Sum of squares length of real forms
For $n,,d\ge 1$$n,d>=1 let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in \mathbb {R}[x_1,\ldots ,x_n]$R[x1,,xn] is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give $p(3,2d)\in {d+1,,d+2\$p(3,2d){d+1,d+2} in the ternary case. Assuming a conjecture of Iarrobino Kanev on dimensions of tangent spaces to catalecticant varieties, we show that $p(n,2d)\sim const\cdot d^(n 1)/2}$p(n,2d)~const*d(n 1)/2 for $d\rightarrow \infty $d and all $n\ge 3$$n>=3. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing $p(3,6)=4$$p(3,6)=4 and $p(4,4)=5$$p(4,4)=5.