Algebraic Dynamics, Model Theory, and Number Theory.

Mathematics Colloquium

Time and place

1 PM on Tuesday, March 5th, 2013;

Alice Medvedev (UC Berkeley)

Abstract

Model Theory, a branch of mathematical logic, is a new and useful way to approach number-theoretic conjectures about "special points" and "special subvarieties," such as the Manin-Mumford Conjecture. Algebraic Dynamics, the study of discrete dynamical systems in the category of algebraic geometry, supplies natural generalizations of these conjectures. We use model-theoretic ideas to settle some cases of these dynamical generalizations of number-theoretic conjectures.

Our key result is a complete characterization of invariant subvarieties for coordinate-wise polynomial dynamical systems on affine space, in characteristic zero. That is, we work over a field K of characteristic zero (such as the complex numbers) and study the iteration of a function

F(x₁, x₂, ... x_n) = ( f₁(x₁), f₂(x₂), ... f_n(x_n) ) for some univariate polynomials f_i over K. Model-theoretic ideas reduce the question of invariant subvarieties in cartesian powers of K to the question of invariant curves in K×K. Refining Ritt's Theorem about composition of polynomials allows us to classify these invariant plane curves.

Our classification implies that, barring obvious obstructions from linear f_i, such dynamical systems always have K-rational points with Zariski-dense forward orbits, a generalization of a case of Zhang's Conjecture. We also prove the dynamical analog of the Manin-Mumford conjecture for the very special case when all f_i are defined over the integers, and for some prime p all f_i are congruent to x^p modulo p.