Hermitian analogues of Hilbert's 17-th problem
type: Item
instance of: scholarly article
title: Hermitian analogues of Hilbert's 17-th problem
zbMATH Open document ID: 1218.32001
author: John P. D'Angelo
published in: Advances in Mathematics
publication date: 2011-03-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.2479
review text: Hilbert's 17th problem, solved affirmatively by Artin in 1927, says that if \(p\) is a real polynomial in several variables which is nonnegative on \({\mathbb{R}}^n\), then there exists a polynomial \(q\) such that \(q^2 p\) is a sum of squares of polynomials. In the present paper the author surveys results and applications obtained during the last 15 years largely by him and with coauthors (Catlin, Putinar, Varolin, Lebl) of hermitian analogues to this result. See the Zentralblatt reviews we cite below. Consider a polynomial \(r(z,\bar{w})=\sum_{\alpha,\beta} c_{\alpha,\beta} z^\alpha \bar{w}^\beta,\) where \(z=(z_1,z_2,\ldots, z_n)\) and \(w=(w_1,w_2,\ldots, w_n)\) are in \(\mathbb{C}^n\). Then \(r\) is called hermitian symmetric (h.s.) if \(r(z,\bar{w})=\overline{r(w,\bar{z})}\) or, equivalently, if the matrix \(C=(c_{\alpha,\beta})\) is hermitian or, equivalently, if \(r(z,\bar{z})\) is real for every \(z\in \mathbb{C}^n.\) A hermitian symmetric polynomial is nonnegative if \(r(z,\bar{z})\geq 0.\) In analogy with quadratic forms the formulation with hermitian matrices suggests to introduce signature pairs, see [\textit{D. Grundmeier}, Int. J. Math. 22, No. 3, 311--343 (2011; Zbl 1218.32017)] and [\textit{J. P. D'Angelo} and \textit{J. Lebl}, J. Geom. Anal. 21, No. 3, 599--619 (2011; Zbl 1228.32037)]: A h.s. polynomial \(r:\mathbb{C}^n\times \mathbb{C}^n \rightarrow \mathbb{C}\) has the signature pair \({\mathbf s}(r)=(A,B)\) if and only if one can write \(r(z,\bar{z})=\sum_{j=1}^A |f_j(z)|^2- \sum_{j=1}^B |g_j(z)|^2 =\| f(z)\| ^2-\| g(z)\| ^2,\) with linearly independent holomorphic polynomials \(f_1,\dots ,f_A, g_1,\dots ,g_B\) or, equivalently, if and only if the underlying hermitian matrix has \(A\) positive and \(B\) negative eigenvalues. One also defines the rank of \(r\) by \({\mathbf r}(r)=A+B.\) The author introduces the following sets: \(\mathcal{P}_1=\mathcal{P}_1(n)\) -- the set of all nonnegative h.s. polynomials in \(n\) variables; more generally \(…
reviewed by: Alexander Kovačec
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32V15
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 32Q40
zbMATH DE Number: 5865630
zbMATH Keywords: Hermitian symmetric function
zbMATH Keywords: quotients of squared norms
zbMATH Keywords: positivity conditions
zbMATH Keywords: pullback property
zbMATH Keywords: Bergman kernel
zbMATH Keywords: signature pairs
zbMATH Keywords: Cauchy-Riemann (CR) geometry
zbMATH Keywords: proper holomorphic mappings
zbMATH Keywords: holomorphic line bundle
zbMATH Keywords: isometric embedding
zbMATH Keywords: collapsing rank
MaRDI profile type: Publication
OpenAlex ID: W2031274894
arXiv ID: 1012.2479
cites work: Q4537717
cites work: Isometric imbedding of complex manifolds
cites work: A stabilization theorem for Hermitian forms and applications to holomorphic mappings
cites work: Positivity conditions for bihomogeneous polynomials
cites work: An isometric imbedding theorem for holomorphic bundles
cites work: Boundary behavior of rational proper maps
cites work: Several Complex Variables and the Geometry of Real Hypersurfaces
cites work: Q4531080
cites work: PROPER HOLOMORPHIC MAPPINGS, POSITIVITY CONDITIONS, AND ISOMETRIC IMBEDDING
cites work: COMPLEX VARIABLES ANALOGUES OF HILBERT'S SEVENTEENTH PROBLEM
cites work: Q5485830
cites work: Hermitian symmetric polynomials and CR complexity
cites work: Positivity conditions for Hermitian symmetric functions
cites work: Extending proper holomorphic mappings of positive codimension
cites work: SIGNATURE PAIRS FOR GROUP-INVARIANT HERMITIAN POLYNOMIALS
cites work: Q4758141
cites work: ``Positive'' noncommutative polynomials are sums of squares.
cites work: Q3645522
cites work: IMBEDDING CAUCHY-RIEMANN MANIFOLDS INTO A SPHERE
cites work: Imbedding Strictly Pseudo Convex Domains Into a Ball
cites work: Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls
cites work: Q2716044
cites work: Zur Darstellung definiter Funktionen als Summe von Quadraten. (To the representation of definite functions as the sum of squares)
cites work: Sums of Hermite squares on pseudoconvex boundaries
cites work: On the representation of hermitian forms as sums of squares
cites work: Q4496287
cites work: Uniform denominators in Hilbert's seventeenth problem
cites work: The factorization problem for nonnegative operator valued functions
cites work: Q3601991
cites work: Lectures on the \(L^2\)-Sobolev theory of the \(\bar\partial\)-Neumann problem
cites work: EFFECTIVE ISOMETRIC EMBEDDINGS FOR CERTAIN HERMITIAN HOLOMORPHIC LINE BUNDLES
cites work: Geometry of Hermitian algebraic functions. Quotients of squared norms
DOI: 10.1016/J.AIM.2010.12.013
description: scientific article