Sum of squares length of real forms

TL;DR

This paper establishes new lower bounds for the minimal number of squares needed to represent sums of squares of forms, providing exact values in specific cases and conjectural asymptotic behavior for higher degrees.

Contribution

It introduces stronger lower bounds for sum of squares length, determines exact values for certain forms, and conjectures asymptotic growth based on algebraic geometry.

Findings

Lower bounds for p(n,2d) are significantly improved.

Exact values p(3,6)=4 and p(4,4)=5 are established.

Asymptotic behavior p(n,2d) ~ const * d^{(n-1)/2} is conjectured for large d.

Abstract

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 $p(4,4)=5$.