On the zeroes of certain periodic functions over valued fields of positive characteristic
نویسندگان
چکیده
Let C be an algebraically closed field of positive characteristic p and complete with respect to a non-archimedean absolute value | . | and Λ ⊂ C a discrete Fp-submodule. Suppose there exists an Fp-basis {λ0, λ1, . . .} of Λ such that 0 < |λ0| < |λ1| < · · · and |λi| −→ ∞. For k ∈ N define the meromorphic function Ck,Λ(z) = ∑ λ∈Λ 1 (z − λ)k on C. We show that all the zeroes x of Ck,Λ satisfy (∗) |x| = |λi| 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. MSC: primary 11R58, secondary 11F52, 14G22
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