Several complex variables and geometry
I authored the book Several Complex Variables and the Geometry of Real Hypersurfaces; this book describes areas closely related to my research interests. In recent years I have been interested in positivity conditions for real-analytic real-valued functions of several complex variables. For example, David Catlin and I gave the following necessary and sufficient condition for a bihomogeneous polynomial $p$ of $n$ complex variables to be positive away from the origin. There is an integer $d$ so that $p$ is the quotient of squared norms of homogeneous holomorphic polynomial mappings, the numerator vanishes only at the origin, and the denominator is the d-th power of the Euclidean norm. We have extended this result to an isometric embedding theorem for holomorphic bundles. See Math Research Letters 6 (1999).
In studying group invariant proper holomorphic mappings between balls in different dimensions I discovered a triangle of integers bearing a neat relationship to Pascal's triangle. Here are the first few rows! Can you figure out the rest?
1 1
1 3 1
1 5 5 1
1 7 14 7 1
1 9 27 30 9 1
1 11 44 77 55 11 1
1 13 65 156 182 91 13 1
1 15 90 275 450 378 140 15 1
Hint: Solve the recurrence defined as follows:
p_0 = x
p_1 = x^3 + 3 xy
p_{n+2} = (x^2 + 2y)p_{n+1} - y^2 p_n
I have also discovered generalizations of this triangle where the coefficients are sometimes negative but the coefficients have remarkable combinatorial and number theoretic properties.
My current interests include complex variables analogues of Hilbert's seventeenth problem; when is a nonnegative real-analytic function a squared norm of a holomorphic mapping? More generally when is it the quotient of squared norms? I am also interested in the interpretation of such questions in terms of metrics on bundles.
I have recently published a book called Inequalities from Complex Analysis. It is in the Carus monograph series. This book considers questions such as this: Given a polynomial on complex Euclidean space that is positive on some set, does it agree with the quotient of squared norms of holomorphic polynomials on that set? The book begins with the definition of the complex numbers, and ends with recent research.
Here is a typical result in the book: If a polynomial function is positive on the unit sphere in complex Euclidean space, then it agrees with a squared norm of a holomorphic mapping there.