arXiv subjects

Alexei Belov-Kanel

Publications and source records attributed to Alexei Belov-Kanel.

12 recordsLinked to original sources

arXiv2025

On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds

In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{irr}$ and the Sigma index $σ$. The Sigma index was defined through the following relationship: \[ σ(G)=\sum_{uv\in E(G)}\left( d_u(G)-d_v(G) \right)^2. \] We establish a precise formula for the Albertson index of a tree $T$ of order $n$ with a Fibonacci degree sequence $\mathscr{D} = (F_3, \dots, F_n)$. Additionally, we derive bounds for the minimum and maximum Albertson indices ($\irr_{\min}$ and $\irr_{\max}$) across various tree structures. Propositions and lemmas provide upper and lower bounds, incorporating parameters such as the maximum degree $ Δ$, minimum degree $δ$. We further relate the Albertson index to the second Zagreb index $M_2(T)$ and the forgotten index $F(T)$, establishing a new upper bound.

math.CO↗

arXiv2025

Bounds on Trees with Topological Indices Among Degree Sequence

In this paper, we investigate The relationship between the Albertson index and the first Zagreb index for trees. For a tree $T=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges, we provide several bounds and exact formulas for these two topological indices, and we show that the Albertson index $\irr(T)$ and the first Zagreb index $M_1(T)$ satisfy the association \[ \operatorname{irr}(T)=d_1^2+d_n^2+(n-2)\left(\frac{Δ+ δ}{2}\right)^2+\sum_{i=2}^{n-1} d_i+d_n - d_1-2n-2.\] Our goal of this paper is provide a topological indices, Albertson index, Sigma index among a degree sequence $\mathscr{D}=(d_1,\dots,d_n)$ where it is non-increasing and non-decreasing of tree $T$.

math.CO↗

arXiv2025

Centralizers in Free Associative Algebras and Generic Matrices

This paper is concerned with the completion of the proof of the Bergman centralizer theorem by using generic matrices based on our previous quantization proof \cite{KBRZh}. Additionally, we establish that the algebra of generic matrices with characteristic coefficients is integrally closed.

math.RA↗

arXiv2025

Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems

We examine the problem of the linearity of an algebraic torus action in the associative setting. We prove the free algebra analog of a classical theorem of BialynickiBirula, which establishes linearity of maximal torus action. Additionally, we formulate and prove linearity theorems for specific classes of regular actions, and provide a framework for constructing non-linearizable actions, analogous to the work of Asanuma. This framework has applications in the study of the Associative Cancellation Conjecture. Furthermore, we show the existence of two non-isomorphic algebras, whose free products with a polynomial ring are isomorphic.

math.AG↗

arXiv2021

On Automorphisms of the Tame Polynomial Automorphism Group in Positive Characteristic

In this paper we prove that over algebraically closed field $K$ of positive characteristic $\neq 2$ every automorphism of the group of origin-preserving automorphisms of the polynomial algebra $K[x_1,\ldots, x_n]$ ($n>3$) which fixes every diagonal matrix preserves, up to composition with a linear inner automorphism, every tame automorphism.

math.AG↗

arXiv2018

Representability of affine algebras over an arbitrary field

In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and providing a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. In this paper, utilizing ideas from that work, we give a full exposition of Belov's theorem that relatively free affine PI-algebras over an arbitrary field are representable. (Kemer proved the theorem over an infinite field.)

math.RA↗

arXiv2013

Specht's problem for associative affine algebras over commutative Noetherian rings

In a series of papers \cite{BRV1}, \cite{BRV2}, \cite{BRV3} we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "$\bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $Γ$," i.e., a non-identity of the representable algebra $A$ arising from $Γ$, whose ideal of evaluations in $A$ is closed under multiplication by $\bar q$-powers of the characteristic coefficients of matrices corresponding to the generators of $A$, where $\bar q$ is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring $C$ involves localizing at finitely many elements a kind of $C$, and reducing to the field case by a local-global principle.

math.RA↗

arXiv2011

Structure of Zariski-closed algebras

The objective of this paper is to describe the structure of Zariski closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a version of Wedderburn's principal theorem, as well as a more explicit description using representations, in terms of "gluing" in Wedderburn components. Finally, we construct "generic" Zariski closed algebras, whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras. Special attention is given to infinite dimensional algebras over finite fields.

math.RA↗

arXiv2011

Full quivers of representations of algebras

We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to \Zcd\ algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: {\it Frobenius} (along the diagonal) and, more generally {\it proportional} Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.

math.RA↗

arXiv2011

On the lifting of the Nagata automorphism

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K $. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF }$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF )}$.

math.AC↗

arXiv2005

The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that any endomorphism of the $n$-th Weyl algebra (the algebra of polynomial differential operators in $n$ variables) is invertible. We prove that the Jacobian conjecture in dimension $2n$ implies the Dixmier conjecture in rank $n$. Together with a well-known implication in the opposite direction, it shows that the stable Jacobian and Dixmier conjectures are equivalent. The main tool of the proof is the reduction to finite characteristic. After the paper was finished we have learned that the main result was already published by Y.Tsuchimoto in Osaka Journal of Mathematics Volume 42, Number 2 (June 2005). His proof is different.

math.RA↗

arXiv2005

Automorphisms of the Weyl algebra

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic.

math.RA↗