There are Significantly More Nonnegative Polynomials than Sums of Squares

Details There are Significantly More Nonnegative Polynomials than Sums of Squares There are Significantly More Nonnegative Polynomials than Sums of Squares

2003-09-07

arxiv.org/abs/math/0309130v1

Mathematics

Algebraic Geometry

11 pages, Latex

Scientific paper

We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for the volumes of the bases. If the degree is greater than 2 then we show that the ratio of the volumes of the bases, raised to the power reciprocal to the ambient dimension, tends to 0 as the number of variables tends to infinity.

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