Counting Theorems for Algebraic Relations
Abstract
Let X be a set definable in a sharply o-minimal structure. We consider the problem of counting the number of points where X intersects algebraic varieties V over Q of dimension k < codim X, as a function of T := deg(V) + h(V), where h(V) is the log-height of V. In particular, we conjecture that after removing a suitable "algebraic part", this number grows polynomially in T -- a generalization of Wilkie's conjecture. We show that this full conjecture implies some open problems in algebraic independence theory. We also formulate a weaker conjecture stating that all intersections above are contained in a poly(T) amount of balls of radius e^{-T}. We then consider the case where X (subset of C^n) is a (compact piece of a) trajectory of a polynomial differential equation satisfying a variant of Nesterenko's D-property. Our main theorem is a proof of the weakened conjecture for such curves when k < sqrt(n) - 1.
Keywords
Cite
@article{arxiv.2604.15189,
title = {Counting Theorems for Algebraic Relations},
author = {Gal Binyamini and Noriko Hirata-Kohno and Makoto Kawashima and Yuval Salant},
journal= {arXiv preprint arXiv:2604.15189},
year = {2026}
}