Real Algebraic and Analytic Geometry
Submission: 2009, December 30.
Recently Lasserre gave sufficient conditions in terms of the coefficients for a polynomial f of degree 2d (d >= 1) in n variables to be a sum of squares of polynomials. Exploiting this result, we are able to determine, for any polynomial f of degree 2d whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f-r is a sum of squares of polynomials. The existence of such a number r was proved earlier by Marshall, but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite. Similar arguments are applied to results of Fidalgo and Kovacec, to determine other numbers r such that f-r is a sum of squares and other lower bounds for f.
Mathematics Subject Classification (2000): 12D99 14P99 90C22.
Keywords and Phrases: Positive polynomials, sums of squares, optimization.
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