arXiv subjects

Dmitri Piontkovski

Publications and source records attributed to Dmitri Piontkovski.

At least 19 recordsLinked to original sources

arXiv2026

Growth in noncommutative algebras and entropy in derived categories

A noncommutative projective variety is defined, after Artin and Zhang, by a graded coherent algebra A, where the category of coherent sheaves is the quotient qgr(A) of the category of finitely presented graded modules by the subcategory of torsion modules. We consider the categorical and polynomial entropies of the Serre twist, that is, of the degree shift functor on the bounded derived category of qgr(A). These two types of entropy can be viewed as analogues of the dimension of the noncommutative variety. We relate these invariants with the growth of the algebra. For algebras of finite global dimension, the entropies are bounded above by the growth entropy and the Gelfand--Kirillov dimension of the algebra. Moreover, these equalities hold for regular algebras, as well as for coordinate rings of smooth projective varieties. However, the polynomial entropy is zero for monomial algebras of polynomial growth, so in this case the inequality is strict.

math.RA↗

arXiv2025

Multilinear nilalgebras and the Jacobian theorem

If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed by Dotsenko.

math.RA↗

arXiv2025

Degree of freedom count in linear gauge invariant PDE systems

We consider not necessarily Lagrangian partial linear differential equations (PDE) with constant coefficients. Einstein proposed a definition of the"strength" of such a field theory that defines its degree of freedom (DoF). Einsteinian strength is based on the asymptotic number of free Taylor series coefficients of bounded degree in the general solution of the PDE system. The direct count of DoF in this way is a complex and technically demanding process. Proceeding from Einsteinian strength of equations and making use of commutative algebra tools, we deduce another DoF count recipe which is formulated in terms of orders of the field equations, their gauge symmetries and gauge identities. This DoF count recipe covers the case of reducible gauge field theories, and it is easy to use. We begin with interpreting the matrix of the system as a linear map between polynomial modules. First, proceeding from Einsteinian definition, we derive an explicit formula for DoF as the multiplicity of a certain extension module. Second, we prove another explicit formula for DoF in terms of orders of equations and gauge generators. A notable consequence of this formula is that two Hermitian conjugate systems have identical DoF. Every classical field theory defines the BRST complex which has the natural grading known as the ghost number. We equip this complex with another grading, which we call differential order. This grading is 1 for every space time-derivative, while all thefields, ghosts, and antifields are assigned with this degree in a certain way, depending on their ghost number and the orders of equations and gauge generators. We compute the Euler characteristic of the BRST complex with respect to this new grading. This provides homological interpretation of DoF for linear gauge system as minus the residue at infinity of the logarithmic derivative of the Euler characteristics for its BRST complex.

math-ph↗

arXiv2023

Entropy of monomial algebras and derived categories

Let A be a finitely presented associative monomial algebra. We study the category qgr(A) which is a quotient of the category of graded finitely presented A-modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category. We calculate the categorical entropy of the Serre twist functor this derived category and show that it is equal to logarithm of the entropy of the algebra A itself. Moreover, we relate these two kinds of entropy with the topological entropy of the Ufnarovski graph of A and the entropy of the path algebra of the graph. If A is a path algebra of some quiver, the categorical entropy is equal to the logarithm of the spectral radius of the quiver's adjacency matrix.

math.RA↗

arXiv2022

Acceptability Judgements via Examining the Topology of Attention Maps

The role of the attention mechanism in encoding linguistic knowledge has received special interest in NLP. However, the ability of the attention heads to judge the grammatical acceptability of a sentence has been underexplored. This paper approaches the paradigm of acceptability judgments with topological data analysis (TDA), showing that the geometric properties of the attention graph can be efficiently exploited for two standard practices in linguistics: binary judgments and linguistic minimal pairs. Topological features enhance the BERT-based acceptability classifier scores by $8$%-$24$% on CoLA in three languages (English, Italian, and Swedish). By revealing the topological discrepancy between attention maps of minimal pairs, we achieve the human-level performance on the BLiMP benchmark, outperforming nine statistical and Transformer LM baselines. At the same time, TDA provides the foundation for analyzing the linguistic functions of attention heads and interpreting the correspondence between the graph features and grammatical phenomena.

cs.CL↗

arXiv2022

Noncommutative Grassmannian of codimension two has coherent coordinate ring

A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different from the recent proofs of the coherence of some graded 3-dimensional Calabi--Yau algebras and is based on properties of a PBW-basis of the algebra.

math.RA↗

arXiv2022

Betti numbers of attention graphs is all you really need

We apply methods of topological analysis to the attention graphs, calculated on the attention heads of the BERT model ( arXiv:1810.04805v2 ). Our research shows that the classifier built upon basic persistent topological features (namely, Betti numbers) of the trained neural network can achieve classification results on par with the conventional classification method. We show the relevance of such topological text representation on three text classification benchmarks. For the best of our knowledge, it is the first attempt to analyze the topology of an attention-based neural network, widely used for Natural Language Processing.

cs.CL↗

arXiv2022

Artificial Text Detection via Examining the Topology of Attention Maps

The impressive capabilities of recent generative models to create texts that are challenging to distinguish from the human-written ones can be misused for generating fake news, product reviews, and even abusive content. Despite the prominent performance of existing methods for artificial text detection, they still lack interpretability and robustness towards unseen models. To this end, we propose three novel types of interpretable topological features for this task based on Topological Data Analysis (TDA) which is currently understudied in the field of NLP. We empirically show that the features derived from the BERT model outperform count- and neural-based baselines up to 10\% on three common datasets, and tend to be the most robust towards unseen GPT-style generation models as opposed to existing methods. The probing analysis of the features reveals their sensitivity to the surface and syntactic properties. The results demonstrate that TDA is a promising line with respect to NLP tasks, specifically the ones that incorporate surface and structural information.

