2013 / June Volume 8 No.2
Signature Pairs of Positive Polynomials
Published Date |
2013 / June
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Title | Signature Pairs of Positive Polynomials |
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Pagination | 169-192 |
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.
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AMS Subject Classification |
12D15, 14P10, 15A63, 32H99
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Received |
2013-04-19
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Accepted |
2013-04-20
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