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Sum of squares length of real forms

For $n,\,d\ge1$ let $p(n,2d)$ denote the smallest number $p$ such that every sum of squares of forms of degree $d$ in $\mathbb{R}[x_1,\dots,x_n]$ is a sum of $p$ squares. We establish lower bounds for these numbers 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 $d\to\infty$ and all $n\ge3$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing $p(3,6)=4$ and $p(4,4)=5$.