On real Waring decompositions of real binary forms

TL;DR

This paper presents an algorithm for obtaining real Waring decompositions of real binary forms, constructing a semialgebraic family of solutions and highlighting differences from the complex case.

Contribution

It introduces a novel algorithm for real Waring decompositions of binary forms and constructs a semialgebraic family of such decompositions.

Findings

Algorithm successfully decomposes binary forms of length up to degree.

Constructs a semialgebraic family of decompositions for given forms.

Highlights differences between real and complex Waring decompositions.

Abstract

The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial $p$ of degree $d$ as a finite sum of $d$-{th} powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any given real binary form $p$ of length at most its degree. In fact, we construct a semialgebraic family of Waring decompositions for $p$. Some examples are shown to highlight the difference between the real and the complex case.