Catlin's Boundary Systems for Sums of Squares Domains

For any given sum of squares domain in $\mathbb{C}^n,$ we reduce the complexity in Catlin's multitype techniques by giving a complete normalization of the geometry. Using this normalization result, we present a more elementary proof of the equality of the Catlin multitype and the commutator multitype for such domains when both invariants are finite. Finally, we reformulate algebraically Catlin's machinery for the commutator multitype computation at the origin for any given sum of squares domain in $\mathbb{C}^n$.