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The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

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Abstract

In this paper, we conjecture a formula for the value of the Pythagoras number for real multivariate sum of squares polynomials as a function of the (total or coordinate) degree and the number of variables. The conjecture is based on the comparison between the number of parameters and the number of conditions for a corresponding low-rank representation. This is then numerically verified for a number of examples. Additionally, we discuss the Pythagoras number of (complex) multivariate Laurent polynomials that are sum of square magnitudes of polynomials on the \(n\)-torus. For both types of polynomials, we also propose an algorithm to numerically compute the Pythagoras number and give some numerical illustrations.

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Authors and Affiliations

  1. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 , Heverlee, Belgium

    Thanh Hieu Le, Laurent Sorber & Marc Van Barel

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  1. Thanh Hieu Le
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  2. Laurent Sorber
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  3. Marc Van Barel
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Correspondence to Marc Van Barel.

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The research was partially supported by the Research Council KU Leuven, project OT/10/038 (Multi-parameter model order reduction and its applications), PF/10/002 Optimization in Engineering Centre (OPTEC), and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). Laurent Sorber is supported by a Flanders Institute of Science and Technology (IWT) doctoral scholarship. The scientific responsibility rests with its author(s)

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Le, T.H., Sorber, L. & Van Barel, M. The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials. Calcolo 50, 283–303 (2013). https://doi.org/10.1007/s10092-012-0068-y

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  • DOI: https://doi.org/10.1007/s10092-012-0068-y

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