Question

Let $f_1, f_2, \ldots, f_m \in \mathbb{Z}[x_1, \ldots, x_n]$. Assume $f_1(X) = f_2(X) = \ldots = f_m(X) = 0$ have no solutions over $\mathbb{C}^n$, then by Hilbert's Nullstellensatz, there exists polynomials $g_1, \ldots, g_m \in \mathbb{C}[x_1, \ldots, x_n]$ such that $1 = f_1 g_1 + \dots f_m g_m$.

In this case, can we always find $g_1, g_2, \ldots, g_m \in \mathbb{Q}[x_1, \ldots, x_n]$ such that $1 = f_1 g_1 + \dots f_m g_m$? In words, if the coefficients of $f_1, \ldots, f_m$ are all integers, can $g_1, g_2, \ldots, g_m$ be taken as polynomials with integer coefficients, such that $f_1 g_1 + \ldots +f_m g_m \in \mathbb{Z}^{+}$?

Added later: I checked Wikipedia on Hilbert's Nullstellensatz. Sorry, it seems to be a stupid question, and the answer is YES.

Answer 1

Yes. Given the degrees of the $g_i$, the equation $1 = \sum_i f_i g_i$ is tantamount to a system of linear equations in the coefficients of the $g_i$, and those linear equations have rational coefficients. Once such a system has a complex solution it automatically has a rational solution.

Answer 2

The answer is definitely yes. The argument is very simple: the ring extension $$\mathbb Q[X_1,\dots,X_n]\subset \mathbb C[X_1,\dots,X_n]$$ is faithfully flat.