Linear diophantine equations with cyclic coefficient matrices and its applications to Riemann surfaces

Abstract

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equationswhere Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.