Volumes of certain loci of polynomials and their applicatoins

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Title: Volumes of certain loci of polynomials and their applicatoins
Author: Sethuraman, Swaminathan
Abstract: To prove that a polynomial is nonnegative on Rn , one can try to show that it is a sum of squares of polynomials (SOS ) . The latter problem is now known to be reducible to a semi -definite programming (SDP ) computation that is much faster than classical algebraic methods , thus enabling new speed -ups in algebraic optimization . However , exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem . Blekherman was recently able to show that for degree k polynomials in n variables with k = 4 fixed those that are SOS occupy a vanishingly small fraction of those that are nonnegative on Rn , as n - > 1 . With an eye toward the case of small n , we refine Blekherman'[s bounds by incorporating the underlying Newton polytope , simultaneously sharpening some of his older bounds along the way . Our refined asymptotics show that certain Newton polytopes may lead to families of polynomials where efficient SDP can still be used for most inputs .
URI: http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2009 -05 -787
Date: 2010-01-16

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Volumes of certain loci of polynomials and their applicatoins. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2009 -05 -787 .

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