New constructions for integral sums of squares formulae (Q1908577)

\textit{A. Hurwitz}' pioneering work [Gött. Nachr. 1898, 309-316 (1898; JFM 29.0177.01)] solved the long standing problem of showing that the identity \[ (x_1^2+ \cdots + x_n^2) (y^2_1+ \cdots +y^2_n) =z^2_1 + \cdots + z^2_n, \] (where \(z_j\) are bilinear in \(x_j\), \(y_j\) with real or integral coefficients) is solvable iff \(n=1, 2, 4, 8\). In the more general identity (Hurwitz-Radon) \[ (x^2_1+ \cdots + x^2_r)(y^2_1 + \cdots + y^2_s)=z^2_1 + \cdots + z^2_n, \tag{*} \] \(z_j\) as above, \(r\) and \(s\) are given and one seeks the minimum value of \(n\) (for any such \(r,s,n\) for which (*) holds, we say the triple \((r,s,n)\) is admissible over \(\mathbb{R}\) or \(\mathbb{Z})\). It is well known (Yuzvinsky, Yiu) that the problem of finding admissible triples over \(\mathbb{Z}\) is equivalent to that of constructing consistently signed interlate matrices. In 1992, \textit{T. L. Smith} and \textit{P. Yiu} [Bol. Soc. Mat. Mex., II. Ser. 37, 479-495 (1992; Zbl 0830.11018)], using this concept of signed interlate matrices, established some of the best known upper bounds for \(n\) whenever \(10\leq r\leq s\) and \(17\leq s\leq 32\). In the present paper, the author improves on six of these bounds and gets bounds for some larger values of \(r\) and \(s\).