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László Remete

Publications and source records attributed to László Remete.

17 recordsLinked to original sources
arXiv2026

Asymptotic formulas for sums of elements from a multiplicative group

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $Γ$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(α)$ denote the absolute exponential height of an algebraic number $α$. Fix non-zero elements $a_1,\ldots , a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\ldots , x_k)\inΓ$ with $H(a_1x_1+\cdots +a_kx_k)\leq X$ as $X\to\infty$ such that no non-empty subsum of $a_1x_1+\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to\infty$ for the number of non-negative integers $n$ with $H(u_n)\leq X$, where $\{ u_n\}$ is a linear recurrence sequence.

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arXiv2024

On the index of power compositional polynomials

The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $θ$ is the index $[\mathbb{Z}_K:\mathbb{Z}[θ]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(θ)$. If $[\mathbb{Z}_K:\mathbb{Z}[θ]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1, \operatorname{deg} h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$ with $\operatorname{rad}(k)\mid \operatorname{rad}(A)$, the power compositional polynomial $f(x^k)$ is monogenic if and only if $f(x)$ is monogenic, provided that $f(x^k)$ is irreducible. At the end of the paper, we give infinite families of polynomials as examples.

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arXiv2022

Hermite equivalence of polynomials

In this paper, we resurrect a long-forgotten notion of equivalence for univariate polynomials with integral coefficients introduced by Hermite in the 1850s. We show that the Hermite equivalence class of a polynomial has a very natural interpretation in terms of the invariant ring and invariant ideal associated with the polynomial. We apply this interpretation to shed light on the relationship between Hermite equivalence and more familiar notions of polynomial equivalence, such as ${\rm GL}_2(\mathbb{Z})$- and $\mathbb{Z}$-equivalence. Specifically, we prove that ${\rm GL}_2(\mathbb{Z})$-equivalent polynomials are Hermite equivalent and, for polynomials of degree $2$ or $3$, the converse is also true. On the other hand, for every $n\geq 4$, we give infinite collections of examples of polynomials $f,g\in \mathbb{Z}[X]$ of degree $n$ that are Hermite equivalent but not ${\rm GL}_2(\mathbb{Z})$-equivalent.

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arXiv2021

Integral basis of pure fields with square-free parameter

Let $m\neq0,\pm1$ and $n\geq 2$ be integers. The ring of algebraic integers of the pure fields of type $\mathbb{Q}(\sqrt[n]{m})$ is explicitly known for $n=2,3,4$. It is well known that for $n=2$, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of $m$ modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.\\ In this paper we explicitly give an integral basis of the field $\mathbb{Q}(\sqrt[n]{m})$, where $m\neq\pm1$ is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of $\mathbb{Q}(\sqrt[n]{m})$ is repeating periodically in $m$ with period length depending on $n$.

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arXiv2020

Power integral bases in cubic and quartic extensions of real quadratic fields

Investigations of monogenity and power integral bases were recently extended from the absolute case (over Q) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded to calculate generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of cubic and quartic fields with real quadratic fields. In case the quartic field is totally complex, we present an especially simple algorithm. We illustrate our method with two detailed examples.

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arXiv2018

Totally real Thue inequalities over imaginary quadratic fields

Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities \[ |F(x,y)|\leq K \;\; (x,y\in Z_M) \] to the resolution of absolute Thue inequalities of type \[ |F(x,y)|\leq k \;\; (x,y\in Z). \] We illustrate our method with an explicit example.

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arXiv2018

Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.

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arXiv2018

Binomial Thue equations and power integral bases in pure quartic fields

It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating power integral bases. In this paper we determine power integral bases in pure quartic fields by reducing the corresponding index form equation to quartic binomial Thue equations. We apply recent results on the solutions of binomial Thue equations of type $a^4-mb^4=\pm 1\; ({\rm in}\; a,b\in Z)$ as well as we perform an extensive calculation by a high performance computer to determine "small" solutions (with $\max(|a|,|b|)<10^{500}$) of binomial Thue equations for $m<10^7$. Using these results on binomial Thue equations we characterize power integral bases in infinite subfamilies of pure quartic fields. For $1< m< 10^7$ we give all generators of power integral bases with coefficients $<10^{1000}$ in the integral basis.

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arXiv2018

Non-monogenity in a family of octic fields

Let $m$ be a square-free positive integer, $m\equiv 2,3 \; (\bmod \; 4)$. We show that the number field $K=Q(i,\sqrt[4]{m})$ is non-monogene, that is it does not admit any power integral bases of type $\{1,α,\ldots,α^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations. Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. We calculate the index of elements as polynomials depending on the parameter, factor these polynomials and consider systems of congruences according to the factors.

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arXiv2018

Power integral bases in a family of sextic fields with quadratic subfields

Let $M=Q(i\sqrt{d})$ be any imaginary quadratic field with a positive square-free $d$. Consider the polynomial \[ f(x)=x^3-ax^2-(a+3)x-1, \] with a parameter $a\in Z$. Let $K=M(α)$, where $α$ is a root of $f$. This is an infinite parametric family of sextic fields depending on two parameters, $a$ and $d$. Applying relative Thue equations we determine the relative power integral bases of these sextic fields over their quadratic subfields. Using these results we also determine generators of (absolute) power integral bases of the sextic fields.

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arXiv2018

Solving binomial Thue equations

We consider binomial Thue equations of type $x^n-my^n=\pm 1$ in $x,y\in Z$. Optimizing the method of Peth\H o we perform an extensive calculation by a high performance computer to determine all solutions with $\max(|x|,|y|)<10^{500}$ of binomial Thue equations for $m<10^7$ for exponents $n=3,4,5,7,11,13,17,19,23,29$.

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arXiv2018

Calculating power integral bases by solving relative Thue equations

In our recent paper we gave an efficient algorithm to calculate "small" solutions of relative Thue equations (where "small" means an upper bound of type $10^{500}$ for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

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arXiv2018

Calculating power integral bases by using relative power integral bases

Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $Q$. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.

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arXiv2018

Integral bases and monogenity of composite fields

We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, of degrees 6 up to 12. The non-monogenity of the number fields is stated very often as a consequence of the non-existence of the solutions of the index form equation. Up to our knowlegde it is not at all feasible to solve the index form equation in these high degree fields, especially not in a parametric form. On the other hand our method implies directly the non-monogenity in almost all cases. We obtain our results in a parametric form, characterizing these infinite parametric families of composite fields.

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arXiv2018

Integral bases and monogenity of the simplest sextic fields

Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $α$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=Q(α)$ are called simplest sextic fields and are well known in the literature. Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases.

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arXiv2018

Integral bases and monogenity of pure fields

Let $m$ be a square-free integer ($m\neq 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\leq n\leq 9$ we show that the period length is $n^2$. We explicitly describe the integral bases, and for $n=3,4,5,6,8$ we explicitly calculate the index forms of $K$. This enables us in many cases to characterize the monogenity of these fields. Using the explicit form of the index forms yields a new technic that enables us to derive new results on monogenity and to get several former results as easy consequences. For $n=4,6,8$ we give an almost complete characterization of the monogenity of pure fields.

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