Let N(k) be the smallest integer n such that there exists an identity
(x_1^2 + ... + x_k^2) (y_1^2+ ... + y_k^2) = f_1^2+ ... + f_n^2,
with f_1,...,f_n being polynomials with integer coefficients in the variables x_1,...,x_k and y_1,...,y_k.

Our motivation for considering N(k) is proving lower bounds for non-commutative circuits.
We will describe this connection, and show that N(k) is at least c k^{6/5} for a constant c > 0.