arXiv subjects

Alexander Guterman

Publications and source records attributed to Alexander Guterman.

At least 19 recordsLinked to original sources

arXiv2026

Relation graphs of the sedenion algebra

Let $\mathbb{S}$ denote the algebra of the sedenions, and $\Gamma_O(\mathbb{S})$ denote its orthogonality graph. We observe that any pair of zero divisors in $\mathbb{S}$ produces a double hexagon in $\Gamma_O(\mathbb{S})$. The set of vertices of a double hexagon can be extended to a basis of $\mathbb{S}$ which has a convenient multiplication table. We describe explicitly the set of vertices of an arbitrary connected component of $\Gamma_O(\mathbb{S})$ and find its diameter. We then establish the bijection between the connected components of $\Gamma_O(\mathbb{S})$ and lines in the imaginary part of the octonions. Finally, we consider the commutativity graph of the sedenions and discover that all elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies between $3$ and $4$.

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arXiv2026

Additive preservers of permanent rank

The permanent rank of a matrix $A$ is the size of the maximal square submatrix in $A$ which has a nonzero permanent. In this paper we characterize additive transformations $\Phi$ which preserve matrices of per-rank-one. Under an additional assumption that $\Phi$ is surjective or that it preserves per-rank-one in both directions we prove that $\Phi$ is a composition of a multiplication with diagonal matrices and permutation matrices from both sides, transposition, and injective endomorphism of the base field.

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arXiv2026

Characterizing uniform hypergraphs via Seidel matrix and Seidel energy

The Seidel energy is defined as the sum of the absolute values of the eigenvalues of the Seidel matrix of a hypergraph. We first characterize the k-uniform hypergraphs of fixed order n with minimum and maximum Frobenius norms of Seidel matrices and then derive bounds for the Seidel energy. Building on these results, we obtain a negative answer to the hypergraph analogue of Haemers Conjecture by showing that the complete k-uniform hypergraph does not, in general, minimize Seidel energy. Motivated by the theory of hypoenergetic and non-hypoenergetic graphs, we define Seidel hypoenergetic and Seidel non-hypoenergetic hypergraphs and prove that almost all k-uniform hypergraphs are Seidel non-hypoenergetic.

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arXiv2026

Extremal problems on the $p$-Seidel energy of graphs

Let $G$ be a graph with vertex set $\{v_1,\dots,v_n\}$. The Seidel matrix of $G$ is an $ n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $v_i$ and $v_j$ are adjacent, and otherwise is $1$. The $p$-Seidel energy of the graph $G$ is defined as the sum of the absolute values of the $p$-th powers of all eigenvalues of the Seidel matrix of $G$ and introduced in [European Journal of Combinatorics, (86) (2020), 103078]. In this article, we characterize the graph that minimizes the $p$-Seidel energy among all graphs with fixed order $n$, for $p>2$. We also characterize the graph that maximizes the $p$-Seidel energy among all graphs with fixed order $n$, for $0 2$, we characterize the graph that minimizes the $p$-Seidel energy among all $r$-regular graphs with fixed order $n$, where $n$ is a prime power with $n\equiv 1\pmod 4$, $r=\frac{n-1}{2}$. For every $p>2$, we also characterize the graph that maximizes the $p$-Seidel energy among all $r$-regular graphs with fixed order $n=2r$. Finally, we pose several open problems concerning the $p$-Seidel energy for different values of $p$.

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arXiv2026

On the computations of the Cullis' determinant

The Cullis' determinant is a generalization of the ordinary determinant for rectangular matrices. It is defined as the alternating sum of maximal minors of a given matrix. In this paper we express the Cullis' determinant of a matrix $X$ as the Pfaffian of the matrix obtained from $X$ by matrix multiplication and transposition. Relying on this result, we present an efficient polynomial-time division-free algorithm for calculating the Cullis' determinant of a given matrix with entries belonging to the commutative ring. We provide an asymptotical analysis of its arithmetical complexity in comparison to the definition-based algorithm. In addition, we derive formulas for horizontal expansion of the Cullis' determinant which complements the existing formula for Laplace expansion along the columns of a matrix.

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arXiv2025

Linear varieties and matroids with applications to the Cullis' determinant

Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the ordinary determinant. Furthermore, for certain values of $n$ and $k$, we explicitly describe such vector spaces of maximal dimension. Namely, we establish that if $k$ is odd, $n \ge k + 2$ and $\dim(V) = (n-1)k$, then $V$ is equal to the space of all $n\times k$ matrices $X$ such that alternating row sum of $X$ is equal to zero. Our proofs rely on the following observations from the matroid theory that have an independent interest. First, we provide a notion of matroid corresponding to a given linear variety. Second, we prove that if the linear variety is transformed by projections and restrictions, then the behaviour of the corresponding matroid is expressed in the terms of matroid contraction and restriction. Third, we establish that if $M$ is a matroid, $I^*$ its coindependent set $M|S$ and its restriction on a set $S$, then the union of $I^*\setminus S$ with every cobase of $M|S$ is coindependent set of $M$.

