$n$-Valued Groups, Kronecker Sums, and Wendt's Matrices

Abstract

The article presents results on the well-known problem concerning the structure of integer polynomials $p_n(z; x, y)$, which define multiplication laws in $n$-valued groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. We show that the $n$-valued multiplication in the group $\mathbb{G}_n$ is realized in terms of the eigenvalues of the Kronecker sum of companion Frobenius matrices for polynomials of the form $t^n - x$ in the variable $t$. The notion of a Wendt $(x, y, z)$-matrix is introduced. When $x = (-1)^n$, $y = z = 1$, one recovers the classical Wendt matrix, whose determinant is used in number theory in connection with Fermat's Last Theorem. It is shown that for each positive integer $n$, the polynomial $p_n$ is given by the determinant of a Wendt $(x, y, z)$-matrix. Iterations of the $n$-valued multiplication in the group $\mathbb{G}_n$ lead to polynomials $p_n(z; x_1, \dots, x_m)$. We prove the irreducibility of the polynomial $p_n(z; x_1, \dots, x_m)$ over various fields. For each $n$, we introduce the notion of classes of symmetric $n$-algebraic $n$-valued groups. The group $\mathbb{G}_n$ belongs to one of these classes. For $n = 2, 3$, a description of the universal objects of these classes is obtained.