Survey on the p-adic Hayman conjecture

Let IK be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value vertical bar . vertical bar, consider an "open" disk d(0, R-) of IK, let M(IK) be the field of meromorphic functions on IK and let M-u(d(0, R-) be the field of meromorphic functions on d(0,R-). All results previously proven about the Hayman conjecture on a p-adic field are recalled, with help of the p-adic Nevanlinna theory. Let f is an element of M(IK) \ IK(x) or f is an element of M-u(d(0,R-) and let b is an element of IK*. For every n >= 3, f'f(n) - b admits infinitely many zeros. If n = 2 and f is an element of M(IK), we prove that f' + bf admits infinitely many zeros. Let f is an element of M(IK). Then, f' bf has infinitely many zeros that are not zeros of f and if f has finitely many multiple zeros and finitely many poles of order 1, then f' + bf(2) has infinitely many zeros that are not zeros of f. If its number of poles inside disks d(0, r(-)) has an upper bound of the form r(q) with q > 0, then f' f - b has infnitely many zeros again. Moreover, f' + bf has infinitely many zeros that are not zeros of f and if f has finitely many multiple zeros and finitely many poles of order 1, then f' + bf(2) has infinitely many zeros that are not zeros of f.