O-minimal structures and unlikely intersections
We prove a uniform bound on the complexity of intersections of definable families with algebraic subvarieties, giving a new route to unlikely-intersection statements in the spirit of the Zilber–Pink conjecture.
1 · Definability
An o-minimal structure is a system of subsets of real space closed under the usual operations and admitting finite cell decompositions. The observation that the graph of the exponential function is definable in the structure generated by it turned model theory into a tool for Diophantine geometry.
2 · The uniformity theorem
Our main theorem bounds the degree of intersection of a definable family with algebraic subvarieties, uniformly over the family and with constants depending only on the definable data. The proof uses the cell decomposition theorem quantitatively, through a combinatorial count of cells.
3 · Arithmetic consequences
Combined with the Pila–Wilkie counting theorem, the uniformity yields new cases of the Zilber–Pink conjecture for products of abelian varieties with definable analytic subvarieties of controlled complexity.