Semialgebraic decomposition of real binary forms of a given degree's space

TL;DR

This paper presents a constructive method and an adaptation of Sylvester's Algorithm to find real Waring decompositions of binary forms, classifying forms by their real rank using semialgebraic sets.

Contribution

It introduces a new constructive approach for real binary forms and adapts Sylvester's Algorithm to determine minimal decompositions and real rank.

Findings

A constructive method for real binary form decomposition

Adaptation of Sylvester's Algorithm for real case

Decomposition of the space into semialgebraic sets by real rank

Abstract

The Waring Problem over polynomial rings asks for how to decompose an homogeneous polynomial of degree $d$ as a finite sum of $d^{th}$ powers of linear forms. First, we give a constructive method to obtain a real Waring decomposition of any given real binary form with length at most its degree. Secondly, we adapt the Sylvester's Algorithm to the real case in order to determine a Waring decomposition with minimal length and then we establish its real rank. We use bezoutian matrices to achieve a minimal decomposition. We consider all real binary forms of a given degree and we decompose this space as a finite union of semialgebraic sets according to their real rank. Some examples are included.