Counting Theorems for Algebraic Relations

TL;DR

This paper explores counting intersection points between definable sets in o-minimal structures and algebraic varieties, proposing conjectures on growth rates and their implications for algebraic independence.

Contribution

It formulates a conjecture on polynomial growth of intersection counts and proves a weakened version for certain polynomial differential equation trajectories.

Findings

Conjecture that intersection counts grow polynomially with T after removing an algebraic part.

Proved the weakened conjecture for trajectories of polynomial differential equations with specific properties.

Showed that the full conjecture implies open problems in algebraic independence theory.

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.