Question 116
The polynomial of seven variables
$$
Q(x_1, x_2, \dots, x_7) = (x_1 + x_2 + \dots + x_7)^2 + 2(x_1^2 + x_2^2 + \dots + x_7^2)
$$
is represented as the sum of seven squares of polynomials with non-negative integer coefficients:
$$
Q(x_1, \dots, x_7) = P_1(x_1, \dots, x_7)^2 + P_2(x_1, \dots, x_7)^2 + \dots + P_7(x_1, \dots, x_7)^2.
$$
Find all possible values of $P_1(1, 1, \dots, 1)$.
$$
Q(x_1, x_2, \dots, x_7) = (x_1 + x_2 + \dots + x_7)^2 + 2(x_1^2 + x_2^2 + \dots + x_7^2)
$$
is represented as the sum of seven squares of polynomials with non-negative integer coefficients:
$$
Q(x_1, \dots, x_7) = P_1(x_1, \dots, x_7)^2 + P_2(x_1, \dots, x_7)^2 + \dots + P_7(x_1, \dots, x_7)^2.
$$
Find all possible values of $P_1(1, 1, \dots, 1)$.