitely many multiple poles, then $f'$ takes all values in $K$ infinitely many times. We then study applications to a meromorphic function $f$ such that $f'+bf^2$ has finitely many zeros, a problem linked to the Hayman conjecture on a p-adic field.