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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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  • $\begingroup$ The polynomials $X^p - X$ and $p(X^p-X)$ provide a counterexample; is primitivity intended? And "multiplicity" of root doesn't have any meaning mod $m$ for $m$ not prime (esp. not square free), but is it intended that the polynomials are irreducible over $\mathbf{Q}$? Anyway, the Jacobson property of finite type $\mathbf{Z}$-algebras is a useful "arithmetic" version of the Nullstellensatz (but it has no arithmetic content, since it is a general fact with $\mathbf{Z}$ replaced by any Dedekind domain with infinitely many primes, or any Jacobson ring at all). $\endgroup$
    – user54268
    Commented Aug 26, 2014 at 5:01
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    $\begingroup$ @user54268 Hm, do $x^2 - x$ and $2(x^2 - x)$ have the same roots mod $4$? (Rhetorical question, yes, but can your proposed counterexample be fixed?) $\endgroup$
    – Todd Trimble
    Commented Jun 5, 2016 at 12:21

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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.

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  • $\begingroup$ Dietrich: I was the one who asked the question there. I am somehow curious whether there is some big picture result that generalizes the result from Gowers book. Don't worry, I have already googled for some time $\endgroup$
    – Polynomial ring
    Commented Aug 25, 2014 at 21:10
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    $\begingroup$ True, but the deep content of arithmetic Nullstellensätze is really about degree and height bounds. Indeed, the first part (existence of $a, g_1,\dots,g_s$) follows directly from the geometric Nullstellensatz. $\endgroup$
    – ACL
    Commented Jun 5, 2016 at 20:40

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