arXiv subjects

Daniel Smertnig

Publications and source records attributed to Daniel Smertnig.

At least 19 recordsLinked to original sources

arXiv2026

Mahler series with multiplicative coefficient sequences

We prove that every Mahler series, over a field of characteristic $0$, with multiplicative coefficients is regular in the sense of Allouche and Shallit. We also obtain an explicit characterization of such series. This yields a joint extension of the characterization of rational series with multiplicative coefficients (by Bézivin and Bell--Bruin--Coons) and of multiplicative automatic sequences (by Konieczny--Lemańczyk--Müllner). Both of these results are used in our characterization, so we do not obtain new proofs of these special cases.

math.NT↗

arXiv2026

Divide and Transfer: Non-Unique Factorizations Beyond Commutativity

Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we first explore these notions in the classical setting of commutative Dedekind domains, where monoids of zero-sum sequences appear as a natural combinatorial model. We then adapt these ideas to the setting of noncommutative Dedekind prime rings using module-theoretic methods. Going a step further, we discuss Rump and Yang's recent divisor theory for ideals in hereditary noetherian prime rings, where divisors can be visualized in a diagrammatic calculus.

math.RA↗

arXiv2026

Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata

In this work we consider two rich subclasses of weighted automata over fields: polynomially ambiguous weighted automata and copyless cost register automata. Primarily we are interested in understanding their expressiveness power. Over the field of rationals and $1$-letter alphabets, it is known that the two classes coincide; they are equivalent to linear recurrence sequences (LRS) whose exponential bases are roots of rationals. We develop a tool we call Pumping Sequence Families, which, by exploiting the simple single-letter behaviour of the models, yields two pumping-like results over arbitrary fields with unrestricted alphabets, one for each class. As a corollary of these results, we present examples proving that the two classes become incomparable over the field of rationals with unrestricted alphabets. We complement the results by analysing the zeroness and equivalence problems. For weighted automata (even unrestricted) these problems are well understood: there are polynomial time, and even NC$^2$ algorithms. For copyless cost register automata we show that the two problems are \textsc{PSpace}-complete, where the difficulty is to show the lower bound.

cs.FL↗

arXiv2026

Factoriality and Class Groups of Upper Cluster Algebras and Finite Laurent Intersection Rings: A Computational Approach

We study factoriality and the class groups of locally acyclic cluster algebras. To do so, we introduce a new class of rings called finite Laurent intersection rings (FLIRs), which includes locally acyclic cluster algebras, full-rank upper cluster algebras, and certain generalized upper cluster algebras and Laurent phenomenon algebras. Our main results are algorithms to compute the class group of an explicit FLIR, to determine factoriality, and to compute all factorizations of a given element. The algorithms are based on multivariate polynomial factorizations, avoiding computationally expensive Gröbner basis calculations.

math.AC↗

arXiv2025

Primes and absolutely or non-absolutely irreducible elements in atomic domains

We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles.

math.AC↗

arXiv2025

Monoid algebras and graph products

In this note, we extend results about unique $n^{\textrm{th}}$ roots and cancellation of finite disconnected graphs with respect to the Cartesian, the strong and the direct product, to the rooted hierarchical products, and to a modified lexicographic product. We show that these results also hold for graphs with countably many finite connected components, as long as every connected component appears only finitely often (up to isomorphism). The proofs are via monoid algebras and generalized power series rings.

math.CO↗

arXiv2024

Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata

We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup representations by invertible transformations. The characterizations use results on unit equations from Diophantine number theory (by Evertse, van der Poorten, and Schlickewei in characteristic zero, and by Derksen and Masser in positive characteristic). Specialized to finitely generated groups in characteristic zero, one of our main theorems recovers an improvement of a very recent similar characterization by Corvaja, Demeio, Rapinchuk, Ren, and Zannier that was motivated by the study of the bounded generation (BG) property. In positive characteristic, we get a characterization of linear BG groups, recovering a theorem of Abért, Lubotzky, and Pyber from 2003. Our motivation comes from weighted finite automata (WFA) over a field. For invertible WFA we show that $M$-ambiguity, finite ambiguity, and polynomial ambiguity are characterized by arithmetic properties. We discover a full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity. In the invertible case, this is a far-reaching generalization of a recent result by Bell and the second author, characterizing unambiguous WFA, that resolved a 1979 conjecture of Reutenauer. As a consequence, using the computability of the (linear) Zariski closure of a finitely generated matrix semigroup, the $M$-ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.

