On the maximum rank of a real binary form

TL;DR

This paper characterizes when a real binary form of degree d has maximum rank d, showing it corresponds exactly to forms with d real roots, thus clarifying the structure of forms with maximal rank.

Contribution

It provides a precise criterion linking the number of real roots of a binary form to the real roots of its directional derivatives, answering a question by Comon and Ottaviani.

Findings

Forms with d real roots have maximum rank d.

The interior of the locus of rank d forms is exactly the set with d real roots.

The criterion involves the roots of directional derivatives of the form.

Abstract

We show that a real homogeneous polynomial f(x,y) with distinct roots and degree d greater or equal than 3 has d real roots if and only if for any (a,b) not equal to (0,0) the polynomial af_x+bf_y has d-1 real roots. This answers to a question posed by P. Comon and G. Ottaviani, and shows that the interior part of the locus of degree d binary real binary forms of rank equal to d is given exactly by the forms with d real roots.