On the zeroes of certain periodic functions over valued fields of positive characteristic

Abstract: Let C be an algebraically closed field of positive characteristic p and complete with respect to a non-archimedean absolute value and a discrete -submodule. Suppose there exists an -basis of Λ such that and . For define the meromorphic function on C. We show that all the zeroes x of satisfy for some i. Furthermore, the number (counted with multiplicities) of zeroes for which (⁎) holds depends only on i and the p-adic expansion coefficients of k, but not on Λ. [Copyright &y& Elsevier]