arXiv subjects

Joel Louwsma

Publications and source records attributed to Joel Louwsma.

12 recordsLinked to original sources

arXiv2025

Transfers and Unstable Degrees in the $\mathbb{A}^{1}$-Brouwer Degrees Package for Macaulay2

We describe a significant update to the Macaulay2 package A1BrouwerDegrees. We extend several methods in the previous version of the package to the setting of finite étale algebras, allowing the computation of transfers along finite étale extensions. Additionally, we implement a number of new features for the computation of unstable $\mathbb{A}^{1}$-Brouwer degrees and manipulation of classes in the unstable Grothendieck--Witt group.

math.AG↗

arXiv2025

Generalized chip firing and critical groups of arithmetical structures on trees

Chip firing provides a way to study the sandpile group (also known as the Jacobian) of a graph. We use a generalized version of chip firing to bound the number of invariant factors of the critical group of an arithmetical structure on a graph. We also show that, under suitable hypotheses, critical groups are additive under wedge sums of graphs with arithmetical structures. These results allow us to relate the number of invariant factors of critical groups associated to any given tree to decompositions of the tree into simpler trees. We use this to classify those trees for which every arithmetical structure has cyclic critical group. Finally, we show how to construct arithmetical structures on trees with prescribed critical groups. In particular, every finite abelian group is realized as the critical group of some arithmetical structure on a tree.

math.CO↗

arXiv2025

On solutions of $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$

We give an algorithm that produces all solutions of the equation $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$, where $k$ is a fixed positive integer that is not a power of $2$, $a$ is an element of $\{0,1,2\}$ that can vary from term to term, and $b$ is a nonnegative integer that can vary from term to term. We also completely characterize the pairs $(k,n)$ for which this equation has a nontrivial solution in integers of this form.

math.NT↗

arXiv2024

$\mathbb{A}^1$-Brouwer degrees in Macaulay2

We describe the Macaulay2 package "A1BrouwerDegrees" for computing local and global $\mathbb{A}^1$-Brouwer degrees and studying symmetric bilinear forms over the complex numbers, the real numbers, the rational numbers, and finite fields of characteristic not equal to 2.

math.AG↗

arXiv2024

Critical groups of arithmetical structures on star graphs and complete graphs

An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group. We show how to determine the critical group of an arithmetical structure on a star graph or complete graph in terms of the entries of the arithmetical structure. We use this to investigate which finite abelian groups can occur as critical groups of arithmetical structures on these graphs.

math.CO↗

arXiv2024

Critical groups of arithmetical structures under a generalized star-clique operation

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly.

math.CO↗

arXiv2024

On Arithmetical Structures on Complete Graphs

An arithmetical structure on the complete graph $K_n$ with $n$ vertices is given by a collection of $n$ positive integers with no common factor each of which divides their sum. We show that, for all positive integers $c$ less than a certain bound depending on $n$, there is an arithmetical structure on $K_n$ with largest value $c$. We also show that, if each prime factor of $c$ is greater than $(n+1)^2/4$, there is no arithmetical structure on $K_n$ with largest value $c$. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on $K_n$.

math.NT↗

arXiv2023

Rational numbers with odd greedy expansion of fixed length

Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as long as the sum is less than $n/d$. It is an open question whether this expansion always has finitely many terms. Given a fixed positive integer $n$, we find all reduced fractions with numerator $n$ whose odd greedy expansion has length $2$. Given $m-1$ odd positive integers, we find all rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators. Given $m-2$ compatible odd positive integers, we find an infinite family of rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators.

math.NT↗

arXiv2020

Arithmetical structures on bidents

An arithmetical structure on a finite, connected graph $G$ is a pair of vectors $(\mathbf{d}, \mathbf{r})$ with positive integer entries for which $(\operatorname{diag}(\mathbf{d}) - A)\mathbf{r} = \mathbf{0}$, where $A$ is the adjacency matrix of $G$ and where the entries of $\mathbf{r}$ have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of $(\operatorname{diag}(\mathbf{d}) - A)$. In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with $n$ vertices and show that this number grows at the same rate as the Catalan numbers as $n$ increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.

math.CO↗

arXiv2015

Digital Fixed Points, Approximate Fixed Points, and Universal Functions

A. Rosenfeld introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).

math.CO↗

arXiv2014

Stable commutator length in Baumslag-Solitar groups and quasimorphisms for tree actions

This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces. In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2,3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions. Returning to Baumslag-Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).

math.GR↗

arXiv2011

Immersed surfaces in the modular orbifold

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.

math.GT↗