Ternary cyclotomic numbers and ternary Jacobi sums

Cyclotomic numbers and Jacobi sums, introduced over two centuries ago by Gauss and Jacobi, respectively, are pivotal in number theory and find wide applications in combinatorial designs, coding theory, cryptography, and information theory. The cyclotomic problem, focused on determining all cyclotomic numbers, or equivalently evaluating all Jacobi sums of a given order, has been a subject of extensive research. This paper explores their trivariate counterparts, termed "ternary cyclotomic numbers" and "ternary Jacobi sums", highlighting the fundamental properties that mirror those of the classical cases. We show the ternary versions of Fourier series expansions, two symmetry properties, and a summation equation. We further demonstrate that ternary Jacobi sums, with at least one trivial variable, can be evaluated in terms of classical Jacobi sums of the same order. These properties are established through elementary methods that parallel those utilized in classical cases. Based on these properties, then we offer explicit calculations for all ternary Jacobi sums and ternary cyclotomic numbers of order $e=2$, and near-complete results for order $e=3$, with the exception of the elusive integer $J_3(1,1,2)$ for us.