arXiv subjects

Artūras Dubickas

Publications and source records attributed to Artūras Dubickas.

12 recordsLinked to original sources

arXiv2026

Subsets of abelian groups closed under addition or subtraction

In this article, we first describe all nonempty sets of integers S with the property that for all n and m in S, not necessarily distinct, the set {n-m,n+m} intersected with S consists of a single element. These are the sets with at most two elements, one of which is 0, and the infinite sets {rk}, where r is a fixed positive integer and k runs over all integers not divisible by 3. In the later sections, we solve the analogous problem for subsets of abelian groups. We also discuss, but do not completely solve, the analogous problem for nonabelian groups.

math.GR↗

arXiv2024

Counting integer polynomials with several roots of maximal modulus

In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also count reducible and irreducible polynomials in that set separately. Our results imply, for instance, that the number of monic integer irreducible polynomials of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is approximately $2H$ for odd $n$, while for even $n$ there are more than $H^{n/8}$ of such polynomials.

math.NT↗

arXiv2023

Euclidean minima of algebraic number fields

In this paper, we use some of our previous results to improve an upper bound of Bayer-Fluckiger, Borello and Jossen on the Euclidean minima of algebraic number fields. Our bound depends on the degree $n$ of the field, its signature, discriminant and the Hermite constant in dimension $n$.

math.NT↗

arXiv2022

Counting decomposable polynomials with integer coefficients

A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree and bounded height. Moreover, we obtain asymptotic formulas for the number of decomposable monic polynomials of even degree. For example, the number of monic sextic integer polynomials which are decomposable and of height at most $H$ is asymptotic to $(16ζ(3)-5/4)H^3$ as $H \to \infty$.

math.NT↗

arXiv2019

Linear relations with conjugates of a Salem number

In this paper we consider linear relations with conjugates of a Salem number $α$. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer $α+1/α$. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is $8$, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is $6$.

math.NT↗

arXiv2018

Multiplicative dependence of the translations of algebraic numbers

In this paper, we first prove that given pairwise distinct algebraic numbers $α_1, \ldots, α_n$, the numbers $α_1+t, \ldots, α_n+t$ are multiplicatively independent for all sufficiently large integers $t$. Then, for a pair $(a,b)$ of distinct integers, we study how many pairs $(a+t,b+t)$ are multiplicatively dependent when $t$ runs through the integers. For such a pair $(a,b)$ with $b-a=30$ we show that there are $13$ integers $t$ for which the pair $(a+t,b+t)$ is multiplicatively dependent. We conjecture that $13$ is the largest value of such translations for any $(a,b)$, where $a \ne b$, prove this for all pairs $(a,b)$ with difference at most $10^{10}$, and, assuming that the $ABC$ conjecture is true, show that for any such pair $(a,b)$, $a \ne b$, there is an absolute bound $C_1$ (independent of $a$ and $b$) on the number of such translations $t$.

math.NT↗

arXiv2018

The distance to square-free polynomials

In this paper, we consider a variant of Turán's problem on the distance from an integer polynomial in $\mathbb{Z}[x]$ to the nea\-rest irreducible polynomial in $\mathbb{Z}[x]$. We prove that for any polynomial $f \in \mathbb{Z}[x]$, there exist infinitely many square-free polynomials $g\in \mathbb{Z}[x]$ such that $L(f-g) \le 2$, where $L(f-g)$ denotes the sum of the absolute values of the coefficients of $f-g$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) \le 1$. For this, for each integer $d \geq 16$ we construct infinitely many polynomials $f \in \mathbb{Z}[x]$ of degree $d$ such that neither $f$ itself nor any $f(x) \pm x^k$, where $k$ is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.

math.NT↗

arXiv2015

Positive density of integer polynomials with some prescribed properties

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer $n$ and any set of $n$ distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree $n$, height at most $H$ and Galois group $S_n$ whose roots are close to the given $n$ points.

math.NT↗

arXiv2015

Counting degenerate polynomials of fixed degree and bounded height

In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.

math.NT↗

arXiv2015

Counting and testing dominant polynomials

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height $H$ among all quadratic integer polynomials tends to $(41+6 \log 2)/72$ as $H \to \infty$. Finally, we will design some algorithms to test whether a given polynomial with integer coefficients is dominant or not without finding the polynomial roots.

math.NT↗

arXiv2014

There are no two non-real conjugates of a Pisot number with the same imaginary part

We show that the number $α=(1+\sqrt{3+2\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $α_1,α_2,α_3,α_4$ satisfy the additive relation $α_1+α_2=α_3+α_4$. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations $α_1 = α_2 + α_3+α_4$ or $α_1 + α_2 + α_3 + α_4 =0$ cannot be solved in conjugates of a Pisot number $α$. We also show that the roots of the Siegel's polynomial $x^3-x-1$ are the only solutions to the three term equation $α_1+α_2+α_3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $α_1=α_2+α_3$.

math.NT↗

arXiv2008

Refinement Equations and Spline Functions

In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial {\bf C}^{\infty} solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.

math.NA↗