math.GR↗

arXiv2024

A monoid-theoretical approach to infinite direct-sum decompositions of modules

Let $\mathcal C$ be a class of modules over a ring $R$, closed under direct sums over index sets of cardinality $κ$ and isomorphisms, and such that the isomorphism classes form a set. The monoid of modules $V(\mathcal C)$ encodes the behavior of finite direct-sum decompositions of modules in $\mathcal C$. We endow $V(\mathcal C)$ with an additional operation reflecting $κ$-indexed direct sums, and study the resulting $κ$-monoid $V^κ(\mathcal C)$. The braiding-property and an equivalent universal property, allow us to show: if every module in $\mathcal C$ is a direct sum of modules generated by strictly fewer than $λ$ many elements, then all relations on $V^κ(\mathcal C)$ are induced by relations between direct sums indexed by sets of cardinality strictly less than $λ$. A theorem of Kaplansky states that every projective module is a direct sum of countably generated modules. We augment this, showing that also all relations between infinite direct sums of projective modules are induced from those between countable direct sums of countably generated projective modules. If every projective module over a ring $R$ is a direct sum of finitely generated projective modules, then the monoid of finitely generated projective modules $V(R)$ completely determines the $κ$-monoid $V^κ(R)$. Together with the realization result of Bergman and Dicks, this characterizes the $κ$-monoids appearing as $V^κ(R)$ for a hereditary ring. In general, the $\aleph_0$-monoid $V^{\aleph_0}(R)$ fully determines $V^κ(R)$. Herbera and Příhoda's characterization of monoids of countably generated projective modules $V^*(R)$ over semilocal noetherian rings, yields a characterization of $V^κ(R)$ for these rings. We also characterize two-generated $\aleph_0$-monoids that appear as $V^{\aleph_0}(R)$ for hereditary rings $R$.

math.RA↗

arXiv2023

Duality of Lattices Associated to Left and Right Quotients

We associate lattices to the sets of unions and intersections of left and right quotients of a regular language. For both unions and intersections, we show that the lattices we produce using left and right quotients are dual to each other. We also give necessary and sufficient conditions for these lattices to have maximal possible complexity.

cs.FL↗

arXiv2023

Computing the linear hull: Deciding Deterministic? and Unambiguous? for weighted automata over fields

The (left) linear hull of a weighted automaton over a field is a topological invariant. If the automaton is minimal, the linear hull can be used to determine whether or not the automaton is equivalent to a deterministic one. Furthermore, the linear hull can also be used to determine whether the minimal automaton is equivalent to an unambiguous one. We show how to compute the linear hull, and thus prove that it is decidable whether or not a given automaton over a number field is equivalent to a deterministic one. In this case we are also able to compute an equivalent deterministic automaton. We also show the analogous decidability and computability result for the unambiguous case. Our results resolve a problem posed in a 2006 survey by Lombardy and Sakarovitch.

cs.FL↗

arXiv2023

On noncommutative bounded factorization domains and prime rings

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $λ(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.

math.RA↗

arXiv2022

$D$-finite multivariate series with arithmetic restrictions on their coefficients

A multivariate, formal power series over a field $K$ is a Bézivin series if all of its coefficients can be expressed as a sum of at most $r$ elements from a finitely generated subgroup $G \le K^*$; it is a Pólya series if one can take $r=1$. We give explicit structural descriptions of $D$-finite Bézivin series and $D$-finite Pólya series over fields of characteristic $0$, thus extending classical results of Pólya and Bézivin to the multivariate setting.

math.CO↗

arXiv2021

A height gap theorem for coefficients of Mahler functions

We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are \emph{gaps} in the possible growths. In proving this height gap theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an arbitrary ground field of characteristic zero, a $k$-Mahler function is $k$-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas.

math.NT↗

arXiv2021

Lattices over Bass rings and graph agglomerations

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring $R$ through the factorization theory of the corresponding monoid $T(R)$. Results of Levy-Wiegand and Levy-Odenthal together with a study of the local case yield an explicit description of $T(R)$. The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations--a natural class of monoids serving as combinatorial models for the factorization theory of $T(R)$. As a consequence, the monoid $T(R)$ is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of $T(R)$ and characterize when $T(R)$ is half-factorial. (Factoriality, that is, torsion-free Krull-Remak-Schmidt-Azumaya, is characterized by a theorem of Levy-Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

math.AC↗

arXiv2021

Noncommutative rational Pólya series

A (noncommutative) Pólya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^\times$. We show that rational Pólya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.

math.CO↗

arXiv2019

Definite orders with locally free cancellation

We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.

math.NT↗

arXiv2019

Factoriality and class groups of cluster algebras

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-$1$ prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss--Leclerc--Schröer.

math.AC↗

arXiv2019

On basic and Bass quaternion orders

A quaternion order O over a Dedekind domain R is Bass if every R-superorder is Gorenstein, and O is basic if it contains an integrally closed quadratic R-order. In this article, we show that these conditions are equivalent in local and global settings: a quaternion order is Bass if and only if it is basic. In particular, we show that the property of being basic is a local property of a quaternion order.

math.RA↗