I read that Hilbert used the nullstellensatz in algebraic number theory rather than in algebraic geometry. What did he use it for? Today, is the nullstellensatz used in algebraic number theory or other areas apart from algebraic geometry?

## **closed** as not a real question by Andrés E. Caicedo, Andy Putman, Martin Brandenburg, Pete L. Clark, J.C. Ottem Mar 15 '11 at 9:40

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I answer: 'is the Nullstellensatz used in other areas?'

Yes, for example, a variant is used in Combinatorics.

There is a Combinatorial Nullstellensatz (due to N. Alon). The rough idea is to translate certain combinatorial problems into statements on the (non-)vanishing of systems of polynomial equations, and then to analyse this system of polynomial equations.

A question related to it was somewhat recently asked on this site by gowers where you can find details on it.

youthink "algebraic number theory" is, and why do you think it is distinct from "algebraic geometry"? And if you want to know what Hilbert used the Nulstellensatz for, what kind of answer are you hoping for? What level of technicality would be more useful or less useful to you, O Creature of Inconstant Name? $\endgroup$ – Yemon Choi Mar 15 '11 at 9:50