A quantitative version of the Catlin-D'Angelo-Quillen theorem

TL;DR

This paper provides a quantitative enhancement of the Catlin-D'Angelo-Quillen theorem, establishing an explicit upper bound on the power needed to express positive bi-homogeneous forms as sums of squares, depending on dimension, degree, and form properties.

Contribution

It introduces a concrete upper bound on the minimal power required in the sum of squares representation for positive bi-homogeneous forms, refining the original qualitative theorem.

Findings

Derived an explicit bound C_f (n+m)^3 log(n)^3 for the power needed

Bound depends on the form's degree, dimension, and minimum value on the sphere

Provides a quantitative criterion for sum of squares decomposition

Abstract

A theorem proved by Quillen and by Catlin and D'Angelo states that a bi-homogeneous form on a multidimensional complex space which is positive away from zero can be written as a sum of squares of absolute values of polynomials once it is multiplied by the norm raised to a sufficiently high even power. In this note we provide a quantitative version of this theorem by giving an upper bound on the minimal power. This bound is roughly C_f (n+m)^3 log(n)^3, where n is the dimension and m the degree of the form, and C_f is a multiplicative constant depending only on f, inversely proportional to the minimum of f on the sphere.