arXiv subjects

Paulius Drungilas

Publications and source records attributed to Paulius Drungilas.

4 recordsLinked to original sources

arXiv2026

No three algebraic conjugates of degree sixteen sum to zero

Let $d$ be the smallest positive integer, not a multiple of $3$, for which there exists an algebraic number $\al$ of degree $d$ over $\mathbb{Q}$ whose three algebraic conjugates add to zero. We prove that $d=20$. This is derived from the following result: for any linear relation $\sum_{j=1}^d a_j \al_j=0$ with coefficients $a_j\in\mathbb{Z}$ among the conjugates $\al_j$ of an algebraic number of degree $d=p^m$, where $p$ is a prime number, $m \geq 1$, the sum $\sum_{j=1}a_j$ is divisible by $p$. If $d=2p^m$, $p\geq 3$ and $\sum_{j=1}^d|a_d| < p$, then $\sum_{j=1}a_j$ is an even number.

math.NT↗

arXiv2026

Linear relations of four conjugates of an algebraic number of degree eight

We characterize all algebraic numbers $α$ of degree $8$ for which there exist four distinct algebraic conjugates $α_1$, $α_2$, $α_3$, $α_4$ of $α$ satisfying the linear relation $α_{1}=α_{2}+α_{3}+α_{4}$. Analogous characterization is obtained for the linear relation $α_{1}+α_{2}+α_{3}+α_{4}=0$. In particular, when an algebraic number $α$ of degree $8$ has a non-even minimal polynomial and possesses exactly six distinct linear relations of the form $α_{i_1}+α_{i_2}+α_{i_3}+α_{i_4}=0$, we prove that $α$ is a sum of a quadratic and a quartic algebraic number.

math.NT↗

arXiv2026

No three algebraic conjugates of degree sixteen sum to zero

Let $d$ be the smallest positive integer, not divisible by $3$, for which there exists an algebraic number over $\mathbb{Q}$ of degree $d$ whose some three algebraic conjugates sum to zero. Employing the classification of vertex-transitive graphs on 16 vertices of degree 6, we prove that $d\neq 16$. This, combined with results obtained by Dubickas, Smyth and Stong \cite{DubickasSmyth2006}, Dubickas and Jankauskas \cite{DubickasJankauskas2015} and Virbalas \cite{Virbalas2025a}, implies that $d=20$.

math.NT↗

arXiv2016

On Newman and Littlewood multiples of Borwein polynomials

A Newman polynomial has all the coefficients in $\{ 0,1\}$ and constant term 1, whereas a Littlewood polynomial has all coefficients in $\{-1,1\}$. We call $P(X)\in\mathbb{Z}[X]$ a Borwein polynomial if all its coefficients belong to $\{ -1,0,1\}$ and $P(0)\neq 0$. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle $|z|=1$ has a non-zero multiple in $\mathbb{Z}[X]$ with coefficients in a finite set $\mathcal{D} \subset \mathbb{Z}$, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.

math.NT↗