Bezout-type Theorems for Differential Fields

TL;DR

This paper establishes bounds on the number of solutions to algebraic differential systems over differentially closed fields, improving previous estimates from doubly-exponential to singly-exponential, with applications in diophantine geometry.

Contribution

It proves Bezout-type theorems for differential equations, providing sharper bounds on solutions and demonstrating their applications in counting transcendental points and intersections in algebraic geometry.

Findings

Number of solutions bounded by a singly-exponential function of system parameters.

Improved bounds in diophantine problems involving algebraic subvarieties.

Applications to counting lattice points and intersections in algebraic and differential contexts.

Abstract

We prove analogs of the Bezout and the Bernstein-Kushnirenko-Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.