Tags » Polynomial Method

Chevalley-Warning Theorem and Blocking Sets

The classical Chevalley-Warning theorem gives us a sufficient condition for a system of polynomial equations over a finite field to have common solutions. Affine blocking sets… 635 more words

Finite Geometry

Balls in Bins

When I was learning combinatorics for the first time (I was probably 16) there was this problem about distributing balls in bins (or urns) that I came across. 535 more words

Combinatorics

The Kakeya problem

The original Kakeya needle problem  is to  find the least amount of area required to continuously rotate a unit line segment in the (Euclidean) plane by a full rotation… 888 more words

Finite Geometry

Two proofs of the Schwartz-Zippel lemma

The fact that a univariate polynomial over a field of degree has at most zeroes is well known. It follows from the so called Factor theorem, if and only if . 864 more words

Finite Geometry

The Bombieri-Stepanov proof of the Hasse-Weil bound

Let be a finite field of order , and let be an absolutely irreducible smooth projective curve defined over (and hence over the algebraic closure of that field). 2,808 more words

Expository

The Lang-Weil bound

Let be a finite field, with algebraic closure , and let be an (affine) algebraic variety defined over , by which I mean a set of the form… 5,465 more words

Expository

Short Proofs of Two Hard Theorems

This summer I’ve been studying problems in geometric combinatorics such as the joints problem and the Erdös distinct distance problem. The joints problem was open for 20 years and the distinct distance problem was open for over 60 years before they were resolved by a new technique called the polynomial method. 1,473 more words

Expository