Sums of squares representations on singular loci

Ngoc Hoang Anh Mai; Victor Magron

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arXiv · 2303.05081

Sums of squares representations on singular loci

Ngoc Hoang Anh Mai·Victor Magron

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Abstract

The problem of characterizing a real polynomial $f$ as a sum of squares of polynomials on a real algebraic variety $V$ dates back to the pioneering work of Hilbert in [Mathematische Annalen 32.3 (1888): 342-350]. In this paper, we investigate this problem with a focus on cases where the real zeros of $f$ on $V$ are singular points of $V$. By using optimality conditions and irreducible decomposition, we provide a positive answer to the following essential question of polynomial optimization: Are there always exact semidefinite programs to compute the minimum value attained by a given polynomial over a given real algebraic variety? Our answer implies that Lasserre's hierarchy, which is known as a bridge between convex and non-convex programs with algebraic structures, has finite convergence not only in the generic case but also in the general case. As a result, we constructively prove that each hyperbolic program is equivalent to a semidefinite program.

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Ngoc Hoang Anh Mai, Victor Magron. 2023-03-09. Sums of squares representations on singular loci. https://arxiv.org/abs/2303.05081

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