Ullemar's formula for the jacobian of the complex moment mapping

Abstract

The complex moment sequence μ ( P ) is assigned to a univalent polynomial P ( z ) by the Cauchy transform of the domain , where is the unit disk. We establish the representation of the Jacobian in terms of roots of the derivative . Combining this result with the special decomposition for the Hurwitz determinants, we prove a formula for , which was previously conjectured by Ullemar. As a consequence, we show that the boundary of the class of all locally univalent polynomials in is contained in the union of three irreducible algebraic surfaces.

Keywords:

• Complex moments
• Locally univalent polynomials
• Toeplitz determinants
• 2000 Mathematics Subject Classification: Primary 30E20, 30C20; Secondary 11C20, 46G25

Acknowledgments

The authors wish to thank Björn Gustafsson for bringing their attention to the present theme and for fruitful discussions. We are grateful to Harold Shapiro for his helpful comments and the referee for careful reading and suggestions which led to an improvement of the article. This article is supported by the Göran Gustafsson Foundation and the Russian Foundation for Basic Research, Grant 03-01-00304.