The length of a sum of squares sigma in a ring R is the smallest natural number k such that sigma can be realized as a sum of k squares in R. For a set S contained in R, the pythagoras number of S, denoted by PS , is the maximum value of length over all sigma belonging to S. This dissertation is motivated by the following simple question: if R is the ring of polynomials in n indeterminates over the reals and S is the span of forms of degree 2 d, then what is PS ?

By parametrizing the set of sums of k squares, we obtain a new formulation of the problem: when is the image of a quadratic map 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 quadratic image is equivalent to convexity of its compact intersection with an affine hyperplane. We prove a relationship between convexity of a quadratic image and rank of the derivative of a quadratic map. We then show how an arbitrary quadratic map can be modified so that its image is preserved and the result on the derivative may be exploited. A necessary condition for quadratic convexity is thus derived.