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Tag Archives: finite fields
A coding theoretic application of the AlonFüredi theorem
The AlonFüredi theorem is something that I have written a lot about in this blog. I spent a considerable amount of time on this theorem during my PhD. In fact, it’s generalisation that I obtained and it’s applications in finite … Continue reading
Bilinear forms and diagonal Ramsey numbers
The recent breakthrough of Conlon and Ferber has shown us that algebraic methods can be used in combination with probabilistic methods to improve bounds on multicolour diagonal Ramsey numbers. This was already shown for the offdiagonal Ramsey numbers by Mubayi … Continue reading
Improved lower bounds for multicolour diagonal Ramsey numbers
Big news in combinatorics today: David Conlon and Asaf Ferber have posted a 4page preprint on arXiv that gives exponential improvements in the lower bounds on multicolour diagonal Ramsey numbers, when the number of colours is at least (also see … Continue reading
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
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
Incidence Bounds and Interlacing Eigenvalues
The Szemerédi–Trotter theorem is one of the central results in discrete geometry which gives us a (tight) bound on the number of incidences, i.e., the number of pointline pairs with the point lying on the line, between finite sets of points and lines … Continue reading
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
4 Comments
A timeline of the polynomial method upto combinatorial nullstellensatz
Over the past 3040 years, the socalled polynomial method has developed into a powerful tool in combinatorics and (additive) number theory. There has been a lot of recent interest in it after Dvir’s paper on the Kakeya conjecture, where he … Continue reading
Posted in Combinatorics, Finite Geometry, Polynomial Method
Tagged combinatorics, finite fields, finite geometry, polynomials
3 Comments
On Zeros of a Polynomial in a Finite Grid: the AlonFuredi bound
My joint paper with Aditya Potukuchi, Pete L. Clark and John R. Schmitt is now up on arXiv: arXiv:1508.06020. This work started a few months back when I emailed Pete and John, pointing out an easy generalization of ChevalleyWarning theorem using something known as … Continue reading
Posted in Combinatorics, Finite Geometry, Polynomial Method
Tagged AlonFuredi, combinatorics, finite fields, polynomials
5 Comments
ChevalleyWarning Theorem and Blocking Sets
The classical ChevalleyWarning theorem gives us a sufficient condition for a system of polynomial equations over a finite field to have common solutions. Affine blocking sets are sets of points in an affine geometry (aka affine space) that intersect every hyperplane. … Continue reading