Some old and new results on zeros of the derivative of a p-adic meromorphic function

摘要

Let K be a complete algebraically closed field of characteristic 0 and let f be a transcendental meromorphic function in K. A conjecture suggests that f' takes every values infinitely many times. We can prove that statement when there exists a constant d such that number of multiple poles inside the disk |x| < r is less than r~d for all r > 1, what was published in a previous paper. Moreover, here we can prove that another family of functions whose zeros and poles satisfy certain conditions verify the conjecture. That works for functions in the whole field K but also for functions in an open disk. Several applications are given to entire functions g in K such that g' divides g, to links between residues and zeros of functions admitting primitives and finally to the p-adic Hayman conjecture in the cases that are not yet solved.