Structures of the Length Seven Power Sum Decompositions of Ternary Quartics

TL;DR

This paper explores the structure of length seven Waring decompositions of ternary quartics to better understand tensor rank and its computational complexity implications.

Contribution

It introduces technical tools for organizing length seven Waring decompositions, aiming to establish a foundation for an induction approach to tensor rank.

Findings

Developed methods to analyze Waring decompositions of ternary quartics.

Provided insights into the maximum symmetric rank for specific forms.

Laid groundwork for future tensor rank determination techniques.

Abstract

Motivated by the search for a deeper understanding of tensor rank, in view of its computational complexity applications, we investigate a possible path to determine the maximum symmetric rank in given degree and dimension. We work in terms of Waring rank of forms, and aiming to set up a firm basis for an induction procedure we examine some technical tools to organize length seven Waring decompositions of ternary quartics, that may turn to be fundamental.