Symmetric rank of some reducible forms

TL;DR

This paper investigates the symmetric rank of specific reducible forms, establishing new lower bounds, classifying quartic normal forms, and analyzing their ranks and decompositions.

Contribution

It introduces a new lower bound for the symmetric rank of reducible forms and classifies quartic forms, advancing understanding of polynomial decompositions.

Findings

New lower bound for symmetric rank of reducible forms

Classification of quartic normal forms and their ranks

Examples of polynomials with maximal and generic ranks

Abstract

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special case of degree 4, we give a list of normal forms for such quartics, and we apply our general result to compute the rank of almost all of them. These families of quartics provide examples of polynomials of generic, supergeneric and even maximal rank, as well as an unexpected number of decompositions.