Quadratic Convexity and Sums of Squares

Quadratic Convexity and Sums of Squares Martin Ames Harrison The length of a sum of squares σ in a ring R is the smallest natural k such that σ can be realized as a sum of k squares in R. For a set S ⊆ R, the pythagoras number of S, denoted by P(S), is the maximum value of length over all σ ∈ S. This dissertation is motivated by the following simple question: if R = R[x1, . . . , xn] and S = R[x1, . . . , xn]2d (the span of forms of degree 2d), then what is P(S)? By parametrizing the set of sums of k squares, we obtain a new formulation of the problem: when is the image A(R) of a quadratic map A : R → R convex? We prove several results on the structure of quadratic images and of the set of quadratic maps in general. In particular, we give conditions under which convexity of A(R) is equivalent to convexity of its compact intersection with an affine hyperplane. We prove, given assumptions on A(R), a relationship between convexity of A(R) and rank of the derivative of A. We then show how an arbitrary quadratic map A can be modified so that convexity of A(R) is preserved and the result on the derivative may be exploited. A necessary condition for quadratic convexity is thus derived.