On the zeroes of certain periodic functions over valued fields of positive characteristic
Let \({\mathbb C}_p\) be an algebraically closed field of finite characteristic \(p\), complete with respect a non-arhimedean absolute value \(|\cdot |\). Consider \(\Lambda \subseteq {\mathbb C}_p\) a discrete \({\mathbb F}_p\)-submodule such that there exists an \({\mathbb F}_p\)--basis \(\{\lambda_0,\lambda_1,\dots\}\) of \(\Lambda\) such that \(0<|\lambda_0|<|\lambda_1|< \cdots \) and \(|\lambda_i|\to\infty\). For a natural number \(k\) define the meromorphic function \[ C_{k,\Lambda}(z):=\sum_{\lambda\in \Lambda}\frac{1} {(z-\lambda)^k} \] on \({\mathbb C}_p\). A weak form of the main result of the paper establishes that all zeroes of \(C_{k,\Lambda}\) lie on critical spheres \(S_{i,\Lambda}=\partial B(0,|\lambda_i|)=\{ y\in {\mathbb C}_p\mid |y|=|\lambda_i|\}\) of \(\Lambda\). The strong form of the result gives an accurate account of the number of zeroes of \(C_{k,\Lambda}\) on \(S_{i,\Lambda}\), which depends only on \(i\) and \(k\), but not on \(\Lambda\). The paper is organized as follows. In Section 2 the authors collect some facts about \(p\)-adic expansions. In Section 3 power sums on elements of finite sublattices are discussed. Goss polynomials attached to \(\Lambda\) are introduced in Section 4. Section 5 is devoted to the study of the fundamental domain \({\mathfrak F}_{\Lambda}\). The proof of the main result, Theorem 6.1, is given in Section 6. The proof is based on properties of power sums and non-archimedean contour integration. In the last section the authors give examples and applications.
distribution of zeros of \(p\)-adic functions
scientific article; zbMATH DE number 6126843
scientific article