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Matvey Borodin

Publications and source records attributed to Matvey Borodin.

12 recordsLinked to original sources
arXiv2025

Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields

We present new criteria that obstruct an isogeny class of abelian varieties over a finite field with a given Weil polynomial from containing a Jacobian of a genus-3 hyperelliptic curve. Based on our analysis of the Weil polynomials of three-dimensional abelian varieties over finite fields up to $\mathbb{F}_{25}$ using the data in the L-functions and Modular Forms Database, we conjecture a collection of apparent obstructions. We provide a survey of known and conjectured results related to this problem, and a detailed statistical analysis of these findings. We conjecture that two of these obstructions classify all isogeny classes asymptotically as $q \to \infty$.

math.NT↗
arXiv2025

On Weil Polynomials of Hyperelliptic Curves over Finite Fields of Characteristic 2

We present new conditions which obstruct the existence of hyperelliptic Jacobians in isogeny classes of abelian varieties over finite fields of characteristic 2. We show that Weil polynomials of Jacobians cannot have coefficients in certain residue classes modulo 2, extending the approach of Costa et al. in arXiv:2002.02067. We prove that for 3- and 4-dimensional abelian varieties over $\mathbb{F}_{2^n}$, as $n \rightarrow\infty$, the parities of the Weil coefficients asymptotically equidistribute. Further, we show that these obstructions disqualify $\frac12$ of all 3-dimensional isogeny classes and $\frac 58$ of all 4-dimensional isogeny classes from containing a hyperelliptic Jacobian. Additionally, we present a practical enumeration algorithm which generates all isomorphism classes of hyperelliptic curves of arbitrary genus over almost any finite field of characteristic 2 based on existing algorithms over $\mathbb{F}_2$. Our analysis shows the runtime to be $\tilde{O}(2^{n(2g-1)})$ expected, and $\tilde{O}(2^{n(2g+2)})$ worst case. This runtime improvement renders the algorithm practical for fields other than $\mathbb{F}_2$.

math.NT↗
arXiv2023

The Orbits of the Action of the Cactus Group on Arc Diagrams

The cactus group $J_n$ is the $S_n$-equivariant fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves with marked points. This group plays the role of the braid group for the monoidal category of Kashiwara crystals attached to a simple Lie algebra. Following Frenkel, Kirillov and Varchenko, one can identify the multiplicity set in a tensor product of $\mathfrak{sl}_2$-crystals with the set of arc diagrams on a disc, thus allowing a much simpler description of the corresponding $J_n$-action. We address the problem of classifying the orbits of this cactus group action. Namely, we describe some invariants of this action and show that in some (fairly general) classes of examples there are no other invariants. Furthermore, we describe some additional relations, including the braid relation, that this action places on the generators of $J_n$.

math.CO↗
arXiv2023

The Ideal of Vanishing Polynomials and the Ring of Polynomial Functions

Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial functions. In particular, we provide several results on generating vanishing polynomials. We first analyze the ideal of vanishing polynomial over $\mathbb{Z}_n$, the ring of the integers modulo $n$. We then establish an isomorphism between the vanishing polynomials of a ring and the vanishing polynomials of the constituent rings in its decomposition. Lastly, we generalize our results to study the ideal of vanishing polynomials over arbitrary commutative rings.

math.AC↗
arXiv2023

Results on Vanishing Polynomials and Polynomial Root Counting

We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close connections to both ring theory and the technical applications of polynomials, along with numerous applications to other mathematical and engineering fields. We first determine the minimum degree of monic vanishing polynomials over a specific infinite family of rings of a specific form and consider a generalization of the notion of a monic vanishing polynomial over a subring. We then present a partial classification of the ideal of vanishing polynomials over general commutative rings with identity of prime and prime square orders. Finally, we prove some results on rings that have a finite number of roots and propose a technique that can be utilized to restrict the number of roots polynomials can have over certain finite commutative rings.

math.AC↗
arXiv2021

Sequences of the Stable Matching Problem

In this paper, we begin by discussing different types of preference profiles related to the stable marriage problem. We then introduce the concept of soulmates, which are a man and a woman who rank each other first. Inversely, we examine hell-pairs, where a man and a woman rank each other last. We generate sequences enumerating preference profiles of different types. We also calculate sequences related to the egalitarian cost, or "quality", of a matching. In total, we introduce and discuss 30 new sequences related to the stable marriage problem and discuss 6 sequences that are already in the OEIS.

math.HO↗
arXiv2021

The Stable Matching Problem and Sudoku

Are you having trouble getting married? These days, there are lots of products on the market for dating, from apps to websites and matchmakers, but we know a simpler way! That's right -- your path to coupled life isn't through Tinder: it's through Sudoku! Read our fabulous paper where we explore the Stable Marriage Problem to help you find happiness and stability in marriage through math. As a bonus, you get two Sudoku puzzles with a new flavor.

math.HO↗
arXiv2020

Chip-Firing and Fractional Bases

We study a particular chip-firing process on an infinite path graph. At any time when there are at least $a+b$ chips at a vertex, $a$ chips fire to the left and $b$ chips fire to the right. We describe the final state of this process when we start with $n$ chips at the origin.

math.CO↗
arXiv2020

The No-Flippancy Game

We analyze a coin-based game with two players where, before starting the game, each player selects a string of length $n$ comprised of coin tosses. They alternate turns, choosing the outcome of a coin toss according to specific rules. As a result, the game is deterministic. The player whose string appears first wins. If neither player's string occurs, then the game must be infinite. We study several aspects of this game. We show that if, after $4n-4$ turns, the game fails to cease, it must be infinite. Furthermore, we examine how a player may select their string to force a desired outcome. Finally, we describe the result of the game for particular cases.

math.CO↗
arXiv2019

It's Common Knowledge

We discuss some old common knowledge puzzles and introduce a lot of new common knowledge puzzles.

math.HO↗
arXiv2019

Variants of Base 3 over 2

We discuss two different systems of number representations that both can be called 'base 3/2'. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a connection between these systems and 3-free sequences.

math.NT↗