Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials

Abstract

Hilbert famously showed that polynomials in n n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n n ) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert’s results are not uniform in n n : unsurprisingly, polynomials in n n variables will exhibit greater complexity as n n increases. However, an array of recent work has shown that in a certain regime—namely, that where the number of polynomials and their degrees are fixed—the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman’s conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman’s conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables.