First talk of the Fall 2024 logic seminar.
Title: Complex Polynomials up to Interdefinability
Abstract: Let P and Q be polynomial functions in several variables over the complex numbers. Say that P defines Q if the graph of Q is definable with parameters in the first-order language with P as its only non-logical symbol (interpreted on the complex numbers in the obvious way). Then say that P and Q are interdefinable if each defines the other. For example, one can check that x+y and x-y are interdefinable, but x+y and xy are not.
In this talk, we will first experiment with some toy examples of this notion of interdefinability. Then, based on joint work with Chieu-Minh Tran, I will present a complete classification of multivariable complex polynomials up to interdefinability. (The surprise: there are not as many classes as you might expect). Time permitting, I will describe the proof strategy, which uses a mix of results from logic, combinatorics, and algebraic geometry.