Construction of identities for sums of squares (Q2043527)

In the paper under review, the author recalled earlier and more recent results of sum-of-squares formulas, and constructed a formula of size \([r+r',2ss',2nn']\) from sum-of squares of sizes \([r,s,n]\) and \([r',s',n']\). More precisely, a sum-of-squares formula of size \([r,s,n]\) is an equation of the type \[(x_1^2+\cdots+x_r^2)\cdot (y_1^2+\cdots+y_s^2)=z_1^2+\cdots+z_n^2,\eqno (1)\] where \(X=(x_1,\cdots,x_r)^T\) and \(Y=(y_1,\cdots,y_s)^T\) are systems of independent indeterminates, and each \(z_k=z_k(X,Y)\) is a bilinear form in \(X\) and \(Y\) with coefficients in a given field \(K\). Denoting \(Z=(z_1,\cdots,z_n)^T\) and writing \(Z=(x_1A_1+\cdots+x_rA_r)Y\) for some \(n\times s\) matrices \(A_i\) with entries in \(K\), (1) is equivalent to the following system of ``Hurwitz Equations'' [\textit{A. Hurwitz}, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1898, 309--316 (1898; JFM 29.0177.01)] for those \(n\times s\) matrices \(A_1,\cdots,A_r:\) \[A_i^TA_i=1_s,1\leq i\leq r;\] \[A_i^TA_j+A_j^TA_i=0,1\leq i,j\leq r~\text{ with~}i\neq j.\] In order to prove the main result, the author defines that two matrices \(A,B\) are \textit{amicable} if \(A^TB=B^TA\). Furthermore, for a \([p,s,n]\)-formula given by the \(n\times s\) matrices \(A_1,\cdots,A_p\), and a \([q,s,n]\)-formula given by the \(n\times s\) matrices \(B_1,\cdots, B_q,\) the author defines that those formulas of sizes \([p,s,n]\) and \([q,s,n]\) are \textit{amicable} if \(A_i^TB_k=B_k^TA_i\) for every \(i,k.\) To proceed, one needs the following lemma with the same proof as Lemma 6 (extending the ``doubling lemma'' of [\textit{C. Zhang} and \textit{H.-L. Huang}, SIGMA, Symmetry Integrability Geom. Methods Appl. 13, Paper 064, 6 p. (2017; Zbl 1426.11027)] to the amicable systems) in the paper under review. Lemma 1. A \([p,s,n]\) formula yields amicable \([p+1,2s,2n]\) and \([1,2s,2n].\) Now one prepares for the proof of the main theorem (see Theorem 7 below). Let an \([r,s,n]\)-formula be given by the \(n\times s\) matrices \(A_1,\cdots, A_r,\) and \(B\) is another matrix of that size. Similarly, let a \([p,q,m]\)-formula be given by the \(m\times q\) matrices \(C_1,\cdots,C_p\), and \(D\) is another matrix of that size. Consider the following system of \(nm\times sq\) matrices: \[A_j\otimes D~\text{ for~}1\leq j\leq r,~\text{ and~}B\otimes C_k~\text{ for~}1\leq k\leq p.\] To yield an \([r+p,sq,nm]\)-formula, one needs to verify the Hurwitz Equations: \[D^TD=1_q\eqno (2)\] \[B^TB=1_s\eqno (3)\] and \[(A_j^TB)\otimes(D^TC_k)+(B^TA_j)\otimes(C_k^TD)=0~\text{ for~every~}j,k.\eqno (4)\] If one assumes that \(A_1,\cdots,A_r,B\) form an \([r+1,s,n]\)-formula, so that (3) is satisfied and \(A_j^TB+B^TA_j=0\) for every \(j\), then condition (4) becomes \[(A_j^TB)\otimes(D^TC_k-C_k^TD)=0,\] which is satisfied if \(D\) is amicable with each \(C_k\). The above shows that if furthermore amicable \([p,q,m]\) (represented by \(C_1,\cdots,C_p\)) and \([1,q,m]\) (represented by \(D\)) exist, so that (2) and (4) are satisfied, then the \([r+1,s,n]\)-formula assumed above yields an \([r+p,sq,nm]\)-formula. This is stated as the following proposition. Proposition 8. Suppose amicable formulas \([p,q,m]\) and \([1,q,m]\) exist. Then an \([r+1,s,n]\)-formula yields an \([r+p,sq,nm]\)-formula. Theorem 7. If formulas \([r,s,n]\) and \([r',s',n']\) exist, then there is an \([r+r',2ss',2nn']\)-formula. Proof. By Lemma 1, \([r',s',n']\) yields amicable formulas \([r'+1,2s',2n']\) and \([1,2s',2n']\). Then, by Proposition 8, the \([r,s,n]\)-formula yields an \([r+r',2ss',2nn']\)-formula.