Conference abstracts

Session S01 - Computational Algebra and Applications of Algebra

July 28, 16:30 ~ 16:55

## Sums of squares and the geometry of syzygies

### Mauricio Velasco

### Universidad de la República (CURE) y Universidad de los Andes, Uruguay / Colombia - mvelasco@cmat.edu.uy

Let $X\subseteq \mathbb{P}^n$ be a reduced scheme over the reals defined by an ideal $I_X$. We show that the number of steps for which the minimal free resolution of $I_X$ is linear is a lower bound for the next-to-minimal rank of extreme rays of the cone dual to the sums of squares in $X$. As a consequence, we obtain:

(1) A complete classification of totally real reduced schemes for which nonnegative quadratic forms are sums of squares.

(2) New certificates of exactness for semidefinite relaxations of polynomial optimization problems on projective varieties.

Joint work with Greg Blekherman (Georgia Tech, USA) and Rainer Sinn (Georgia Tech, USA).