Sums of squares of k-term forms

Abstract

In this paper we study the cones corresponding to sums of squares of $n$-ary $d$-ic forms with at most $k$ terms. We show that these are strictly nested as $k$ increases, leading to the usual sum of squares cone. We also discuss the duals of these cones. For $n\ge 3$, we construct indefinite irreducible $n$-ary $d$-ic forms with exactly $k$ terms for $2\le k\le \left(\genfrac{}{}{0ex}{}{n+d-1}{n-1}\right)$.