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.

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.