Problem 1.1

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.

Let $0\ne f\in\mathbb Z[X,Y]$, let $d=\deg f$, and let $H=\max|c_{ij}|$ be its coefficient height. Assume its zero set in $\mathbb Z^2$ is finite.

Statement recorded in the supplied source.

Find an upper bound $B(d,H)$ for $\max\{|x|,|y|\}$ for every integer solution of $f(x,y)=0$, depending only on $d,H$. The question asks for a determinable bound, not merely the existence of a maximum for each fixed $f$. [S2]

Short English statement

Give a degree-and-coefficient bound for every integer solution of a plane equation known to have only finitely many integer solutions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700001 (AMR-086-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Problem 1.1. https://arxiv.org/abs/math/0312440.