On the Catlin multitype of sums of squares domains

TL;DR

This paper demonstrates that for sum of squares domains with finite D'Angelo type, the polynomial model derived from the Catlin multitype retains the sum of squares structure and is invariant under the defining ideal, using Kolar's algorithm.

Contribution

It establishes the invariance of the multitype for such domains and reformulates Kolar's algorithm in terms of ideals, providing explicit polynomial transformations.

Findings

Polynomial model of sum of squares domains is also a sum of squares domain.

Multitype is an invariant of the ideal of holomorphic functions defining the domain.

Reformulation of Kolar's algorithm in terms of ideals and explicit polynomial transformations.

Abstract

For a sum of squares domain of finite D'Angelo 1-type at the origin, we show that the polynomial model obtained from the computation of the Catlin multitype at the origin of such a domain is likewise a sum of squares domain. We also prove, under the same finite type assumption, that the multitype is an invariant of the ideal of holomorphic functions defining the domain. Both results are proven using Martin Kolar's algorithm for the computation of the multitype introduced in [13]. Given a sum of squares domain, we rewrite the Kolar algorithm in terms of ideals of holomorphic functions and also introduce an approach that explicitly constructs the homogeneous polynomial transformations used in the algorithm.