arXiv subjects

Ana Jurasić

Publications and source records attributed to Ana Jurasić.

8 recordsLinked to original sources

arXiv2026

Four-square polynomial triples: quadratic classification and equal-degree obstructions

We study triples of distinct nonconstant polynomials $a,b,c$ over $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$ for which $$ ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 $$ are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadratic polynomials in $\mathbb{R}[X]$, the one arising from a recent construction of Jurasić, and that none exists in $\mathbb{Q}[X]$ or $\mathbb{Z}[X]$. Moreover, for arbitrary nonconstant entries over $\mathbb{R}$ or $\mathbb{Q}$, we prove that the degree pattern $(2n,2m,2l)$ with $1\le n\le m\le l$ satisfies either $l=m$ or $l>n+m$. For the case $l=m$, we derive factorization systems and reducibility restrictions. In the equal-degree case over $\mathbb{Q}$, we further show that every entry has at least two distinct irreducible factors. As an application, we show that in a natural regular parametrized family of polynomial Diophantine triples in $\mathbb{Q}[X]$ the fourth expression $abc+1$ is never a square when the parameter is constant or the first entry is a square.

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arXiv2026

Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple

A set $\{a,b,c\}$ of distinct positive integers is called a Diophantine triple if $ab+1$, $ac+1$, and $bc+1$ are perfect squares. Dujella formulated the conjecture that the only integral $x$-coordinates on the attached elliptic curve $$y^2=(ax+1)(bx+1)(cx+1)$$ are $0,d_-,d_+$, together with $-1$ when the triple contains $1$, where $d_{\pm}=a+b+c+2abc \pm2\sqrt{(ab+1)(ac+1)(bc+1)}$. The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with $x\ne-1$ make $ax+1$, $bx+1$, and $cx+1$ all perfect squares? We construct infinitely many triples $\{5,115,c\}$ admitting an integral point with $x\ne-1$ for which all three factors are nonsquares.

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arXiv2026

Regularity of Diophantine quadruples over $\mathbb{Q}(i)[X]$

We prove that every Diophantine quadruple $\{a,b,c,d\}$ over $\mathbb{Q}(i)[X]$ that contains at least one non-constant polynomial is regular; that is, $$(a+b-c-d)^2=4(ab+1)(cd+1).$$ This result is consistent with the corresponding results over $\mathbb{Z}[i][X]$ and $\mathbb{R}[X]$, but contrasts with the situation over $\mathbb{C}[X]$.

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arXiv2026

On exotic Diophantine triples in $\mathbb{R}[X]$

Originally, an exotic Diophantine triple is a set $\{a,b,c\}$ of distinct nonzero rational numbers for which \[ a+1,\quad b+1,\quad c+1,\quad ab+1,\quad ac+1,\quad bc+1,\quad abc+1 \] are all perfect squares. We prove that there is no such triple in $\mathbb{R}[X]$, with at least one nonconstant element, if none of $a,b,c$ is equal to $1$. Equivalently, under the distinct nonzero convention, every exotic Diophantine triple in $\mathbb{R}[X]$ with a nonconstant element must contain the element $1$.

math.NT↗

arXiv2026

Four squares from three real polynomials

We construct three nonconstant polynomials $a,b,c\in \mathbb{R}[X]$ such that \[ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 \] are all squares in $\mathbb{R}[X]$. The entries in the construction are quadratic, so the construction has the smallest possible positive degrees. By composition with nonconstant polynomials, this gives infinitely many examples over $\mathbb{R}[X]$ with all entries nonconstant.

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arXiv2026

The extensibility of the Diophantine triple $\{2, b, c\}$

The aim of this paper is to consider the extensibility of the Diophantine triple $\{2,b,c\}$, where $2<b<c$, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of $c$'s (depending on $b$). As corollary, for example, we prove that for $b/2-1$ prime, all Diophantine quadruples $\{2,b,c,d\}$ with $2<b<c<d$ are regular.

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arXiv2017

A polynomial variant of a problem of Diophantus and its consequences

We prove that every Diophantine quadruple in $\mathbb{R}[X]$ is regular. More precisely, we prove that if $\{a, b, c, d\}$ is a set of four non-zero polynomials from $\mathbb{R}[X]$, not all constant, such that the product of any two of its distinct elements increased by $1$ is a square of a polynomial from $\mathbb{R}[X]$, then $$(a+b-c-d)^2=4(ab+1)(cd+1).$$ One consequence of this result is that there does not exist a set of four non-zero polynomials from $\mathbb{Z}[X]$, not all constant, such that a product of any two of them increased by a positive integer $n$, which is not a perfect square, is a square of a polynomial from $\mathbb{Z}[X]$. Our result also implies that there does not exist a set of five non-zero polynomials from $\mathbb{Z}[X]$, not all constant, such that a product of any two of them increased by a positive integer $n$, which is a perfect square, is a square of a polynomial from $\mathbb{Z}[X]$.

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arXiv2015

Elementary resolution of a family of quartic Thue equations over function fields

We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-λY)+Y^4=ξ,\end{eqnarray*} where the solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter $λ\in\mathbb{C}[T]$ is some non-constant polynomial and $0\neqξ\in\mathbb{C}$. It is a function field analogue of the family solved by Mignotte, Pethő and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are very elementary (in particular compared to previous results on parametrized Thue equations over function fields).

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