Аннотация:
In 1971, V. M. Buchstaber and S. P. Novikov proposed a construction motivated by the theory of characteristic classes.
This construction describes a multiplication such that the product of any pair of points is a multiset of $n$ points.
An axiomatic definition of $n$-valued groups, the results of their algebraic theory, and topological applications
were obtained in a subsequent series of works by V. M. Buchstaber. Currently, the theory of $n$-valued
(formal, finite, discrete, topological, and algebro-geometric) groups and their applications in various areas
of mathematics and mathematical physics are being developed by a number of authors.
In this talk, for each $n$, the notion of classes of symmetric $n$-algebraic $n$-valued groups will be introduced.
For $n = 2$ and $3$, a description of the universal objects in these classes will be presented.
An important class of $n$-algebraic $n$-valued groups is given by the groups $\mathbb{G}_n$ over the field
of complex numbers $\mathbb{C}$.
We show that the $n$-valued multiplication $x*y = [z_1,...,z_n]$ in $\mathbb{G}_n$ is realized in terms
of the eigenvalues of the Kronecker sum of the Frobenius companion matrices of the polynomials $t^n - x$
and $t^n - y$ in the variable $t$.
We introduce $(n \times n)$-matrices $W_n(z; x,y)$ such that for any $n$ their determinant is
an integer-valued homogeneous symmetric polynomial $p_n(z; x,y)$ defining the operation $x*y$.
The matrix $W_n(1; (-1)^n, 1)$ is the classical Wendt matrix, which was introduced in 1894
in connection with Fermat’s Last Theorem.
Groups ${\mathbb G}_n$ and polynomials $p_n(z;x,y)$ arise and play an important role in various fields
of mathematics and mathematical physics.
In this talk we will present results that open up a new approach to the well-known problem
of the Fermat equation for the Kummer tower of cyclotomic fields.
This talk is based on the results of the preprint arXiv: 2505.04296, V. Buchstaber, M. Kornev,
$n$-Valued Groups, Kronecker Sums, and Wendt's $(x,y,z)$-Matrices.
