Mike Stillman: Ranks of Homogeneous Polynomials, Macaulay Inverse Systems, and Computational Algebra: Quaternary Quartics

Abstract

The rank of a homogeneous polynomial \(F\) of degree \(d\) is the smallest number such that \(F\) can be written as the sum of \(d\)-th powers of this number of linear forms.

We will first introduce the main tools and players in this game, including rank, Macaulay inverse systems, and graded Betti numbers. These methods are powerful and have many applications, perhaps to problems that you are working on! In this talk, we will apply these methods to investigate the ranks of low degree polynomials, culminating in a stratification of the space of quartic polynomials in 4 variables, which gives a better picture of the possible ranks of such forms. We will keep this all down to earth with examples done via my computer algebra system Macaulay2.

In this talk, we will assume no previous knowledge of these notions!

The parts of this talk that are new are joint with: G. Kapustka, M. Kapustka, K. Ranestad, H. Schenck and B. Yuan (in a massive paper arxiv 2111.05817).