Waring loci and the Strassen conjecture

TL;DR

This paper characterizes Waring loci for various forms and introduces a Waring loci version of Strassen's Conjecture, providing proofs in multiple cases and advancing understanding of polynomial decompositions.

Contribution

It offers a complete description of Waring loci for key families of forms and proposes a new version of Strassen's Conjecture related to Waring loci, with partial proofs.

Findings

Complete description of Waring loci for quadrics, monomials, binary forms, and plane cubics.

Introduction of a Waring loci version of Strassen's Conjecture.

Proofs of the conjecture in several cases.

Abstract

The Waring locus of a form F is the collection of the degree one forms appearing in some minimal sum of powers decomposition of F. In this paper, we give a complete description of Waring loci for several family of forms, such as quadrics, monomials, binary forms and plane cubics. We also introduce a Waring loci version of Strassen's Conjecture, which implies the original conjecture, and we prove it in many cases.