8 papers
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…
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…
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…
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…
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…
Sails for universal quadratic forms
VÃtÄzslav 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…