Hermitian analogues of Hilbert's 17-th problem
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 \(\mathcal{P}_k\) -- the set of all polynomials \(r\) such that for all selections of \(k\) points \(z_i\), the matrix \(r(z_i,z_j)\) is positive semidefinite; \(\mathcal{P}_\infty\) -- the set of all squared norms, i.e., polynomials so that \(r(z,\bar{z})= \| F(z)\| ^2= \sum_{j=1}^m |f_j(z)|^2\), where the \(f_j(z)\) are holomorphic in several complex variables; \(\mathcal{Q}\) -- the set of all polynomials that are quotients of squared norms, i.e. that are of the form \(r=\frac{\| F \| ^2}{\| G\| ^2}\) or, equivalently, for which \(\| G\| ^2 r=\| F\| ^2\); \(\mathcal{Q'}\) -- the set of all \(r\in \mathcal{P}_1\) so that for some \(s \in \mathcal{P}_1\), there holds a relation \(rs=\| F\|^2\); \(\text{rad}(\mathcal{P}_\infty)\) -- the set of polynomials \(r\) so that for some \(N,\) \(r^N \in \mathcal{P}_\infty\); \(\mathcal{L}\) -- the set of all \(r\geq 0\) so that \(\log(r)\) is plurisubharmonic. (Compare also [\textit{J. P. D'Angelo}, Contemp. Math. 395, 47--58 (2006; Zbl 1098.12001); \textit{J. P. D'Angelo} and \textit{D. Varolin}, Asian J. Math. 8, No. 2, 215--232 (2004; Zbl 1116.32001)].) Inspired by the 17th problem, the author asks for tractable necessary and sufficient conditions that guarantee for a polynomial \(r\) to lie in \(\mathcal{Q},\) in \(\mathcal{Q}',\) and in \(\text{rad}(P_\infty),\) respectively. In Section 2.1, the reader learns of results that hint at that there may be no nice results of the Hilbert-Artin type for as to when a h.s. polynomial is nonnegative. For example, \(\mathcal{P}_1 \not \subseteq \mathcal{Q'},\) and \textit{A. Pfister}'s famous estimates [Invent. Math. 4, 229--237 (1967; Zbl 0222.10022)] for the numbers of sums of squares have no nice analogues. Section 2 justifies the notation \(\mathcal{P}_\infty\) referring to D'Angelo and Varolin [loc. cit.] to verify that this set is the intersection of the (decreasing) sequence of sets \(\mathcal{P}_j,\) \(j=1,2,3,\dots\), and to look up more about the stability index. Lemma 2.1 records a simple but later important result that allows to reduce certain questions involving dimensions \(\geq 2\) to dimension 1. Closure properties for \(\mathcal{Q},\mathcal{Q'},\mathcal{P}_j\) under sums, products and limits are stated. A h.s. polynomial \(r\) is bihomogeneous of total degree \(2m\) if for all \( \lambda \in \mathcal{C}, r(\lambda w,\bar{\lambda}\bar{w})= |\lambda|^{2m}r(w,\bar{w}).\) \textit{J. P. D'Angelo}'s book [Inequalities from complex analysis. The Carus Mathematical Monographs. 28. Washington, DC: The Mathematical Association of America (MAA) (2002; Zbl 0996.30001)] seems to contain relevant background material for this and the next section. Section 3 gives as one of the main results (Theorem 3.1) for a bihomogeneous h.s. polynomial of total degree \(2m\) four conditions all equivalent to that \(r(z,\bar{z})\) assumes a positive minimum value on the sphere. One of these is that there is an integer \(d\) so that the hermitian matrix underlying \(\| z \| ^{2d} r(z,\bar{z})\) is positive definite. Other conditions involve positivity of operators. Unless familiar with the work [Math. Res. Lett. 3, No. 2, 149--166 (1996; Zbl 0858.32010)] by \textit{D. W. Catlin} and \textit{J. P. D'Angelo}, for real algebraic geometers it will come as a surprise that Pólya's theorem concerning real polynomials positive on the hyperplane \(\sum x_i=1\) is a mere corollary to Theorem 3.1. Parts of the theorem where known already to \textit{D. G. Quillen} [Invent. Math. 5, 237--242 (1968; Zbl 0198.35205)] but are given a new proof by Catlin and D'Angelo using the Bergman kernel function. As a consequence of this theorem, one gets Theorem 3.3: If \(r(z,\bar{z})\) is positive on \(S^{2n-1}\), then it agrees with a squared norm there. Section 4 improves on [\textit{J. P. D'Angelo}, loc. cit. (2006)] and gives in Theorem 4.1 for a nonnegative one variable h.s. polynomial \(r\) five equivalent conditions to be a quotient of squared norms. One of these conditions is that \(r\) is in \(\mathcal{Q}'\). In other words, \(\mathcal{Q}'(1)=\mathcal{Q}(1)\). In Section 5, a version of \textit{D. Varolin}'s theorem [Am. J. Math. 130, No. 2, 291--315 (2008; Zbl 1146.32008)] is recorded: assume \(r=\| f\| ^2-\| g\| ^2\) is bihomogeneous with linearly independent components of \(f\) and \(g.\) Then \(r\in \mathcal{Q}\) if and only if there is a \(\lambda <1\) so that \(\| g\| ^2 \leq \lambda \| f\| ^2.\) This theorem and the mentioned Theorem 4.1 and Lemma 2.1 are used to show for any \(n\) the following fundamental fact that the author considers the main new result (Theorem 5.3): A h.s. polynomial \(r\not \equiv 0\) is a quotient of squared norms if and only if it divides a squared norm. In other words: for all \(n,\) \(\mathcal{Q}(n)=\mathcal{Q}'(n).\) For Section 6, see also [\textit{J. P. D'Angelo}, J. Korean Math. Soc. 40, No. 3, 341--371 (2003; Zbl 1044.32010)] and [\textit{F. Forstnerič}, Invent. Math. 95, No. 1, 31--62 (1989; Zbl 0633.32017)]. Consider \(\Omega \subset\mathbb{C}^n\) and \(\Omega' \subset\mathbb{C}^N.\) A holomorphic \(f: \Omega \rightarrow \Omega'\) is proper if \(K\subset \Omega'\) compact implies compact \(f^{-1}(K)\subset \Omega.\) After recording a number of facts for proper holomorphic maps between unit balls \(B_n\) and \(B_N,\) the main theorem of this section shows that if a holomorphic polynomial \(q\in \mathbb{C}^n \rightarrow \mathbb{C}\) does not vanish on the closed unit ball, then there is an integer \(N\) and holomorphic \(p:\mathbb{C}^n \rightarrow \mathbb{C}^N\) so that \(p/q\) is rational proper from \(B_n\) to \(B_N\) and reduced to lowest terms. This and the next two sections seem those most closely related to CR-geometry. Section 7 notes and partially proves the containments \(\text{int}(\mathcal{P}_2)\subset \text{rad}(\mathcal{P}_\infty)\subset \mathcal{P}_2 \cap \mathcal{Q}'\subset \mathcal{P}_2 \subset \mathcal{L} \subset \mathcal{P}_1\), of which the relations with \(\mathcal{P}_2,\) a closed cone in real Euclidean space, is the most striking one. Efforts are made to develop a feeling for the Cauchy-Schwarz inequality \(r(z,\bar{z}) r(w,\bar{w}) \geq |r(z,\bar{w})|^2\) which, if it holds, guarantees \(|r|\in \mathcal{P}_1\) and can be viewed as a curvature condition (see Varolin [loc. cit.]). Section 8, entitled `Isometric embedding for holomorphic bundles', restates Theorem 3.1 in the language of universal line bundles over complex projective space \(\mathbb{P}_n\). In particular, Theorem 8.2 states a result concerning isometric embedding of holomorphic bundles. Finally, Section 9 gives examples of the collapsing rank phenomenon: there are h.s. bihomogeneous polynomials \(q,r\) of total degree \(2^m\) and signatures \textbf{s}\((q)=(2^{m-1}+1,0)\) and \({\mathbf s}(r)=(2^{m-2}+1,2^{m-2})\) but \({\mathbf s}(qr)=(2,0).\) The many examples and relatively short proofs should make for convenient reading of this paper as an introduction to the topics it discusses.
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