36 citations · 235 across the 39 of their papers we have counts for
33 papers · 1 filter
The addition on totally positive integers uniquely determines the totally real number field
Vitezslav Kala, Ritoprovo Roy
Let be a totally real number field. We prove that the totally positive algebraic integers of , viewed as an abstract additive semigroup, uniquely determine…
There are consecutive cubic fields with large class numbers, when ordered by discriminant
Vitezslav Kala, Om Prakash
We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a give…
Kitaoka's Conjecture and sums of squares
Vitezslav Kala, Kristyna Kramer, Jakub Krasensky
We connect the existence of a ternary classical universal quadratic form over a totally real number field with the property that all totally positive multiples of 2 are sums of…
Kitaoka's Conjecture for quadratic fields
Vitezslav Kala, Jakub Krásenský, Dayoon Park +2
We prove that there are at most 13 real quadratic fields that admit a ternary universal quadratic lattice, thus establishing a strong version of Kitaoka's Conjecture for quadratic…
Universality criterion sets for quadratic forms over number fields
Vitezslav Kala, Jakub Krásenský, Giuliano Romeo
In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset of the totally positive integers in a given totally real number field such that if a quadr…
Most totally real fields do not have universal forms or Northcott property
Nicolas Daans, Vitezslav Kala, Siu Hang Man +2
We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott prope…
Universality lifting from a general base field
Vitezslav Kala, Daejun Kim, Seok Hyeong Lee
Given a totally real number field , we show that there are only finitely many totally real extensions of of a fixed degree that admit a universal quadratic form defined over…
There is no 290-Theorem for higher degree forms
Vitezslav Kala, Om Prakash
We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be cha…
Sails for universal quadratic forms
Vítězslav Kala, Siu Hang Man
We establish a new connection between sails, a key notion in the geometric theory of generalised continued fractions, and arithmetic of totally real number fields, specifically, un…
Even better sums of squares over quintic and cyclotomic fields
Vitezslav Kala, Pavlo Yatsyna
We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously…
Universal quadratic forms and Dedekind zeta functions
Vítězslav Kala, Mentzelos Melistas
We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadrati…
Universal quadratic forms and Northcott property of infinite number fields
Nicolas Daans, Vítězslav Kala, Siu Hang Man
We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals , then the set of totally positive integers…
Periodicity of general multidimensional continued fractions using repetend matrix form
Hanka Řada, Štěpán Starosta, Vítězslav Kala
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible struc…
Partitions into powers of an algebraic number
Vítězslav Kala, Mikuláš Zindulka
We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number . We prove that if is real quadratic, then the number of partitions is alwa…
Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
Vítězslav Kala, Ester Sgallová, Magdaléna Tinková
We study a new connection between multidimensional continued fractions, such as Jacobi--Perron algorithm, and additively indecomposable integers in totally real cubic number fields…
Real quadratic fields with a universal quadratic form of given rank have density zero
Vitezslav Kala, Pavlo Yatsyna, Błażej Żmija
We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More gene…
Universal quadratic forms and indecomposables in number fields: A survey
Vítězslav Kala
We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discu…
On continued fraction partial quotients of square roots of primes
Vítězslav Kala, Piotr Miska
We show that for each positive integer there exist only finitely many prime numbers such that appears an odd number of times in the period of continued fraction of $\sq…
Consecutive real quadratic fields with large class numbers
Giacomo Cherubini, Alessandro Fazzari, Andrew Granville +2
For a given positive integer , we prove that there are at least integers such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqr…
On Kitaoka's conjecture and lifting problem for universal quadratic forms
Vítězslav Kala, Pavlo Yatsyna
For a totally positive definite quadratic form over the ring of integers of a totally real number field , we show that there are only finitely many totally real field extensions…
Number fields without universal quadratic forms of small rank exist in most degrees
Vítězslav Kala
We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.
Universal quadratic forms, small norms and traces in families of number fields
Vítězslav Kala, Magdaléna Tinková
We obtain good estimates on the ranks of universal quadratic forms over Shanks' family of the simplest cubic fields and several other families of totally real number fields. As the…
Sums of squares in -integers
Vítězslav Kala, Pavlo Yatsyna
In totally real number fields, we characterize the rings of -integers (obtained by inverting a rational integer ) such that all their totally positive elements are represente…
Ternary quadratic forms representing a given arithmetic progression
Tomáš Hejda, Vítězslav Kala
A positive quadratic form is -universal if it represents all the numbers where is a non-negative integer, and almost -universal if it represents a…