A quantitative version of Catlin-D’Angelo–Quillen theorem

Abstract

Let $${f (z,\bar z)}$$ be a positive bi-homogeneous hermitian form on $${\mathbb{C}^n}$$ , of degree m. A theorem proved by Quillen and rediscovered by Catlin and D’Angelo states that for N large enough, $${\langle{z,\bar z}\rangle^Nf(z,\bar z)}$$ can be written as the sum of squares of homogeneous polynomials of degree m + N. We show this works for N ≥ C f ((n + m) log n)3 where C f has a natural expression in terms of coefficients of f, inversely proportional to the minimum of f on the sphere. The proof uses a semiclassical point of view on which 1/N plays a role of the small parameter h.