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Mikhail Kornev

Publications and source records attributed to Mikhail Kornev.

6 recordsLinked to original sources
arXiv2026

Two-Valued Groups, Chazy Equation, Dubrovin-Frobenius Structures, and QYBE

We show that the associativity condition of the universal symmetric 2-algebraic 2-valued group defined by the Buchstaber polynomial admits several mutually equivalent interpretations from the viewpoints of the Chazy equation, Gauss-Manin connections, Dubrovin-Frobenius structures, and the quantum Yang-Baxter equation. These results place the universal 2-valued law in a unified framework linking geometry, algebraic topology, group theory, and mathematical physics.

math.AG↗
arXiv2026

Buchstaber, Ochanine, Krichever, and Witten Genera

We introduce a new class of one-dimensional commutative formal group laws with formal inverse $\bar u=-u$ whose modulus square construction yields Buchstaber's family of polynomials, and prove that the formal group law $F_{\mathrm{Bc}}(u,v)$ is universal for this class over commutative $\mathbb{Z}[1/2]$-algebras. This class is related to, but does not coincide with, the family of formal group laws associated with the Krichever genus. We compute the values of the corresponding Hirzebruch genus $\mathrm{Bc}$ on theta divisors and complex projective spaces, describe its relation to the Ochanine, Krichever, and Witten genera, and show how this construction gives examples not arising from Hirzebruch's elliptic genera of level $n$. We construct a complex-oriented multiplicative cohomology theory $\mathrm{Buc}^*$ such that the complex orientation $\mathrm{U}^*\to\mathrm{Buc}^*$ induces the genus $\mathrm{Bc}$ on coefficient rings, and prove that its localization at the discriminant is naturally isomorphic to the Landweber-exact elliptic cohomology theory $\mathrm{U}^*(-)\otimes_{\Omega_{\mathrm{U}}}\mathbb{Z}[1/2,a_1,a_2,a_3,\Delta^{-1}]$. Finally, we prove that $\mathbb{Z}[a_1,12a_2,360a_3]$ is the smallest subring of $\mathbb{Q}[a_1,a_2,a_3]$ over which the Buchstaber exponential $f_{\mathrm{Bc}}(u)$ is a Hurwitz series. As a corollary, we prove the Hurwitz-integrality statement predicted by Bunkova's coefficient-divisibility conjecture and show that $\mathbb{Z}[g_2/2,6g_3]$ is the minimal Hurwitz coefficient ring of the Weierstrass sigma-function.

math.AT↗
arXiv2025

Algebraic $n$-Valued Monoids on $\mathbb{C}P^1$, Discriminants and Projective Duality

In this work, we establish connections between the theory of algebraic $n$-valued monoids and groups and the theories of discriminants and projective duality. We show that the composition of projective duality followed by the M\"obius transformation $z\mapsto 1/z$ defines a shift operation $\mathbb{M}_n(\mathbb{C}P^1)\mapsto \mathbb{M}_{n-1}(\mathbb{C}P^1)$ in the family of algebraic $n$-valued coset monoids $\{\mathbb{M}_{n}(\mathbb{C}P^1)\}_{n\in\mathbb{N}}$. We also show that projective duality sends each Fermat curve $x^n+y^n=z^n$ $(n\ge 2)$ to the curve $p_{n-1}(z^n; x^n, y^n)=0$, where the polynomial $p_n(z;x,y)$ defines the addition law in the monoid $\mathbb{M}_n(\mathbb{C}P^1)$. We solve the problem of describing coset $n$-valued addition laws constructed from cubic curves. As a corollary, we obtain that all such addition laws are given by polynomials, whereas the addition laws of formal groups on general cubic curves are given by series.

math.GR↗
arXiv2025

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

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.

math.GR↗
arXiv2024

$n$-valued Coset Groups and Dynamics

We obtain asymptotic and exact formulae of growth functions for some families of $n$-valued coset groups. We also describe connections between the theory of $n$-valued groups and Symbolic Dynamics.

math.GR↗