Let $ A $ to be symmetrical $ d times d $ matrix with whole entries such as the quadratic form $ Q (x) = langle Axis, x rangle, x in mathbb {R} ^ d $, is non-negative defined. For who $ d $ does this imply that $ Q $ is the sum of many squares of linear shapes with integer coefficients

$$

Q (x) = sum_ {i = 1} ^ N ( ell_i (x)) ^ 2 quadric text {for some} , N?

$$

For $ d = $ 2 That's right, I know about Sweden's 1995 IMO problem, but it's probably been known for a longer time.

I think I can prove it for other smaller, though less elemental dimensions (using Minkowski's theorem on lattice points in convex bodies: if $ Q $ is positive definite, one can find a linear form $ ell (x) $ such as $ Q- ell ^ 2 $ is always non-negative defined, it amounts to finding a whole point in an ellipsoid), but for the big ones $ d $ this argument fails.