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Multivariate Polynomials

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Positive Trigonometric Polynomials and Signal Processing Applications

Part of the book series: Signals and Communication Technology ((SCT))

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Abstract

Are the notions and results presented in the previous two chapters valid in the multivariate case? The answer is mostly yes, but with some limitations. The notion of Gram matrix is related directly only to sum-of-squares polynomials. Unlike the univariate case, multivariate nonnegative polynomials are not necessarily sum-of-squares. However, positive trigonometric polynomials are sum-of-squares, but the degrees of the sum-of-squares factors may be arbitrarily high, at least theoretically. To benefit from the SDP computation machinery, we must relax the framework from nonnegative polynomials to sum-of-squares polynomials (whose factors have bounded degree). The principle of sum-of-squares relaxations, presented in Sect. 3.5, is central to the understanding of this chapter. It resides in the idea that (many interesting) optimization problems with nonnegative polynomials can be approximated with a sequence of problems with sum-of-squares, implemented via SDP. Larger the order of the sum-of-squares, better the approximation, but higher the complexity. This chapter is rather long, so here is an outline of its content. The first three sections present some important properties of nonnegative and sum-of-squares multivariate polynomials. The Gram matrix (or generalized trace) parameterization of sum-of-squares trigonometric polynomials is introduced in Sect. 3.4. After discussing sum-of-squares relaxations in Sect. 3.5, dealing with sparse polynomials is considered in Sect. 3.6. The similar notions for real polynomials are presented in Sect. 3.7. The connections between pairs of relaxations for trigonometric and real polynomials are investigated in Sect. 3.8. The Gram pair parameterization of sum-of-squares trigonometric polynomials is examined in Sect. 3.9; similarly to the univariate case, as discussed in Sect. 2.8.3, the Gram-pair matrices have half the size of the Gram matrix. Finally, in Sect. 3.10, the previous results are generalized for polynomials with matrix coefficients.

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  1. Department of Automatic Control and Computers, University Politehnica of Bucharest, Bucharest, Romania

    Bogdan Dumitrescu

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Dumitrescu, B. (2017). Multivariate Polynomials. In: Positive Trigonometric Polynomials and Signal Processing Applications. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-53688-0_3

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  • DOI: https://doi.org/10.1007/978-3-319-53688-0_3

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-53688-0

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