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Bezout-type theorems for differential fields

Part of: Discontinuous groups and automorphic forms Model theory Arithmetic problems. Diophantine geometry Differential and difference algebra

Published online by Cambridge University Press:  13 March 2017

Gal Binyamini
Gal Binyamini*
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel email gal.binyamini@weizmann.ac.il
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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.

Keywords

differential equationsBezout theoremdiophantine geometry

MSC classification

Primary: 12H05: Differential algebra
Secondary: 03C60: Model-theoretic algebra 11F03: Modular and automorphic functions 14G05: Rational points

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 867 - 888
Copyright
© The Author 2017 

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