An absolute bound for generalized Diophantine tuples over polynomial rings

Abstract

Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Let $k\ge 2$ be an integer, and let $n\in \mathbb{F}[x]\setminus \{0\}$. We study generalized Diophantine tuples $A\subset \mathbb{F}[x]$ with property $D_k(n)$, meaning that $ab+n$ is a $k$-th power in $\mathbb{F}[x]$ for all distinct elements $a,b\in A$. For $k\ge 18$, we prove that every such tuple satisfies $|A|\le 6$, except for the necessary exceptional family in which $n=s^2$ is a $k$-th power and $A\subset s\mathbb{F}$. This bound is absolute: it is independent of both $n$ and $\mathrm{deg}n$. Our proof develops a new method for studying polynomial Diophantine tuples, combining a determinant criterion, generalizations of the Mason--Stothers theorem, and the Combinatorial Nullstellensatz. We also record a conditional analogue for generalized Diophantine tuples over the integers.

23 pages