Plurisubharmonic Polynomials and Bumping

We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain Ω ⊂ C in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with ∂Ω, at the site of the bumping, are explicitly realised. Generally, when Ω ⊂ C, n ≥ 3, the known methods lead to bumpings with high orders of contact — which are not explicitly known either — at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C. These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity.