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Signature Pairs of Positive Polynomials
by Jennifer Halfpap   Jiri Lebl

Vol. 8 No. 2 (2013) P.169~P.192

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 $r(z,\bar{z}){\lVert {z} \rVert}^{2d}$ 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 $r(z,\bar{z}){\lVert {z} \rVert}^{2d}$ is a squared norm interpolate between polynomials positive on the sphere and those that are Hermitian sums of squares.

KEYWORDS
Hermitian sums of squares, Hilbert's 17th problem, positivity classes, Hermitian symmetric polynomials

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 12D15, 14P10, 15A63, 32H99

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