An application of Hermitian K-theory: sums-of-squares formulas (Q2439232)

A sums-of-squares formula of type \([r, s, n]\) over a field \(F\) of characteristic \(\neq 2\) (with strictly positive integers \(r\), \(s\) and \(n\)) is a formula \[ \left( \sum_{i=1}^r x_i^2 \right) \cdot \left( \sum_{i=1}^s y_i^2 \right) = \left( \sum_{i=1}^n z_i^2 \right) \in F[x_1, \ldots, x_r, y_1, \ldots, y_s] \] where \(z_i = z_i(X,Y )\) for each \(i \in \{1, \ldots, n\}\) is a bilinear form in \(X\) and \(Y\) (with coefficients in \(F\)). Here, \(X =(x_1,\ldots,x_r)\) and \(Y = (y_1,\ldots,y_s)\) are coordinate systems. An old problem posed by \textit{A. Hurwitz} in [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1898, 309--316 (1898; JFM 29.0177.01)] concerns the existence of sums-of-squares formulas. For any \(m \in \mathbb{Z}_{>0}\) let \(\phi(m)\) denote the cardinality of the set \(\{l \in \mathbb{Z} : 0 < l \leq m \text{ and } l \equiv 0, 1, 2 \text{ or } 4 (\mod 8)\}\). In the paper under review the author shows that if a sums-of-squares formula of type \([r, s, n]\) exists over a field \(F\) of characteristic \(\neq 2\), then \(2^{\phi(s-1)-i+1}\) divides \(\choose{n}{i}\) for \(n - r < i \leq \phi(s - 1)\).