The existence of the [r, s, n] sum of square formula:

(x1^2+, ...+xr^2)*(y1^2+, ...+ys^2) = (z1^2+, ...+zn^2)

is related to the existence of an axial map of projective spaces:

P^(r-1)xP^(s-1)-->P^(n-1)

There is a recent work extending this formula to some fields of non-zero characteristic:

http://www.uoregon.edu/~ddugger/ksum.pdf

The existence of the $[r, s, n]$ sum of square formula:

$(x_1^2+ \ldots +x_r^2) \cdot (y_1^2+ \ldots +y_s^2) = (z_1^2+ \ldots +z_n^2)$

is related to the existence of an axial map of projective spaces:

$P^{r - 1} \times P^{s-1} \to P^{n-1}$

There is a recent work extending this formula to some fields of non-zero characteristic:

http://www.uoregon.edu/~ddugger/ksum.pdf

The existence of the [r, s, n] sum of square formula:

(x1^2+, ...+xr^2)*(y1^2+, ...+ys^2) = (z1^2+, ...+xn^2)

is related to the existence of an axial map of projective spaces:

P^(r-1)xP^(s-1)-->P^(n-1)

There is a recent work extending this formula to some fields of non-zero characteristic:

http://www.uoregon.edu/~ddugger/ksum.pdf

The existence of the [r, s, n] sum of square formula:

(x1^2+, ...+xr^2)*(y1^2+, ...+ys^2) = (z1^2+, ...+zn^2)

is related to the existence of an axial map of projective spaces:

P^(r-1)xP^(s-1)-->P^(n-1)

There is a recent work extending this formula to some fields of non-zero characteristic:

http://www.uoregon.edu/~ddugger/ksum.pdf