Bass Note Spectra of Binary Forms

TL;DR

This paper proves that the spectrum of any real isotropic homogeneous binary form of degree three or higher is a continuous interval from zero to a positive maximum, resolving longstanding conjectures in the field.

Contribution

It establishes that the spectrum of such binary forms is always an interval, confirming a conjecture related to spectral gaps and completing Mahler's program for all degrees.

Findings

Spectrum of binary forms is an interval [0, M_P]

Resolves Mordell's conjecture from 1940

Completes Mahler's program for binary forms

Abstract

We show that the spectrum of every $\mathbb{R}-$isotropic homogeneous binary form $P$ of degree $n\geq3$ is an interval of the form $[0,M_P],$ where $M_P$ is some positive constant. This completes the discussion around a conjecture of Mordell from 1940 (disproved by Davenport) regarding the existence of spectral gaps for binary cubic forms and further settles Mahler's program for binary forms of every degree.