Vadim Ponomarenko - Profile on Academia.edu (original) (raw)

Papers by Vadim Ponomarenko

arXiv (Cornell University), Feb 25, 2011

Let G be an abelian group, let S be a sequence of terms s 1 , s 2 , . . . , sn ∈ G not all contai... more Let G be an abelian group, let S be a sequence of terms s 1 , s 2 , . . . , sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let W ⊙ S = {w 1 s 1 + . . . + wnsn : w i a term of W, w i = w j for i = j}, which is a particular kind of weighted restricted sumset. We show that |W ⊙ S| ≥ min{|G| -1, n}, that W ⊙ S = G if n ≥ |G| + 1, and also characterize all sequences S of length |G| with W ⊙ S = G. This result then allows us to characterize when a linear equation where α, a 1 , . . . , ar ∈ Z are given, has a solution (x 1 , . . . , xr) ∈ Z r modulo n with all x i distinct modulo n. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group G ∼ = Cn 1 ⊕ Cn 2 (where n 1 | n 2 and n 2 ≥ 3) having k distinct terms, for any k ∈ [3, min{n 1 + 1, exp(G)}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence.

arXiv (Cornell University), Oct 20, 2021

Length density is a recently introduced factorization invariant, assigned to each element n of a ... more Length density is a recently introduced factorization invariant, assigned to each element n of a cancellative commutative atomic semigroup S, that measures how far the set of factorization lengths of n is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative integers).

Deleted Journal, Dec 23, 2022

Expressing whole numbers as sums of figurate numbers, including tetrahedral numbers, is a longsta... more Expressing whole numbers as sums of figurate numbers, including tetrahedral numbers, is a longstanding problem in number theory. Pollock's tetrahedral number conjecture states that every positive integer can be expressed as the sum of at most five tetrahedral numbers. Here we explore a generalization of this conjecture to negative indices. We provide a method for computing sums of two generalized tetrahedral numbers up to a given bound, and explore which families of perfect powers can be expressed as sums of two generalized tetrahedral numbers.

Several properties of atoms of block monoids are developed. We count the number of atoms of small... more Several properties of atoms of block monoids are developed. We count the number of atoms of small cardinality. We also develop various properties of atoms of maximal cardinality (the Davenport constant). We use these properties to find new bounds on the Davenport constant both in general and for several important classes of groups. We also determine the Davenport constant (with computer assistance) for a variety of specific groups.

Ukrainian Mathematical Journal, Jul 1, 2015

We define semigroup semirings by analogy with group rings and semigroup rings. We study the arith... more We define semigroup semirings by analogy with group rings and semigroup rings. We study the arithmetic properties and determine sufficient conditions under which a semigroup semiring is atomic, has finite factorization, or has bounded factorization. We also present a semigroup-semiring analog (although not a generalization) of the Gauss lemma on primitive polynomials. Напiвгруповi напiвкiльця визначаються по аналогiї з груповими кiльцями та напiвгруповими кiльцями. Вивчено арифметичнi властивостi та отримано достатнi умови, за яких напiвгрупове напiвкiльце є атомним, має скiнченну факторизацiю або має обмежену факторизацiю. Також наведено напiвгрупово-напiвкiльцевий аналог (хоча i не узагальнення) гауссiвської леми про примiтивнi полiноми. The study of group rings is very popular and active, see, e.g., . Similarly, the study of semigroup rings has a rich history, see, e.g., . Also, the study of semirings is very active not only in mathematics but in computer science and control theory, see, e.g., . In this context, it seems natural to study semigroup semirings; however relatively little work (see, e.g., ) has been done in this area. We propose to study the arithmetic properties of semigroup semirings, specifically factorization into irreducibles and primes. As tools we will use tools from semirings as well as from semigroups. This work is organized as follows. Section 1 collects basic facts and tools about semirings and factorization theory. Section 2 introduces semigroup semirings and develops several tools to find atoms. Section 3 introduces content sets and maximal common divisors, which generalize the notion of greatest common divisors. These tools are sufficient to determine necessary conditions for a semigroup semiring to be atomic, finite factorization, and bounded factorization. We also prove a semiring analog of the Gauss lemma on primitive polynomials. Definition 1.1. We call R = (R, +, ×) a semiring if it satisfies the following: (1) (R, +) is a commutative monoid with identity 0; (2) (R, ×) is a commutative monoid with identity 1. We abbreviate × via juxtaposition; (3) for all a ∈ R, a0 = 0; (4) for all a, b, c ∈ R, a(b + c) = (ab) + (ac). Following , we call R an information algebra if it also satisfies the following: (5) for all a, b ∈ R, if a + b = 0 then a = b = 0; (6) for all a, b ∈ R, if ab = 0 then a = 0 or b = 0. For convenience, set R * = R \ {0}. Properties (5), (6) are equivalent to R * being closed under +, ×. Lemma 1.1. Let R be an information algebra, and let a 1 , a 2 , . . . , a k , b 1 , b 2 , . . . , b k ∈ R. Then k i=1 a i b i is nonzero if and only if there is some i ∈ [1, k] with a i and b i each nonzero.

Israel Journal of Mathematics, Sep 20, 2012

Involve, Mar 4, 2021

A numerical semigroup S is an additive subsemigroup of the non-negative integers with finite comp... more A numerical semigroup S is an additive subsemigroup of the non-negative integers with finite complement, and the squarefree divisor complex of an element m ∈ S is a simplicial complex ∆ m that arises in the study of multigraded Betti numbers. We compute squarefree divisor complexes for certain classes numerical semigroups, and exhibit a new family of simplicial complexes that are occur as the squarefree divisor complex of some numerical semigroup element. F ∈∆ (-1) |F | .

Let P be a set of n points on the real line and let k be a fixed positive integer. Assume that fo... more Let P be a set of n points on the real line and let k be a fixed positive integer. Assume that for every x ∈ P the set {y ∈ P | |y -x| ≤ 1} of all points in P at distance at most 1 from x has cardinality that is divisible by k. We show that necessarily n is divisible by k.

arXiv (Cornell University), Jun 28, 2018

A numerical semigroup S is a subset of the non-negative integers containing 0 that is closed unde... more A numerical semigroup S is a subset of the non-negative integers containing 0 that is closed under addition. The Hilbert series of S (a formal power series equal to the sum of terms t n over all n ∈ S) can be expressed as a rational function in t whose numerator is characterized in terms of the topology of a simplicial complex determined by membership in S. In this paper, we obtain analogous rational expressions for the related power series whose coefficient of t n equals f (n) for one of several semigroup-theoretic invariants f : S → R known to be eventually quasipolynomial.

Springer proceedings in mathematics & statistics, 2022

Let A be a m × m nonnegative square matrix and let F (A) denote the number of positive entries in... more Let A be a m × m nonnegative square matrix and let F (A) denote the number of positive entries in A. We consider conditions on A to make the sequence {F (A n )} ∞ n=1 monotone. This is known for F (A) ≤ 3 and F (A) ≥ m 2 -2m + 2; we extend this to F (A) = 4.

We explore the dilated floor function f a (x) = ax and its commutativity with functions of the sa... more We explore the dilated floor function f a (x) = ax and its commutativity with functions of the same form. A previous paper found all a and b such that f a and f b commute for all real x. In this paper, we determine all x for which the functions commute for a particular choice of a and b. We calculate the proportion of the number line on which the functions commute. We determine bounds for how far away the functions can get from commuting. We solve this fully for integer a, b and partially for real a, b.

International Journal of Algebra and Computation, May 1, 2016

The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integ... more The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of n. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary degrees achieved by elements of S, focusing on the case where S in finitely generated (where C(S) is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of C(S) that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for C(S), and present some open questions for further study.

Communications in Algebra, May 19, 2016

We generalize the geometric sequence {a p , a p-1 b, a p-2 b 2 , . . . , b p } to allow the p cop... more We generalize the geometric sequence {a p , a p-1 b, a p-2 b 2 , . . . , b p } to allow the p copies of a (resp. b) to all be different. We call the se- We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

arXiv (Cornell University), Aug 15, 2020

For a commutative cancellative monoid M , we introduce the notion of the length density of both a... more For a commutative cancellative monoid M , we introduce the notion of the length density of both a nonunit x ∈ M , denoted LD(x), and the entire monoid M , denoted LD(M). This invariant is related to three widely studied invariants in the theory of non-unit factorizations, L(x), ℓ(x), and ρ(x). We consider some general properties of LD(x) and LD(M) and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD(M) is rational and there is a nonunit element x ∈ M with LD(M) = LD(x) (such a monoid is said to have accepted length density). While it is wellknown that the much studied asymptotic versions of L(x), ℓ(x) and ρ(x) (denoted L(x), ℓ(x), and ρ(x)) always exist, we show the somewhat surprising result that LD(x) = limn→∞ LD(x n) may not exist. We also give some finiteness conditions on M that force the existence of LD(x).

Pollock’s Generalized Tetrahedral Numbers Conjecture

The American Mathematical Monthly, 2021

The nth tetrahedral number Ten = ( n+2 3 ) represents the sum of the first n triangular numbers. ... more The nth tetrahedral number Ten = ( n+2 3 ) represents the sum of the first n triangular numbers. In the song “The Twelve Days of Christmas,” Ten counts the total number of gifts received after day n. A longstanding conjecture of Pollock (from [4]) is that every positive integer may be expressed as the sum of at most five tetrahedral numbers. To date, only 241 positive integers have been found requiring five tetrahedral numbers (see [3]). Recently, progress has been made (in [1]) on a related conjecture of Pollock from the same 19th century paper. Here we instead consider generalized tetrahedral numbers Ten = (n+2)(n+1)n 6 , defined for all integers n. These are the generalized binomial coefficients ( n+2 3 ) , as popularized in [2]. With these we can prove the following.

The American Mathematical Monthly, 2019

Graph theory was used to analyze a series of small pseudodendrimeric structures. Descriptive indi... more Graph theory was used to analyze a series of small pseudodendrimeric structures. Descriptive indices were developed to characterize the pseudodendrimer graphs. The relative proportion of these dendrimers in typical samples was estimated based on three growth models. Weighted average values for the descriptive indices over typical aggregate samples were found to differ only slightly from values for perfect dendrimers.

Involve, Dec 31, 2011

We provide a variety of results concerning the problem of determining maximal vectors g such that... more We provide a variety of results concerning the problem of determining maximal vectors g such that the Diophantine system M x = g has no solution: conditions for the existence of g, conditions for the uniqueness of g, bounds on g, determining g explicitly in several important special cases, constructions for g, and a reduction for M.

The Mathematical Intelligencer