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Mikuláš Zindulka

Publications and source records attributed to Mikuláš Zindulka.

10 recordsLinked to original sources
arXiv2026

Negative bias in moments of the Legendre family of elliptic curves

We determine the bias in higher moments of the Legendre family of elliptic curves in view of the Negative Bias Conjecture of S. J. Miller. We show that every lower order term is either zero or negative on average for both even and odd moments, and explicitly compute the third and fourth moment. The results about Hurwitz class numbers in arithmetic progressions which we prove in the process may be useful for other applications.

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arXiv2026

Representing rational integers by generalized quadratic forms over quadratic fields

We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.

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arXiv2025

Totally positive elements with $m$ partitions exist in almost all real quadratic fields

In this paper, we study partitions of totally positive integral elements $α$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper bound for the norm of $α$ that can be represented as a sum of indecomposables in at most $m$ ways, completely characterize the $α$'s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with $m$ partitions for $1 \leq m \leq 7$.

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arXiv2025

Counting principal ideals of small norm in the simplest cubic fields

We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm.

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arXiv2024

No proper generalized quadratic forms are universal over quadratic fields

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.

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arXiv2024

Partitions into powers of an algebraic number

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $β$. We prove that if $β$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $β$ is larger than 1. Further, we show that for $β$ satisfying a certain condition, the partition function attains all non-negative integers as values.

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arXiv2023

Partitions in real quadratic fields

We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such that there exists an element with exactly 1-5, 7, and 11 partitions.

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arXiv2023

The Six-Vertex Model with a Non-Standard Boundary Condition

We consider the enumeration of states in the Brubaker-Bump-Friedberg six-vertex model, whose boundary conditions are determined by an integer partition. In general, we find the number of states is a polynomial in the largest part of the partition. By explicating this technique, we also enumerate the states completely for hook shapes and staircases.

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arXiv2023

Artin Twin Primes

We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number $π_{d,g}(x)$ of primes $p \leq x$ such that both $p$ and $p+d$ are Artin primes for $g$. In particular, we identify a class of pairs $(d,g)$ for which $π_{d,g}(x) =0$. Our results suggest that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution.

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