Lower bounds for a polynomial in terms of its coefficients
type: Item
instance of: scholarly article
title: Lower bounds for a polynomial in terms of its coefficients
zbMATH Open document ID: 1205.12001
author: M. Ghasemi
author: Murray Marshall
published in: Archiv der Mathematik
publication date: 2010-11-10T00:00:00Z
review text: The authors give sufficient conditions in terms of the coefficients of polynomials in \(\mathbb{R}[x]=\mathbb{R}[x_1,\dots,x_n]\) in order that these are sums of squares of polynomials (sos), strengthening results of \textit{C. Fidalgo} and \textit{A. Kovacec} [``Positive semidefinite diagonal minus tail forms are sums of squares'', Math. Z. 269, No. 3-4, 629--645 (2011; Zbl 1271.11045)] and \textit{J. B. Lasserre} [Arch. Math. 89, No. 5, 390--398 (2007; Zbl 1149.11018)]. They also determine a lower bound for any polynomial \(f\) whose highest degree term is positive definite. For \(f=\sum_\alpha f_\alpha x^\alpha \in \mathbb{R}[x]\) let \(\Omega(f)=\{\alpha \in \mathbb{Z}_{\geq 0}^n: f_\alpha \neq 0, \text{ and } \alpha\not \in \{0, 2de_1,\dots,2de_n\}\}\) (the \(e_i\) are the standard vectors in \(\mathbb{R}^n\)) and \(\Delta=\Delta(f)=\{\alpha\in \Omega(f): f_\alpha <0 \text{ or } \alpha\not \in (2\mathbb{Z})^n \}.\) Two of the more easily stated main results are theorems 2.1 and 2.3: \quad Suppose \(f\in \mathbb{R}[x]\) is a form of degree \(2d\) and assume that at least one of the following conditions i or ii holds: {i.} For all \(i=1,\dots,n,\) \(f_{2de_i} \geq \sum_{\alpha\in \Delta} |f_\alpha| \frac{\alpha_i}{2d};\) {ii.} \(\min_{i=1,\dots,n} f_{2de_i} \geq \frac{1}{2d} \sum_{\alpha\in \Delta} |f_\alpha| (\alpha^\alpha)^{\frac{1}{2d}};\) then \(f\) is a sum of (binomial) squares. The proofs use theorem 2.3 of Fidalgo and Kovacec [loc. cit.]. From this result for forms similar results with somewhat more complicated formulae for arbitrary polynomials of degree \(2d\) can be derived. In section 3, given a polynomial \(f\) of degree \(2d\), and letting \(f_{2d}\) be its homogeneous part of highest degree (\(2d\)), lower bounds \(r_{\text{L}},\) \(r_{\text{FK}}\) and \(r_{\text{dmt}}\) for \(f_{\text{sos}}=\sup\{r: f-r \text{ is sos}\}\) are obtained; depending on which of the cited author's criteria are used. Hereby one gets lower bounds for \(f_*=\inf\{f(a):a\i…
reviewed by: Alexander Kovačec
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 90C22
zbMATH DE Number: 5814270
zbMATH Keywords: positive polynomials
zbMATH Keywords: sums of squares
zbMATH Keywords: optimization
MaRDI profile type: Publication
full work available at URL: s00013-010-0179-0
OpenAlex ID: W2105100124
cites work: Bounds for the Zeros of Polynomials
cites work: Positive semidefinite diagonal minus tail forms are sums of squares
cites work: Q5585021
cites work: Global optimization with polynomials and the problem of moments
cites work: Sufficient conditions for a real polynomial to be a sum of squares
cites work: Q5452017
cites work: Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization
cites work: Q4428719
DOI: 10.1007/S00013-010-0179-0
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description: scientific article; zbMATH DE number 5814270