Sum of squares length of real forms

Abstract

For $n,\,d\ge 1$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in ${\mathbb {R}}[x_1,\ldots ,x_n]$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give $p(3,2d)\in \{d+1,\,d+2\}$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that $p(n,2d)\sim const\cdot d^{(n-1)/2}$ for $d\rightarrow \infty$ and all $n\ge 3$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing $p(3,6)=4$ and $p(4,4)=5$.