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

A quantitative version of Catlin-D’Angelo–Quillen theorem 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

, Volume 3 (1) – Jul 1, 2012
19 pages

/lp/springer-journals/a-quantitative-version-of-catlin-d-angelo-quillen-theorem-A99hIwJKdj
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-012-0035-4
Publisher site
See Article on Publisher Site

### 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jul 1, 2012