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Category Archives: Polynomial Method
The footprint bound
Studying the set of common zeros of systems of polynomial equations is a fundamental problem in algebra and geometry. In this post we will look at estimating the cardinality of the set of common zeros, when we already know that … Continue reading
Introduction to polynomial method
(The following is a blogfriendly version of Chapter 7 of my PhD thesis, which is an introduction to the socalled polynomial method.) The polynomial method is an umbrella term for different techniques involving polynomials which have been used to solve … Continue reading
The coefficient formula and ChevalleyWarning
We discuss the new simultaneous generalization of ChevalleyWarning and Morlaye’s result on polynomial equations over finite fields obtained by Pete Clark. Continue reading
Applications of AlonFuredi to finite geometry
In a previous post I discussed how the AlonFuredi theorem serves as a common generalisation of the results of Schwartz, DeMillo, Lipton and Zippel. Here I will show some nice applications of this theorem to finite geometry (reference: Section 6 of my … Continue reading
Posted in Combinatorics, Finite Geometry, Polynomial Method
Tagged AlonFuredi, blocking set, polynomials
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The EllenbergGijswijt bound on cap sets
Four days back Jordan Ellenberg posted the following on his blog: Briefly: it seems to me that the idea of the CrootLevPach paper I posted about yesterday can indeed be used to give a new bound on the size of subsets … Continue reading
Posted in Combinatorics, Finite Geometry, Polynomial Method
Tagged capset, Dion Gijswijt, Ernie Croot, Jordan Ellenberg, Peter Pach, Seva Lev
6 Comments
The ErdősGinzburgZiv theorem
Let be a sequence of integers (not necessarily distinct). Then there exists a subsequence of the sum of whose elements is divisible by . This is one of the first problems I saw when learning the pigeonhole principle. And it’s … Continue reading
Posted in Combinatorics, Polynomial Method
Tagged additivecombinatorics, AlonFuredi, combinatorics, polynomials
8 Comments
AlonFuredi, SchwartzZippel, DeMilloLipton and their common generalization
In the post Balls in Bins I wrote about a combinatorial function which denotes the minimum value of the product among all distributions of balls (so ) in bins with the constraints . It turns out that this combinatorial function is linked … Continue reading
Posted in Combinatorics, Polynomial Method
Tagged AlonFuredi, combinatorics, finite fields, polynomials
3 Comments