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

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.

23 pages, 5 figures, 2 algorithms

References in corpus (4)

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• Some new canonical forms for polynomials
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• Real and complex rank for real symmetric tensors with low ranks