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Zrinka Franušić

Publications and source records attributed to Zrinka Franušić.

6 recordsLinked to original sources
arXiv2026

Cardinalities of Jordan Diophantine sets of upper triangular integer matrices

Let $U$ be the ring of upper triangular $2\times2$ integer matrices. For $N\in U$, a Jordan $D(N)$-set is a set of distinct nonzero matrices in $U$ such that $(AB+BA)/2+N$ is a square in $U$ for any two distinct elements $A$ and $B$. We determine all $N$ for which an infinite Jordan $D(N)$-set exists. If no infinite set exists, every such set has at most six elements. If $N$ is not a difference of two squares in $U$, the bound is five. Both bounds are best possible. We also give necessary and sufficient conditions for the existence of infinite sets with nonzero pairwise Jordan products and of infinite sets consisting of nonsingular matrices.

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arXiv2026

$D(N)$-quadruples in upper-triangular $2\times2$ integer matrices

We introduce analogues of Diophantine $D(N)$-$m$-tuples in the noncommutative ring $M_2(\mathbb Z)$ of $2\times2$ integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product $$A\circ B=\frac12(AB+BA).$$ Special attention is devoted to upper-triangular integer matrices $UT_2(\mathbb Z)$, where squares admit a particularly simple description. Motivated by the classical connection between representations of $n$ as a difference of two squares and the existence of $D(n)$-quadruples in commutative rings, we investigate the existence of Jordan $D(N)$-quadruples in $UT_2(\mathbb Z)$.

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arXiv2026

Differences of squares of upper-triangular $2\times 2$ integer matrices

We consider the problem of characterizing upper-triangular matrices $M=\begin{pmatrix}p&r\\0&q\end{pmatrix}\in M_2(\mathbb Z)$ which can be represented in the form $A^2-B^2$ with upper-triangular integer matrices $A$ and $B$ and give a complete criterion in terms of representations of $p$ and $q$ as differences of two squares and an additional divisibility condition on $r$. Also, we give a complete classification of representable matrices in terms of congruence conditions on $p$, $q$, and $r$.

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arXiv2019

On the extensions of the Diophantine triples in Gaussian integers

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple $\{k-1, k+1, 16k^3-4k\}$ in Gaussian integers $\mathbb{Z}[i]$ to a Diophantine quadruple. Similar one-parameter family, $\{k-1, k+1, 4k\}$, was studied in Franušić's previous paper, where it was shown that the extension to a Diophantine quadruple is unique (with an element $16k^3-4k$). The family of the triples of the same form $\{k-1, k+1, 16k^3-4k\}$ was already studied in rational integers. It appeared as a special case while solving the extensibility problem of Diophantine pair $\{k-1, k+1\}$, in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between $k+1$ and $16k^3-4k$ is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.

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arXiv2016

Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

Let $M$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$ and let $ξ$ be a root of polynomial $$f\left( x\right) =x^{4}-2cx^{3}+2x^{2}+2cx+1,$$ where $c\in\mathbb{Z}_{M},$ $c\notin\left\{ 0,\pm2\right\}$. We consider an infinite family of octic fields $K_{c}=M\left( ξ\right)$ with the ring of integers $\mathbb{Z}_{K_{c}}.$ Our goal is to determine all generators of relative power integral basis of $\mathcal{O=}\mathbb{Z}_{M}\left[ ξ\right]$ over $\mathbb{Z}_{M}.$ We show that our problem reduces to solving the system of relative Pellian equations \[ cV^{2}-\left( c+2\right) U^{2}=-2μ,\ \ cZ^{2}-\left( c-2\right) U^{2}=2μ, \] where $μ$ is an unit in $\mathbb{Z}_{M}$. We solve the system completely and find that all non-equivalent generators of power integral basis of $\mathcal{O}$ over $\mathbb{Z}_{M}$ are given by $α=ξ,$ $2ξ-2cξ^{2}+ξ^{3}$ for $\left\vert c\right\vert \geq159108$ and $|c|\leq200$.

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