So far, most of our real-valued functions on Cn have been Hermitian forms such as 〈Lz, z〉. These are linear in z and in z. When L is nonnegative definite, 〈Lz, z〉 = ∥√L (z) ∥2; the squared norm makes it evident why the values are nonnegative. In this chapter we consider more general real-valued functions of several complex variables, and we apply some of the ideas from Hermitian linear algebra in this nonlinear setting. Things become more interesting, even for real-valued polynomials.

Section 1 helps motivate the rest of the Chapter by considering positivity conditions for polynomials in one and several real variables. Section 2 returns us to material on real-valued functions of several complex variables. The rest of the Chapter studies reality and positivity conditions for polynomials.

Real variables analogues

The main goal in this book is Theorem VII.1.1. This result has a much simpler analogue in the setting of several real variables. This simpler result was proved by Pólya around 1925, in conjunction with Artin's solution of Hilbert's seventeenth problem. Pólya's result is interesting even in one variable, where it had been noted earlier by Poincaré. This section provides a self-contained discussion of the one-variable result. Pólya's Theorem (in several variables) will appear as Corollary VII.1.6. See [R] and [HLP] for related results and references; see [HLP] for an elementary proof of Pólya's Theorem.

First consider polynomials in one real variable x with real coefficients, and write R[x] to denote the set of such polynomials. Our emphasis on inequalities suggests two problems:

Problem VI.1.1. Characterize those p ∈ R[x] such that p(x) > 0 for all nonnegative x.