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.



Hilbert's Nullstellensatz on polynomials with integer coefficients
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.


