Georgia Institute of Technology College of Sciences

The structure of sums-of-squares representations of (nonnegative homogeneous) polynomials is one interesting subject in real algebraic geometry. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. In this talk, I will introduce Gram spectrahedron, connection to toric variety, a new result that if a variety $X$ is arithmetically Cohen-Macaulay and a linearly normal variety of almost minimal degree (i.e. $\deg(X)=\text{codim}(X)+2$), then every sum of squares on $X$ is a sum of $\dim(X)+2$ squares.