Speaker
Description
A set $\{a, b, c, d\}$ of four distinct non-zero polynomials in $\mathbb{R}[X]$, which are not all constants, is called a polynomial $D(4)$-quadruple in $\mathbb{R}[X]$ if the product of any two of its distinct elements, increased by 4, is a square of a polynomial in $\mathbb{R}[X]$.
We prove some properties of these sets, and to tackle the problem of regularity of polynomial $D(4)$-quadruples in $\mathbb{R}[X]$, we investigate whether the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ is satisfied by each such polynomial $D(4)$-quadruple in $\mathbb{R}[X]$. Our earlier research focused on the regularity of the polynomial $D(4)$-quadruple in $\mathbb{Z}[i][X]$, and we now compare these results with the recent findings from $\mathbb{R}[X]$.
**References**
1. M. Bliznac Trebješanin, S. Bujačić Babić, *Polynomial
D(4)-quadruples over Gaussian integers*, Glas. Mat. Ser. III, to
appear,
2. A. Dujella, A. Jurasić, *On the size of sets in a
polynomial variant of a problem of Diophantus*, Int. J. Number
Theory \textbf{6} (2010), 1449--1471.
3. A. Filipin, A. Jurasić,
*Diophantine quadruples in $\mathbb{Z}[i][X]$*,
Period. Math. Hungar. \textbf{82} (2021), 198--212.
