Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

Abstract

A solution to the effectiveness problem in Kohn's algorithm for generating subelliptic multipliers is provided for domains that include those given by sums of squares of holomorphic functions (also including infinite sums). These domains are of particular interest due to their relation with complex and algebraic geometry and in particular, seem to include all previously known cases. Furthermore, combined with a recent result of M. Fassina, our effectiveness method allows establishing effective subelliptic estimates for more general classes of domains. Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as $Q\circ\Gamma$, where $\Gamma$ is constructed from pre-multipliers and $Q$ is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic-geometric techniques by Y.-T. Siu and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo. The proposed procedures are purely algebraic and as such can also be of interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.