Old and new applications of the Combinatorial Nullstellensatz in geometry

In previous talks at this seminar Zoltán Lóránt Nagy and Péter Maga
discussed applications of the Combinatorial Nullstellensatz in graph theory
and additive combinatorics. Here we mostly focus on applications in finite
and discrete convex geometry. An almost cover of a discrete set of points
is a collection of hyperplanes that cover all points except one. After showing
the short proofs of some classical results, such as Jamison's theorem and
the Alon-Füredi theorem, I will present some new results obtained with
Gábor Hegedüs. Beacuse of its relevance, at some point I may make a short
detour to additive combinatorics as well. I will also propose some open problems
in the hope that some people visiting the Erdős Center will find them appealing.