The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same roots mod every $m$ iff they differ by a sign. I would like to know the general form of this result, and see some reference where I can read about it and some applications (perhaps). All help is appreciated.
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One "arithmetic version" of the Nullstellensatz states that if $f_1, ..., f_s$ belong to $\mathbb{Z}[X_1,...,X_n]$ without a common zero in $\mathbb{C}^n$, then there exist $a \in \mathbb{Z} \setminus {0}$ and $g_1,...,g_s$ in $\mathbb{Z}[X_1,...,X_n]$ such that $a = f_1g_1 + ... + f_sg_s$. Finding degree and height bounds for $a$ and $g_1, ..., g_s$ has received some attention, see for example here. 

