Question 112
The polynomial
$$
Q(x_1, \dots, x_4) = 4(x_1^2 + x_2^2 + x_3^2 + x_4^2) - (x_1 + x_2 + x_3 + x_4)^2
$$
is represented as a sum of squares of four polynomials of four variables with integer coefficients.
a) Find at least one such representation.
b) Prove that for any such representation at least one of the four polynomials is identically zero.
$$
Q(x_1, \dots, x_4) = 4(x_1^2 + x_2^2 + x_3^2 + x_4^2) - (x_1 + x_2 + x_3 + x_4)^2
$$
is represented as a sum of squares of four polynomials of four variables with integer coefficients.
a) Find at least one such representation.
b) Prove that for any such representation at least one of the four polynomials is identically zero.