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Tag Archives: finite fields
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
2 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
1 Comment
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
3 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
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. Of course in a circle of diameter one we can continuously … Continue reading
Posted in Finite Geometry, Polynomial Method
Tagged combinatorics, finite fields, finite geometry
1 Comment
Two proofs of the SchwartzZippel 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 . But what about a polynomial in variables, , where … Continue reading
Posted in Finite Geometry, Polynomial Method
Tagged combinatorics, finite fields, polynomials
8 Comments