Nonnegative Polynomials and Sums of Squares

By Grigoriy Blekherman. 2010

108 citations recorded by the source corpus.

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

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/45c19793f83c431709db3e60820c385f1ddce274

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In Computational Theory and Mathematics

1. Mauricio Velasco
2. Gregory G. Smith
3. Grigoriy Blekherman
4. Ali Soleyman Jahan

In Numerical Analysis

1. Mauricio Velasco
2. Gregory G. Smith
3. Grigoriy Blekherman

Citations in the corpus

• Sums of squares and varieties of minimal degree (2013, background): “ j=1 p κjp˜ ∗ j 2 − e ∑ j=1 λjp˜∗ j 2 # +κe+1 b˜∗ +κe+2 κe+1 a˜∗ 2 . Since we have κe+1 >0, the Cauchy-Schwartz inequality once more shows thatℓis nonnegative on squares (cf. Theorem 7.1 in [Ble]). By repeating the argument above, we conclude that ℓ∈ Σ∗ X \P ∗ X.8 G. BLEKHERMAN, G.G. SMITH, AND”