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

2017/06/13 by M. Ansola, Ansola, Macarena, A. Dı́az-Cano +3
Computer Science · Engineering · Mathematics · #14P10 #15A69 #15A72 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.1706.04207

openalex publication_date 2017/06/13 · openalex created_date 2017/06/23 · openalex updated_date 2026/07/28

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.

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