Lower bounds for a polynomial in terms of its coefficients
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 \(f_*=\inf\{f(a):a\in \mathbb{R}^n\}\), since \(f_* \geq f_{\text{sos}}.\) (Estimates of this form lie at the heart of efficient approximations to minima of polynomials by means of semidefinite programming.) These theorems assume \(f_{2d}\) in the interior of the cone of sos forms of degree \(2d.\) In particular a theorem by \textit{M. Marshall} [Can. J. Math. 61, No. 1, 205--221 (2009; Zbl 1163.13019)] showing in these case the existence of \(f_{\text{sos}}\neq -\infty\) is quantified. \quad Section 4 illustrates via nice examples that depending on the polynomial \(f,\) any of the bounds \(r_{\text{L}}, r_{\text{FK}}\) and \(r_{\text{dmt}}\) can occur as the best lower bound for \(f_{\text{sos}}\).
scientific article; zbMATH DE number 5814270
scientific article