Sums-of-squares formulas over algebraically closed fields (Q1693096)

Let \(F\) be a field of characteristic not 2. A sums-of-squares formula of type \([r,s,n]\) over \(F\) is an identity of the form \[ (x^2_1+\cdots+ x^2_r)(y^2_1+\cdots+ y^2_s)= z^2_1+\cdots+ z^2_n, \] where each \(z_i\) is a bilinear expression in the \(x\)'s and \(y\)'s over \(F\). The question of the existence of a sums-of-squares formula is reformulated in terms of algebraic geometry. It is shown that if a sums-of-squares formula of type \([r,s,n]\) exists over any field \(F\), then a sums-of-squares formula of type \([r,s,n]\) exists over some finite field. Furthermore, the author provides an explicit degree bound on this finite field. Thus the existence of a sums-of-squares formula over an algebraically closed field is computable theoretically (though not practically). For the explicit computations a zeta function is used and a result of \textit{E. Bombieri} [Invent. Math. 47, 29--39 (1978; Zbl 0396.14001)].