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...
Ausführliche Beschreibung

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 %$p(4,4)=5%$. Ausführliche Beschreibung