On the Polynomial Quadruples with the Property $D(-1;1)$

Abstract

In this paper we prove, under some assumptions, that every polynomial $D(-1)$-triple in $\mathbb{Z}[X]$ can only be extended to a polynomial $D(-1;1)$-quadruple in $\mathbb{Z}[X]$ by polynomials $d^{\pm}$. More precisely, if $\{a,b,c;d\}$ is a polynomial $D(-1;1)$-quadruple, then $$d=d^{\pm}=-(a+b+c)+2(abc\pm rst)\,,$$ where $r$, $s$ and $t$ are polynomials from $\mathbb{Z}[X]$ with positive leading coefficients that satisfy $ab-1=r^2$, $ac-1=s^2$ and $bc-1=t^2$.