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

Victor Buchstaber, Mikhail Kornev

TL;DR

The work analyzes the structure of the polynomials $p_n(z;x,y)$ that encode $n$-valued group multiplication on $\mathbb{C}$, revealing that they arise as determinants of Wendt-type matrices obtained from Kronecker sums of Frobenius companion matrices for $t^n-x$ and $t^n-y$. It generalizes to $p_n(z;x_1,...,x_m)$ and connects these polynomials to determinants of the so-called Wendt $(x,y,z)$-matrices, discriminants, and resultants, establishing irreducibility results over various fields and providing a discriminant-theoretic interpretation. The paper also classifies symmetric $n$-algebraic $n$-valued groups for small $n$ and constructs a universal framework for such groups, with concrete descriptions for $n=2,3$. Representation-theoretic and dynamical viewpoints are developed via $n$-valued structures on representations and Cartesian products, linking multivalued group theory to linear algebra, invariant theory, and number theory. Overall, the results illuminate deep connections among $n$-valued algebraic structures, determinant representations, and discriminant varieties, with implications for number theory (Fermat-type determinants) and algebraic geometry.

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.

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

TL;DR

The work analyzes the structure of the polynomials that encode -valued group multiplication on , revealing that they arise as determinants of Wendt-type matrices obtained from Kronecker sums of Frobenius companion matrices for and . It generalizes to and connects these polynomials to determinants of the so-called Wendt -matrices, discriminants, and resultants, establishing irreducibility results over various fields and providing a discriminant-theoretic interpretation. The paper also classifies symmetric -algebraic -valued groups for small and constructs a universal framework for such groups, with concrete descriptions for . Representation-theoretic and dynamical viewpoints are developed via -valued structures on representations and Cartesian products, linking multivalued group theory to linear algebra, invariant theory, and number theory. Overall, the results illuminate deep connections among -valued algebraic structures, determinant representations, and discriminant varieties, with implications for number theory (Fermat-type determinants) and algebraic geometry.

Abstract

The article presents results on the well-known problem concerning the structure of integer polynomials , which define multiplication laws in -valued groups over the field of complex numbers . We show that the -valued multiplication in the group is realized in terms of the eigenvalues of the Kronecker sum of companion Frobenius matrices for polynomials of the form in the variable . The notion of a Wendt -matrix is introduced. When , , 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 , the polynomial is given by the determinant of a Wendt -matrix. Iterations of the -valued multiplication in the group lead to polynomials . We prove the irreducibility of the polynomial over various fields. For each , we introduce the notion of classes of symmetric -algebraic -valued groups. The group belongs to one of these classes. For , a description of the universal objects of these classes is obtained.
Paper Structure (10 sections, 35 theorems, 139 equations)

This paper contains 10 sections, 35 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Basic Concepts of the Theory of $n$-Valued Groups
  3. Symmetric $n$-Algebraic $n$-Valued Groups
  4. Kronecker Sums and Frobenius Companion Matrices
  5. Groups $\mathop{\mathrm{\mathbb{G}}}\nolimits_n$ and Kronecker Sums
  6. Groups $\mathop{\mathrm{\mathbb{G}}}\nolimits_n$ and Wendt $(x, y, z)$-matrices
  7. Irreducibility of Polynomials $p_n$
  8. Polynomials $p_n$ and Discriminant Varieties
  9. $n$-Valued Structures on Representations of Groups
  10. Groups $\mathop{\mathrm{\mathbb{G}}}\nolimits^m_n$

Key Result

Proposition 1

The multiplication $\mu$ given by (coset) defines an $n$-valued multiplication on the orbit space $X = G/H$ with identity $e_x = \pi(e_G)$ and inverse $\mathop{\mathrm{\mathrm{inv}}}\nolimits_X(x) = \pi(\mathop{\mathrm{\mathrm{inv}}}\nolimits_G(u))$, where $u\in\pi^{-1}(x)$.

Theorems & Definitions (75)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1: (Coset construction, Buchstaber)
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Remark
  • ...and 65 more