Nonnegative Polynomials and Sums of Squares

Details Nonnegative Polynomials and Sums of Squares Nonnegative Polynomials and Sums of Squares

2010-10-18

arxiv.org/abs/1010.3465v2

Mathematics

Algebraic Geometry

The paper was substantially revised to improve the presentation. The results remain essentially the same, but the presentation

Scientific paper

In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree 2d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares

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