Sums of fourth powers of binary forms

We study homogeneous polynomials that can be written as sums of (2s)-th powers of degree d forms. Similar to sums of squares these form full-dimensional convex cones for any s,d. The smallest integer k such that any such form (of fixed degree and number of variables) has a length k representation is called the (2s)-Pythagoras number. We show that all even higher Pythagoras numbers tend to infinity for a fixed number of variables (at least three) as the degree increases.

We then study the cone of binary octics that can be written as sums of fourth powers of quadratics to investigate the case of binary forms. This is the smallest case such that we do not consider sums of squares nor powers of linear forms. The 4-Pythagoras number is shown to be 3 or 4 and we also determine the convex structure of this cone.

(joint with Tomasz Kowalczyk)