$n$-Valued Groups, Kronecker Sums, and Wendt Matrices
Date:
In the first part of the talk, for each $n$ the notion of classes of symmetric $n$-algebraic $n$-valued groups will be introduced. For $n = 2, 3$, a description of the universal objects of these classes will be given. The second part of the talk is devoted to the $n$-valued groups $G_n$ over the field of complex numbers $\mathbb{C}$. It will be shown that the $n$-valued addition $x * y = [z_1, \ldots, z_n]$ in $\mathbb{G}_n$ is realized in terms of the eigenvalues of the Kronecker sum of the companion matrices of the Frobenius polynomials $t^n - x$ and $t^n - y$ in the variable $t$, where $z = t^n$. $(n \times n)$-matrices $W_n(z; x,y)$ will be introduced, whose determinants are integral polynomials $p_n(z; x,y)$ defining the law $x*y$. It will be shown that $p_n(x; (-1)^n, (-1)^n)$ is the characteristic polynomial of the classical Wendt matrix, which was introduced in 1894 in connection with Fermat’s Last Theorem. As corollaries, results on the structure of the polynomials $p_n(z; x,y)$ will be presented.
The talk is based on the results of the preprint arXiv:2505.04296.