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 Kolář’s algorithm for the computation of the multitype introduced in Kolář (Int Math Res Not (IMRN) 18:3530–3548, 2010). Given a sum of squares domain, we rewrite the Kolář algorithm in terms of ideals of holomorphic functions and also introduce an approach that explicitly constructs the homogeneous polynomial transformations used in the algorithm.
Highlights
Domains defined by sums of squares of holomorphic functions constitute a very important class in the field of several complex variables as they connect in a very natural way complex analysis with algebraic geometry
We provide an answer to this question by showing that the multitype of a sum of squares domain can be computed from the related ideal of holomorphic functions via the restatement of the Kolár algorithm at the level of ideals
For the correspondence between polynomial changes of variables and elementary row and column operations on the Levi matrix associated to the domain, we introduce a variant of the notion of dependence compared to the standard one in linear algebra as follows: Let APj be the Levi matrix of the leading polynomial Pj at step j of the Kolár algorithm
Summary
Domains defined by sums of squares of holomorphic functions constitute a very important class in the field of several complex variables as they connect in a very natural way complex analysis with algebraic geometry. There exists an allowable polynomial transformation on Pj with respect to the variable zk if and only if the kth row of APj is dependent Using this characterization of the polynomial transformations and the restatement of the Kolár algorithm in terms of ideals of holomorphic functions, the row reduction algorithm connects nicely the notion of simplifying the Jacobian module associated to a sum of squares domain with elementary row operations on the complex Jacobian matrix of the same domain. By employing this algorithm at every step of the Koláralgorithm for the computation of the multitype, we are able to construct the weighted homogeneous polynomial transformations needed in the Koláralgorithm. This article constitutes part of the author’s Ph.D thesis at Trinity College Dublin under the supervision of Andreea Nicoara
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