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arXiv2025

Linear maps preserving the Cullis' determinant. I

This paper is the first in the series of papers devoted to the explicit description of linear maps preserving the Cullis' determinant of rectangular matrices with entries belonging to an arbitrary ground field which is large enough. The Cullis' determinant is defined for every matrix of size $n\times k$, where $n \ge k \ge 1$ and is equal to the ordinary determinant if $n = k$. In this paper we solve the linear preserver problem for the Cullis' determinant for $k \ge 4, n \ge k + 2$ and $n + k$ is even. It appears that in this case all linear maps preserving the Cullis' determinant are non-singular and could be represented by two-sided matrix multiplication. Note that the cases where $n = k$ or $n = k + 1$ admit slightly different description allowing (sub)matrix transposition and were completely studied before: the case where $n = k$ is a classical linear preserver problem for the ordinary determinant and was solved by Frobenius; the complete characterisation for the case where $n = k + 1$ was obtained in the previous paper by the authors.

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arXiv2025

Linear maps preserving the Cullis' determinant. II

This paper is the second in the series of papers devoted to the explicit description of linear maps preserving the Cullis' determinant of rectangular matrices with entries belonging to an arbitrary ground field which is large enough. In this part we solve the linear preserver problem for the Cullis' determinant defined on the spaces of matrices of size $n\times k$ with $k \ge 4,\; n \ge k + 2$ and $n + k$ is odd. In comparison with the case when $n + k$ is even, in this case linear maps preserving the Cullis' determinant could be singular and are represented as a sum of two linear maps: first is two-sided matrix multiplication and second is any linear map whose image consists of matrices, all rows of which are equal.

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arXiv2024

Birkhoff-James classification of norm's properties

For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $\Gamma(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $\Gamma_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $\Gamma(\mathcal X)$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $\Gamma_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $\Gamma_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

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arXiv2023

On the lengths of descendingly flexible and descendingly alternative algebras

We introduce the classes of descendingly flexible and descendingly alternative algebras over an arbitrary field $\mathbb{F}$. We suggest a new method based on the sequence of differences between the dimensions of the linear spans of words, which allows us to obtain upper bounds on the lengths of these algebras. We also present an example of an algebra of arbitrarily large dimension such that these bounds are achieved on it asymptotically.

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arXiv2023

On the lengths of Okubo algebras

We compute the lengths of two particular cases of (possibly non-unital) composition algebras, namely, standard composition algebras and Okubo algebras over an arbitrary field $\mathbb{F}$. These results finish the complete description of lengths of symmetric composition algebras and finite-dimensional flexible composition algebras.

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arXiv2023

On linear preservers of permanental rank

Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By $\Lambda^{k}$ and $\Lambda^{\leq k}$ we denote the subsets of matrices $A \in {\rm Mat}_{n}(\mathbb{F})$ with ${\rm prk}\,(A) = k$ and ${\rm prk}\,(A) \leq k$, respectively. In this paper for each $1 \leq k \leq n-1$ we obtain a complete characterization of linear maps $T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm Mat}_{n}(\mathbb{F})$ satisfying $T(\Lambda^{\leq k}) = \Lambda^{\leq k}$ or bijective linear maps satisfying $T(\Lambda^{\leq k}) \subseteq \Lambda^{\leq k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then $\Lambda^{k}$ is Zariski dense in $\Lambda^{\leq k}$ and apply this to describe such bijective linear maps satisfying $T(\Lambda^{k}) \subseteq \Lambda^{k}$.

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arXiv2023

Integrability of matrices

The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in number theory and dynamics.

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arXiv2022

Maximal Generalized Rank in Graphical Matrix Spaces

In this note we prove two extensions of a recent combinatorial characterization due to Li, Qiao, Wigderson, Wigderson and Zhang (arXiv:2206.04815) of the maximal dimension of bounded rank subspaces of the graphical matrix space associated with a bipartite graph. Our first result shows that the above characterization remains valid for a wide class of generalized rank functions, including e.g. the permanental rank. Our second result extends the characterization to bounded rank subspaces of the graphical alternating matrix space associated with a general graph.

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arXiv2022

Roots and Critical Points of Polynomials over Cayley--Dickson Algebras

We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial $f(x)$ are also roots of its companion polynomial $C_f(x)$ (defined to be the norm of $f(x)$). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of $f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$, and we also prove that all roots of $f'(x)$ are contained in the snail of $f(x)$, as defined by Ghiloni and Perotti for quaternions. The latter two results generalize the classical Gauss--Lucas theorem to the locally-complex Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on real polynomials to this setting.

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arXiv2022

Values of the length function for nonassociative algebras

We study realizable values of the length function for unital possibly nonassociative algebras of a given dimension. To do this we apply the method of characteristic sequences and establish sufficient conditions of realisability for a given value of length. The proposed conditions are based on binary decompositions of the value and algebraic constructions that allow to modify length function of an algebra. Additionally we provide a classification of unital algebras of maximal possible length in terms of their basis.

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arXiv2022

Steady growth of length function and Malcev algebras

We introduce and investigate the algebras of steadily growing length, that is the class of algebras, where the length is bounded by a linear function of the dimension. In particular we show that Malcev algebras belong to this class and establish the exact upper bound for its length.

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arXiv2022

Algebras of slowly growing length

We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite dimensional Lie algebras as well as many other important classical finite dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras, and Zinbiel algebras. An exact upper bounds for the length of these algebras is proved. To do this we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function.

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