arXiv subjects

Pavlo Yatsyna

Publications and source records attributed to Pavlo Yatsyna.

At least 19 recordsLinked to original sources

arXiv2026

Height lower bounds for elements of highly composite rings

Let $\mathbb{Q}^{(d)}$ be the composite field of all number fields of degree at most $d$. In 2001 Bombieri and Zannier proved that $\mathbb{Q}^{(2)}$ has the Northcott property and asked what happens for $d\geq 3$. Here we study the absolute Weil height for elements in the composite ring of the rings of integers of such number fields. In particular, we consider $\mathbb{Q}^{(3)}$ as the composite field of $\mathbb{Q}^{(2)}$ and a minimal infinite family of cubic fields, and we show that the composite ring of the rings of integers of these fields does have the Northcott property. Our results follow from new height lower bounds, expressed in terms of the degree. Moreover, we introduce a notion of size for subfields of $\mathbb{Q}^{(3)}$. For instance, $\mathbb{Q}^{(3)}$ has size $1$ and the maximal abelian subfield of $\mathbb{Q}^{(3)}$ has size $1/2$. We show that there is a subfield of $\mathbb{Q}^{(3)}$ of size $1$ which has the Northcott property.

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arXiv2026

Lattice point sumsets and asymptotic approximate groups

We establish new quantitative bounds for asymptotic approximate groups arising from finite subsets of lattices and, more generally, semi-linear subsets of abelian groups. Our approach combines Khovanskii's theorem on sumsets with Rogers and Zong bounds for the covering numbers.

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arXiv2026

The Minimal Degree of Salem Numbers with Negative Trace

We show that the minimal degree of a Salem number with prescribed negative trace is bounded below by $T^2/\log T$ and above by $T^2\log T$, up to explicit constants and lower-order terms, where $T$ is the absolute value of the trace. This improves the previous doubly exponential upper bound of McKee and Smyth. Building on their construction, we also prove that for every prescribed negative trace there exists a constant $d_0$ such that Salem numbers of that trace exist in every even degree exceeding $d_0$.

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arXiv2026

Even better sums of squares over quintic and cyclotomic fields

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 unsolved cases of quartic and quintic fields. This fully settles the lifting problem for universal forms in degrees at most 5. The main tool behind the proof is a computationally intensive classification of fields in which every multiple of 2 is the sum of squares. We further extend these results to some real cyclotomic fields of large degrees and prove Kitaoka's conjecture for them.

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arXiv2026

Pythagoras numbers for infinite algebraic fields

We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.

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arXiv2025

Sums of two units in number fields

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, δ\in \mathcal{O}_K^\times$ satisfying $\varepsilon + δ= n$. We show that $\mathcal{N}_K$ is a finite set if $K$ does not contain any real quadratic subfield. In the case where $K$ is a cubic field, we also explicitly classify all solutions to the unit equation $\varepsilon + δ= n$ when $K$ is either cyclic or has negative discriminant.

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arXiv2025

Asymptotics of $n$-universal lattices over number fields

We prove an explicit asymptotic formula for the logarithm of the minimal ranks of $n$-universal lattices over the ring of integers of totally real number fields. We also show that, for any constant $C > 0$ and $n \geq 3$, there are only finitely many totally real fields with an $n$-universal lattice of rank at most $C$, with all such fields being effectively computable. Similarly, for any $n \geq 3$, we show that there are only finitely many totally real fields admitting an $n$-universal criterion set of size at most $C$, with all such fields likewise being effectively computable.

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arXiv2025

Additively indecomposable quadratic forms over totally real number fields

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper bounds for the minimal ranks of $n$-universal quadratic forms. For $\mathbb{Q}(\sqrt{2}),~\mathbb{Q}(\sqrt{3}),~\mathbb{Q}(\sqrt{5}),~\mathbb{Q}(\sqrt{6})$, and $\mathbb{Q}(\sqrt{21})$, we classify, up to equivalence, all classical, additively indecomposable binary quadratic forms.

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arXiv2025

Counting polynomials with positive roots

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying Maclaurin's inequalities. For cubic polynomials, we derive a much more precise asymptotic result. Furthermore, we analyse the arithmetic properties of the discriminants of these polynomials, showing that a positive proportion of cubics have square-free discriminants.

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arXiv2025

Most totally real fields do not have universal forms or Northcott property

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 property, is meager. The main tool is a new theorem on the number of square classes of totally positive units represented by a quadratic lattice of a given rank.

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arXiv2025

Real quadratic fields with a universal quadratic form of given rank have density zero

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 generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients.

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arXiv2025

Failures of integral Springer's Theorem

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.

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arXiv2025

Kitaoka's Conjecture for quadratic fields

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 fields. More generally, we obtain explicit upper bounds on the discriminants of real quadratic fields with a quadratic lattice of rank at most 7 that represents all totally positive multiples of a fixed integer.

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arXiv2024

There are Salem numbers with trace $-3$ and every degree at least $34$

We prove that there exist Salem numbers with trace $-3$ and every even degree $\geq 34$. Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace $-3$ and degree $\leq 30$, our result is optimal up to possibly the single value $32$, for which it is expected there are no such numbers.

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arXiv2023

Equiangular lines in Euclidean spaces: dimensions 17 and 18

We show that the maximum cardinality of an equiangular line system in 17 dimensions is 48, thereby solving a longstanding open problem. Furthermore, by giving an explicit construction, we improve the lower bound on the maximum cardinality of an equiangular line system in 18 dimensions to 57.

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arXiv2022

On Kitaoka's conjecture and lifting problem for universal quadratic forms

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal (namely, those that have a short basis in a suitable sense). Along the way we give a general construction of a universal form of rank bounded by $D(\log D)^{d-1}$, where $d$ is the degree of $K$ over $\mathbb Q$ and $D$ is its discriminant. Furthermore, for any fixed degree we prove (weak) Kitaoka's conjecture that there are only finitely many totally real number fields with a universal ternary quadratic form.

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arXiv2022

On quadratic Waring's problem in totally real number fields

We improve the bound of the $g$-invariant of the ring of integers of a totally real number field, where the $g$-invariant $g(r)$ is the smallest number of squares of linear forms in $r$ variables that is required to represent all the quadratic forms of rank $r$ that are representable by the sum of squares. Specifically, we prove that the $g_{\mathcal{O}_K}(r)$ of the ring of integers $\mathcal{O}_K$ of a totally real number field $K$ is at most $g_{\mathbb{Z}}([K:\mathbb{Q}]r)$. Moreover, it can also be bounded by $g_{\mathcal{O}_F}([K:F]r+1)$ for any subfield $F$ of $K$. This yields a sub-exponential upper bound for $g(r)$ of each ring of integers (even if the class number is not $1$). Further, we obtain a more general inequality for the lattice version $G(r)$ of the invariant and apply it to determine the value of $G(2)$ for all but one real quadratic field.

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arXiv2022

Potential energy of totally positive algebraic integers

Given positive real numbers, we prove two inequalities involving their potential energy and their power sums. We also prove an inequality involving the energy and the discriminant and apply it to deduce a result on totally positive irreducible polynomials.

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