Grigoriy Blekherman

Nonnegative polynomials and sums of squares

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

On maximum, typical and generic ranks

d

Typical Real Ranks of Binary Forms

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

Algebraic boundaries of Hilbert’s SOS cones

We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.

On real typical ranks

We prove a conjecture of Comon and Ottaviani that typical real Waring ranks of bivariate forms of degree d take all integer values between

Sums of squares and varieties of minimal degree

\lfloor \frac{d+2}{2}\rfloor

Positive Gorenstein ideals

and d. That is, we show that for all d and all

Do Sums of Squares Dream of Free Resolutions

\lfloor \frac{d+2}{2}\rfloor \leq m \leq d

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

there exists a bivariate form f such that f can be written as a linear combination of mdth powers of real linear forms and no fewer, and additionally all forms in an open neighborhood of f also possess this property. Equivalently we show that for all d and any

Extreme rays of Hankel spectrahedra for ternary forms

\lfloor \frac{d+2}{2}\rfloor \leq m \leq d