On the Waring rank of binary forms: The binomial formula and a dihedral cover of rank two forms

TL;DR

This paper provides an explicit formula for the Waring rank of binary binomials and characterizes binary forms of degree d with rank two, revealing their structure and enumeration.

Contribution

It introduces a formula for the Waring rank of binary binomials and classifies binary forms of rank two with respect to fixed linear forms.

Findings

Explicit formula for the Waring rank of binary binomials

Classification of binary forms of degree d with rank two

Enumeration of such forms up to scalar multiplication

Abstract

Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients. Firstly, we give an explicit formula for the Waring rank of any binary binomial and several examples to illustrating it. Secondly, we prove that, up to scalar multiplication, there are exactly $\binom{d-1}{2}$ binary forms of degree $d$ with Waring rank two and multiple of three fixed distinct linear forms.