cs.CL↗

arXiv2022

Bernstein algebras that are algebraic and the Kurosh problem

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of train Bernstein algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train). Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the barideal is nil, the Kurosh problem asks whether a finitely generated Bernstein-train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case. By results of Yagzhev, the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.

math.RA↗

arXiv2021

Wilf classes of non-symmetric operads

Two operads are said to belong to the same Wilf class if they have the same generating series. We discuss possible Wilf classifications of non-symmetric operads with monomial relations. As a corollary, this would give the same classification for the operads with a finite Groebner basis. Generally, there is no algorithm to decide whether two finitely presented operads belong to the same Wilf class. Still, we show that if an operad has a finite Groebner basis, then the monomial basis of the operad forms an unambiguous context-free language. Moreover, we discuss the deterministic grammar which defines the language. The generating series of the operad can be obtained as a result of an algorithmic elimination of variables from the algebraic system of equations defined by the Chomsky--Schutzenberger enumeration theorem. We then focus on the case of binary operads with a single relation. The approach is based on the results by Rowland on pattern avoidance in binary trees. We improve and refine Rowland's calculations and empirically confirm his conjecture. Here we use both the algebraic elimination and the direct calculation of formal power series from algebraic systems of equations. Finally, we discuss the connection of Wilf classes with algorithms for the Quillen homology of operads calculation.

math.CO↗

arXiv2020

Context-free languages and associative algebras with algebraic Hilbert series

In this paper, homological methods together with the theory of formal languages of theoretical computer science are proved to be effective tools to determine the growth and the Hilbert series of an associative algebra. Namely, we construct a class of finitely presented associative algebras related to a family of context-free languages. This allows us to connect the Hilbert series of these algebras with the generating functions of such languages. In particular, we obtain a class of finitely presented graded algebras with non-rational algebraic Hilbert series.

math.RA↗

arXiv2019

Noncommutative algebras, context-free grammars and algebraic Hilbert series

In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.

math.RA↗

arXiv2017

Homogeneous finitely presented monoids of linear growth

If a finitely generated monoid M is defined by a finite number of degree-preserving relations, then it has linear growth if and only if it can be decomposed into a finite disjoint union of subsets (which we call "sandwiches") of the form $a b$, where $a,b,w$ are elements of $M$ and $ $ denotes the monogenic semigroup generated by $w$. Moreover, the decomposition can be chosen in such a way that the sandwiches are either singletons or "free" ones (meaning that all elements $a w^n b$ in each sandwich are pairwise different). So, the minimal number of free sandwiches in such a decomposition is a numerical invariant of a homogeneous (and conjecturally, non-homogeneous) finitely presented monoid of linear growth.

math.GR↗

arXiv2017

Growth in varieties of multioperator algebras and Groebner bases in operads

We discuss algorithmic approach to growth of the codimension sequences of varieties of multilinear algebras, or, equivalently, the sequences of the component dimensions of algebraic operads. The (exponentional) generating functions of such sequences are called codimension series of varieties, or generating series of operads. We show that in general there does not exist an algorithm to decide whether the growth exponent of a codimension sequence of a variety defined by given finite sets of operations and identities is equal to a given rational number. In particular, we solve negatively a recent conjecture by Bremner and Dotsenko by showing that the set generating series of binary quadratic operads with bounded number of generators is infinite. Then we recall algorithms which in many cases calculate the codimension series in the form of a defining algebraic or differential equation. For a more general class of varieties, these algorithms give upper and lower bounds for the codimensions in terms of generating functions and asymptotical bounds for the growth of codimensions. The upper bound (based on an operadic version of the Golod--Shafarevich theorem) is just a formal power series satisfying an algebraic equation defined effectively by the generators and the identities of the variety. The first stage of an algorithm for the lower bound is the construction of a Groebner basis of the operad. If the Groebner basis happens to be finite and satisfies mild restrictions, a recent theorem by the author and Anton Khoroshkin guarantees that the desired generating function is either algebraic or differential algebraic. We describe algorithms producing such equations. In the case of infinite Groebner basis, these algorithms applied to its finite subsets give lower bounds for the generating function of the codimension sequence.

math.RA↗

arXiv2017

Algebras of linear growth and the dynamical Mordell-Lang conjecture

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the "only if" part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with $5$ generators is infinite. Our approach is based on a connection with the dynamical Mordell--Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell--Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem--Mahler--Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counter-examples to this theorem in the finite characteristic case give examples of algebras with irrational Hilbert series.

math.RA↗

arXiv2014

A remark on Golod--Shafarevich algebras

We show that a direct limit of surjections of (weak) Golod--Shafarevich algebras is a weak Golod--Shafarevich algebra as well. This holds both for graded and for filtered algebras provided that the filtrations are induced by the filtration of the first entry of the sequence. It follows that the limit is an algebra of exponential growth. An example shows that the assumptions of this theorem cannot be directly weakened.

math.RA↗

arXiv2014

On generating series of finitely presented operads

Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

math.QA↗

arXiv2009

Cohomology of 3-dimensional color Lie algebras

We develop the cohomology theory of color Lie superalgebras due to Scheunert--Zhang in a framework of nonhomogeneous quadratic Koszul algebras. In this approach, the Chevalley--Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra. As an application, we calculate cohomologies with trivial coefficients of 3-dimensional color Lie superalgebras.

math.KT↗