Signature pairs of positive polynomials

Abstract

A well-known theorem of Quillen says that if $r(z,\bar{z})$ is a bihomogeneous polynomial on ${\mathbb{C}}^n$ positive on the sphere, then there exists $d$ such that $r(z,\bar{z}){\Vert z\Vert }^{2d}$ is a squared norm. We obtain effective bounds relating this $d$ to the signature of $r$. We obtain the sharp bound for $d=1$, and for $d>1$ we obtain a bound that is of the correct order as a function of $d$ for fixed $n$. The current work adds to an extensive literature on positivity classes for real polynomials. The classes ${\Psi }_d$ of polynomials for which $r(z,\bar{z}){\Vert z\Vert }^{2d}$ is a squared norm interpolate between polynomials positive on the sphere and those that are Hermitian sums of squares.

17 pages, 4 figures; fixed typos; accepted to Bulletin of the Institute of Mathematics, Academia Sinica, New Series

References in corpus (1)

• Pfister's theorem fails in the Hermitian case

Cited by in corpus (1)

• Sum of Squares Conjecture: the Monomial Case in ${\mathbb{C}}^3$