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Kálmán Győry

Publications and source records attributed to Kálmán Győry.

13 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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arXiv2025

Effective reduction theory of integral polynomials of given non-zero discriminant and its applications

We give a survey on the general effective reduction theory of integral polynomials and its applications. We concentrate on results providing the finiteness for the number of `$\mathbb{Z}$-equivalence classes' and `$GL_2(\mathbb{Z})$-equivalence classes' of polynomials of given discriminant. We present the effective finiteness results of Lagrange from 1773 and Hermite from 1848, 1851 for quadratic resp. cubic polynomials. Then we formulate the general ineffective finiteness result of Birch and Merriman from 1972, the general effective finiteness theorems of Győry from 1973, obtained independently, and of Evertse and Győry from 1991, and a result of Hermite from 1857 not discussed in the literature before 2023. We briefly outline our effective proofs which depend on Győry's effective results on unit equations, whose proofs involve Baker's effective theory of logarithmic forms. Then we focus on our joint paper with Bhargava, Remete and Swaminathan from 2023, where Hermite's finiteness result from 1857 involving `Hermite equivalence classes' is compared with the above-mentioned modern results involving $\mathbb{Z}$-equivalence and $GL_2(\mathbb{Z})$-equivalence, and where it is confirmed that Hermite's result from 1857 is much weaker than the modern results mentioned. The results of Győry from 1973 and Evertse and Győry from 1991 together established a general effective reduction theory of integral polynomials with given non-zero discriminant, which has significant consequences and applications, including Győry's effective finiteness theorems from the 1970's on monogenic orders and number fields. We give an overview of these in our paper. We also give an overview of bounds on the number of times a given order is monogenic or rationally monogenic. In the Appendix we discuss related topics not strictly belonging to the reduction theory of integral polynomials.

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arXiv2023

Multiplicative dependence of rational values modulo approximate finitely generated groups

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are `close' (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field $K$. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $α\in K$ such that $f_1(α),\ldots,f_n(α)$ are multiplicatively dependent modulo such sets.

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arXiv2023

Explicit bounds for the solutions of superelliptic equations over number fields

Let $f$ be a polynomial with coefficients in the ring $O_S$ of $S$-integers of a number field $K$, $b$ a non-zero $S$-integer, and $m$ an integer $\ge 2$. We consider the equation $( \star )$: $f(x) = b y^m$ in $x,y \in O_S$. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of $K, S, f, m$ and the $S$-norm of $b$ for the heights of the solutions $x$ of the equation $( \star)$. Further, we give an explicit bound $C$ in terms of $K, S, f$ and the $S$-norm of $b$ such that if $m > C$ the equation $(\star)$ has only solutions with $y = 0$ or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of Bérczes, Evertse, and Győry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the $S$-norm of $b$ instead of its height.

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arXiv2023

On power values of pyramidal numbers, II

For $m \geq 3$, we define the $m$th order pyramidal number by \[ \mathrm{Pyr}_m(x) = \frac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation $\mathrm{Pyr}_m(x) = y^2$ are found in positive integers $x$ and $y$, for $6 \leq m \leq 100$. In this paper, we consider the question of higher powers, and find all solutions to the equation $\mathrm{Pyr}_m(x) = y^n$ in positive integers $x$, $y$, and $n$, with $n \geq 3$, and $5 \leq m \leq 50$. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.

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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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arXiv2019

Number systems over general orders

Let $\mathcal{O}$ be an order, that is a commutative ring with $1$ whose additive structure is a free $\mathbb{Z}$-module of finite rank. A generalized number system (GNS for short) over $\mathcal{O}$ is a pair $(p,\mathcal{D} )$ where $p\in\mathcal{O}[x]$ is monic with constant term $p(0)$ not a zero divisor of $\mathcal{O}$, and where $\mathcal{D}$ is a complete residue system modulo $p(0)$ in $\mathcal{O}$ containing $0$. We say that $(p,\mathcal{D})$ is a GNS over $\mathcal{O}$ with the finiteness property if all elements of $\mathcal{O}[x]/(p)$ have a representative in $\mathcal{D}[x]$ (the polynomials with coefficients in $\mathcal{D}$). Our purpose is to extend several of the results from a previous paper of Pethő and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order $\mathcal{O}$ and GNS $(p,\mathcal{D})$ over $\mathcal{O}$, the pair $(p,\mathcal{D})$ admits the finiteness property. This is closely related to work of Vince on matrix number systems. Let $\mathcal{F}$ be a fundamental domain for $\mathcal{O} \!\otimes_{\mathbb{Z}}\! \mathbb{R}/\mathcal{O}$ and $p\in \mathcal{O}[X]$ a monic polynomial. For $α\in\mathcal{O}$, define $p_α(x):=p(x+α)$ and $\mathcal{D}_{\mathcal{F} ,p(α)}:= p(α)\mathcal{F}\cap\mathcal{O}$. Under mild conditions we show that the pairs $(p_α,\mathcal{D}_{\mathcal{F},p(α)}\,)$ are GNS over $\mathcal{O}$ with finiteness property provided $α\in\mathcal{O}$ in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that $(p_{-m},\mathcal{D}_{\mathcal{F} ,p(-m)}\,)$ does not have the finiteness property for each large enough positive rational integer $m$.

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arXiv2018

On the smallest number of terms of vanishing sums of units in number fields

Let $K$ be a number field. In the terminology of Nagell a unit $\varepsilon$ of $K$ is called {\it exceptional} if $1-\varepsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ is solvable in units $\varepsilon_1,\varepsilon_2,\varepsilon_3$ of $K$. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer $k$ with $k\geq 3$, denoted by $\ell(K)$, such that the unit equation $\varepsilon_1+\dots+\varepsilon_k=0$ is solvable in units $\varepsilon_1,\dots,\varepsilon_k$ of $K$. If no such $k$ exists, we set $\ell(K)=\infty$. Apart from trivial cases when $\ell(K)=\infty$, we give an explicit upper bound for $\ell(K)$. We obtain several results for $\ell(K)$ in number fields of degree at most $4$, cyclotomic fields and general number fields of given degree. We prove various properties of $\ell(K)$, including its magnitude, parity as well as the cardinality of number fields $K$ with given degree and given odd resp. even value $\ell(K)$. Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions.

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arXiv2018

Mahler's work on Diophantine equations and subsequent developments

We discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, S-unit equations and S-integral points on elliptic curves, and go into later developments concerning the number of solutions to Thue-Mahler equations and effective finiteness results for Thue-Mahler equations. For the latter we need estimates for p-adic logarithmic forms, which may be viewed as an outgrowth of Mahler's work on the p-adic Gel'fond-Schneider theorem. We also go briefly into decomposable form equations, these are certain higher dimensional generalizations of Thue-Mahler equations.

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arXiv2017

S-parts of values of univariate polynomials, binary forms and decomposable forms at integral points

Let $S$ be a finite set of primes. The $S$-part $[m]_S$ of a non-zero integer $m$ is the largest positive divisor of $m$ that is composed of primes from $S$. In 2013, Gross and Vincent proved that if $f(X)$ is a polynomial with integer coefficients and with at least two roots in the complex numbers, then for every integer $x$ at which $f(x)$ is non-zero, we have (*) $[f(x)]_S\leq c\cdot |f(x)|^d$, where $c$ and $d$ are effectively computable and $d<1$. Their proof uses Baker-type estimates for linear forms in complex logarithms of algebraic numbers. As an easy application of the $p$-adic Thue-Siegel-Roth theorem we show that if $f(X)$ has degree $n\geq 2$ and no multiple roots, then an inequality such as (*) holds for all $d>1/n$, provided we do not require effectivity of $c$. Further, we show that such an inequality does not hold anymore with $d=1/n$ and sufficiently small $c$. In addition we prove a density result, giving for every $ε>0$ an asymptotic estimate with the right order of magnitude for the number of integers $x$ with absolute value at most $B$ such that $f(x)$ has $S$-part at least $|f(x)|^ε$. The result of Gross and Vincent, as well as the other results mentioned above, are generalized to values of binary forms and decomposable forms at integral points. Our main tools are Baker type estimates for linear forms in complex and $p$-adic logarithms, the $p$-adic Subspace Theorem of Schmidt and Schlickewei, and a recent general lattice point counting result of Barroero and Widmer.

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arXiv2011

Effective results for unit equations over finitely generated domains

Let A be a commutative domain containing Z which is finitely generated as a Z-algebra, and let a,b,c be non-zero elements of A. It follows from work of Siegel, Mahler, Parry and Lang that the equation (*) ax+by=c has only finitely many solutions in elements x,y of the unit group A* of A, but the proof following from their arguments is ineffective. Using linear forms in logarithms estimates of Baker and Coates, in 1979 Győry gave an effective proof of this finiteness result, in the special case that A is the ring of S-integers of an algebraic number field. Some years later, Győry extended this to a restricted class of finitely generated domains A, containing transcendental elements. In the present paper, we give an effective finiteness proof for the number of solutions of (*) for arbitrary domains A finitely generated over Z. In fact, we give an explicit upper bound for the `sizes' of the solutions x,y, in terms of defining parameters for A,a,b,c. In our proof, we use already existing effective finiteness results for two variable S-unit equations over number fields due to Győry and Yu and over function fields due to Mason, as well as an explicit specialization argument.

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arXiv2011

Multiply monogenic orders

Let O be an order in an algebraic number field K, i.e., a ring with quotient field K which is contained in the ring of integers of K. The order O is called monogenic, if it is of the shape Z[w], i.e., generated over the rational integers by one element. By a result of Győry (1976), the set of w with Z[w]=O is a union of finitely many equivalence classes, where two elements v,w of O are called equivalent if v+w or v-w is a rational integer. An order O is called k times monogenic if there are at least k different equivalence classes of w with Z[w]=O, and precisely k times monogenic if there are precisely k such equivalence classes. It is known that every quadratic order is precisely one time monogenic, while in number fields of degree larger than 2, there may be non-monogenic orders. In this paper we study orders which are more than one time monogenic. Our first main result is, that in any number field K of degree at least 3 there are only finitely many three times monogenic orders. Next, we define two special types of two times monogenic orders, and show that there are number fields K which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of K, we prove that K has only finitely many two times monogenic orders which are not of these types. We give some immediate applications to canonical number systems. Further, we prove extensions of our results for domains which are monogenic over a given domain A of characteristic 0 which is finitely generated over Z.

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