Zero counting and invariant sets of differential equations

TL;DR

This paper introduces the constructible orbits condition for polynomial vector fields, showing that it ensures polynomial growth of intersection counts, with applications to differential equations and rational point bounds.

Contribution

It defines the constructible orbits condition and proves polynomial bounds on intersection counts for systems satisfying this condition, including linear and certain nonlinear differential equations.

Findings

Constructible orbits condition implies polynomial growth of intersection numbers.

Established the condition for linear differential equations over (t) and planar polynomial systems.

Derived a polylogarithmic bound on rational points in projections of trajectories.

Abstract

Consider a polynomial vector field $\xi$ in $\mathbb{C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of degree $d$. We introduce a condition on $\xi$ called \emph{constructible orbits} and show that under this condition $N(K,d)$ grows polynomially with $d$. We establish the constructible orbits condition for linear differential equations over $\mathbb{C}(t)$, for planar polynomial differential equations and for some differential equations related to the automorphic $j$-function. As an application of the main result we prove a polylogarithmic upper bound for the number of rational points of a given height in planar projections of $K$ following works of Bombieri-Pila and Masser.