Positive Polynomials and Varieties of Minimal Degree

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares of quadratic forms. We show more generally that every nonnegative quadratic form on a real projective variety X of minimal degree is a sum of dim(X) + 1 squares of linear forms. This provides a new proof for one direction of a result due to Blekherman, Smith, and Velasco. We explain the geometry behind this generalization and discuss some related questions and open problems. (Joint work with G. Blekherman, R. Sinn, and C. Vinzant)