Arithmetic Conditions for the Invertibility of Circulant Matrices over Finite Fields

Abstract

This paper investigates arithmetic conditions, over finite fields, for the invertibility of circulant matrices whose first row is $(a_0,\ldots,a_{n-1})$, in terms of linear constraints on the entries $a_0,\ldots,a_{n-1}$. The coefficients of such linear constraints are determined by field traces arising from Frobenius orbits of roots of unity, and under additional assumptions they are given by Ramanujan sums reduced modulo the characteristic of the given field. These results provide finite-field analogues of the corresponding criteria over $\mathbb{Q}$, where the relevant Ramanujan-sum linear constraints can be formulated without such additional assumptions, while also showing some differences caused by reduction modulo the characteristic. The constraints can often determine invertibility using aggregate information about the entries of a row, rather than the particular pattern of the entries. We also discuss the polynomial gcd criterion, which gives necessary and sufficient conditions for invertibility and yields further consequences for special families of circulant matrices.