Coercive polynomials: stability, order of growth, and Newton polytopes. (2nd January 2019)

• Record Type: Journal Article
• Authors: Bajbar, Tomáš; Stein, Oliver
• Is Part Of: Optimization. Volume 68:Number 1(2019)
• Page Start: 99
• Page End: 124
• Publication Date: 2019-01-02
• Subjects: Newton polytope -- coercivity -- stability -- order of growth -- error bound -- convenient polynomials; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1426585
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: In this article we introduce a stability concept for the coercivity of multivariate polynomialsf ∈ R [ x ] . In particular, we consider perturbations of f by polynomials up to the so-called degree of stable coercivity, and we analyze this stability concept in terms of the corresponding Newton polytopes at infinity. For coercive polynomialsf ∈ R [ x ] we also introduce the order of coercivity as a measure expressing the order of growth of f, and we identify a broad class of multivariate polynomialsf ∈ R [ x ] for which the order of coercivity and the degree of stable coercivity coincide. For these polynomials we give a geometric interpretation of this phenomenon in terms of their Newton polytopes at infinity, which we call the degree of convenience. We relate our results to the existing literature and we illustrate them with some examples. As applications we show that the gradient maps corresponding to a broad class of polynomials are always subjective, we establish Hölder type global error bounds for such polynomials, and we link our results to the existence of solutions in the calculus of variations.