  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 $p(4,4)=5$.
