Bass Note Spectra of Binary Forms

Abstract: For some homogeneous polynomial $P$ in $k$ variables and some unimodular $k$-dimensional lattice $\Lambda$, what is the smallest value that $\vert P\vert$ assumes on the non-zero vectors of $\Lambda$? The set we obtain by varying $\Lambda$ in the moduli space of unimodular lattices, is referred to as the bass note spectrum of $P$. While this set is fundamental in the geometry of numbers, not many cases are understood. Even in dimension 2, the problem has been solved only when $P$ is $\mathbb{R}$-anisotropic or a quadratic form. In this talk, we will explain Mordell's and Davenport's theorems on the spectra of binary cubic forms and further explain how to resolve this problem for all binary forms of any degree.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar