Degree-only bounds for polynomial D(n)-tuples

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a nonzero $n\in\mathbb Z[X]$, call a set $A=\{a_1,\ldots,a_m\}$ of distinct nonzero polynomials in $\mathbb Z[X]$ a polynomial $D(n)$-tuple if $a_i a_j+n\in\{b(X)^2:b\in\mathbb Z[X]\}$ for all $i<j$, and there is no nonzero $p\in\mathbb Z[X]$ for which every $a_i/p$ and $n/p^2$ is a constant rational integer. This last condition excludes sets obtained by multiplying an integer tuple and its parameter by a common polynomial and its square. Let $P_n$ be the supremum of their cardinalities.

Statement recorded in the supplied source.

Is there a function $B:\mathbb Z_{\ge0}\to\mathbb N$ such that $P_n\le B(\deg n)$ for every nonzero integer polynomial $n$, independent of the sizes of its coefficients? [S2]

Short English statement

Can polynomial Diophantine tuples be bounded in size solely by the degree of the parameter polynomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500010 (AMR-084-0010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Andrej Dujella, Diophantine m-tuples, author-maintained survey (snapshot accessed 9 October 2026), Section 7.2, definition of polynomial tuples and degree-only upper-bound question. https://raw.githubusercontent.com/dujella/dujella.github.io/master/gener.html.