The previous talk in the series is here. The next talk in the series is here.
Noga Alon’s Combinatorial Nullstellensatz shows that under appropriate conditions a polynomial cannot be zero everywhere on a Cartesian product. It has many applications to combinatorial theorems with statements that appear to have nothing to do with polynomials. Here I present the Nullstellensatz and its proof, and give two applications. The first is the Cauchy-Davenport theorem, which asserts that if A and B are two subsets of the finite field 𝔽p (where p is a prime), then the sumset A+B (defined to be the set of all a+b with a ∈ A and b ∈ B) has size at least min{p, |A|+|B|-1}. The second is to a special case of a theorem of Dias da Silva and Yahya Ould Hamidoune, which asserts that the restricted sumset (defined to be the set of all a+b with a ∈ A, b ∈ B, and a not equal to b) has size at least min{p, |A|+|B|-3}.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.