When does a polynomial p(x) of degree n with real coefficients assume positive values whenever the variable x is positive? An easy sufficient condition is that all of the n+1 coefficients be positive. But this condition is clearly not necessary, as shown by the quadratic p(x) = x2 - 2x + 2. A simple but nonobvious necessary and sufficient condition was known to Poincaré, namely, that p(x) be expressible as a quotient of two polynomials, each having all coefficients positive (no missing terms, i.e., terms below the leading term with coefficient 0, are allowed in either polynomial). There is a beautiful classical analytic proof of this result given at the beginning of Chapter 6 in d'Angelo's elegant new Carus Monograph. It involves uniform convergence in a novel way and is a fine example of the usefulness of that important concept for a basic analysis course.
Pólya generalized Poincaré's result to the case of homogeneous polynomials in several real variables in 1925. A related issue is Hilbert's Seventeenth problem, which asks whether a nonnegative polynomial in n real variables can be written as a sum of squares of rational functions. Artin showed that the answer is yes in 1926, but his solution was nonconstructive. The present book discusses the circle of ideas involved in finding appropriate generalizations of these theorems for polynomials in several complex variables. These ideas include complex linear algebra, functional analysis, and some bits of several complex variables. The goal of the book, which is reached in the last two chapters, is to find conditions under which certain kinds of complex polynomials are positive. These conditions and the proofs establishing their connections with positivity are definitely nontrivial and are the result of recent work by the author and David Catlin. Consequently, this volume is one of the more advanced books in the Carus Monograph series.
The first five chapters of this book should be accessible to a good senior undergraduate with some training in basic real and complex analysis. The remaining two chapters are more demanding and are written at the level of a second year graduate student. There are good exercises throughout the book, some of which anticipate ideas to be presented later on. One possible use for this book is as the text for an honors reading course for a well-trained senior. Such a student is not likely to make it all the way through the book, but he/she would be exposed to many useful ideas in complex and functional analysis at a level providing a good preparation for graduate school. The book is also a joy to read for trained mathematicians, since it brings together in a focused way many topics from modern analysis.
I shall now comment briefly on the contents of the seven chapters of this book. Chapter 1 develops the complex numbers from scratch at a leisurely pace. A nice feature is the discussion of alternative representations of C in terms of certain two-by0two real matrices and also as the quotient of the polynomial ring R[x] by the ideal generated by x2 + 1. The author proves that if two polynomials in two variables z and w agree when w = z-conjugate, then they agree for all complex values of z and w. This sort of result, which can be described as treating z and its conjugate as independent variables, is called polarization. It is an interesting and useful device, but I wish that the author had chosen to explain the terminology.
Chapter 2 discusses Hermitian inner products on complex vector spaces and introduces the basic facts about Hilbert space. Here also a concept of polarization occurs, which allows the recovery of the inner product from the induced norm. I had not seen before the connection between polarization and roots of unity (Proposition II.4.1). At the end of the chapter the concept of orthonormal system is applied to derive the generating functions for the Laguerre polynomials. Such peripheral developments, which occur in each chapter, are one of the most attractive features of this book.
Chapter 3 is devoted to the necessary basic material in several complex variables (SCV). Unfortunately, most mathematicians learn almost nothing about SCV unless they enter this active field as a research area. This theory has its own flavor, quite different from that of one complex variable, and draws ideas and techniques from analysis (e.g., partial differential equations), algebra (e.g., algebraic geometry), and topology (e.g., cohomology theory). This chapter provides a brief taste of those bits of SCV that can be approached using ideas from functional analysis. Multi-index notation is introduced, e.g., the representation of z13z22 as zk , where k = (3,2). This simple notational device makes many formulas in SCV less ugly than they would otherwise be, but it can also lead to confusion if readers are not used to it.
The goal in this chapter is introduce basic structures on the Hilbert space A2(D) consisting of all (norm)-square integrable holomorphic functions on a domain D in Cn. When D is a bounded domain, the orthogonal projection from the standard Hilbert space L2(D) onto its closed subspace A2(D) can be computed by integration against the Bergman kernel function. This kernel is a basic object in SCV and is computed explicitly in this chapter when D is the unit ball Bn. There is a pleasant detour made to study the gamma and beta functions, which are needed for some of the other formulas in this chapter.
In Chapter 4 much information about operators on Hilbert space is assembled with special attention to the finite dimensional situation. The complex analog of Strang's Fundamental Theorem of Linear Algebra (see , p. 81) is discussed, as is the theory of least squares. I found the presentation very attractive but definitely challenging. For instance, differentiation of functions with values in Banach spaces is introduced and used in the development. Special attention is devoted to positive definite transformations and to their characterization in terms of the matrix of coefficients. In particular, Theorem IV.5.9 writes a positive Hermitian form as a sum of squares, which is really the quadratic case of the theorem to be reached in the last chapter of the book. Positive definiteness is used to prove some of the classical inequalities associated with the names Hadamard, Hilbert, Herglotz, and Wirtinger.
This study continues in Chapter 5, where the spectral theory of compact Hermitian operators is developed. There is a nice aside concerning fractional integrals and derivatives and also a brief introduction to singular integral operators and pseudodifferential operators. Taken together, the first five chapters provide a readable introduction to a broad range of topics in modern real, complex, and functional analysis.
In the final two chapters use the machinery developed in the earlier part of the book to study positivity of polynomials in SCV. Chapter 6 begins with a discussion of the case of polynomials with real coefficients and is followed by basic definitions and concepts needed to state the main theorem. The context is bihomogeneous real-valued polynomials in several complex variables, i.e., polynomials p(z, z-conjugate), homogeneous of degree k in each variable with the matrix of coefficients Hermitian. Here is one place where multi-index notation can be quite confusing to a novice. A reader needs to work out some examples to make sense of a matrix Amn, where m and n are multi-indices. At the end of the chapter eight different positivity conditions are introduced and studied. It turns out that seven of them are distinct.
Chapter 7 begins with the statement of the main theorem. It provides various criteria for the positivity (when z is nonzero) of a real-valued bihomogeneous polynomial of degree 2m. The main portion of the theorem was first proven by Quillen in 1967 using quite different ideas. Pólya's theorem in the real case is a simple consequence of this result. Various applications of the main theorem are discussed in the middle portion of the chapter. Some of these are sophisticated and cannot be readily appreciated by a reader unfamiliar with ideas in SCV . The chapter concludes with the elegant proof of the main theorem using the tools which have been developed throughout the book.
In conclusion, let me say that I really enjoyed reading this book with the necessary care to prepare this review. The first five chapters are accessible to the broad mathematical community with basic training in analysis and are useful for an honors course at the senior undergraduate level. The entire book offers an attractive but demanding introduction to modern complex analysis at the graduate level.
 Gilbert Strang, Linear Algebra and its Applications, Academic Press, New York, 1976.
Jeffrey Nunemacher (firstname.lastname@example.org) is Professor of Mathematics and Computer Science at Ohio Wesleyan University. His original specialty was several complex variables, but his interests now range over analysis and algebra and their interconnections.