Tag Archives: polynomials

A coding theoretic application of the Alon-Füredi theorem

The Alon-Füredi 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

Posted in Coding Theory, Combinatorics, Extremal Combinatorics, Finite Geometry, Polynomial Method | Tagged , , , , , | 2 Comments

Introduction to polynomial method

(The following is a blog-friendly version of Chapter 7 of my PhD thesis, which is an introduction to the so-called polynomial method.) The polynomial method is an umbrella term for different techniques involving polynomials which have been used to solve … Continue reading

Posted in Combinatorics, Incidence Geometry, Polynomial Method | Tagged , , , , | 2 Comments

The coefficient formula and Chevalley-Warning

We discuss the new simultaneous generalization of Chevalley-Warning and Morlaye’s result on polynomial equations over finite fields obtained by Pete Clark. Continue reading

Posted in Number Theory, Polynomial Method | Tagged , , , , , , | 2 Comments

Applications of Alon-Furedi to finite geometry

In a previous post I discussed how the Alon-Furedi 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 , , | 1 Comment

The Erdős-Ginzburg-Ziv 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 , , , | 8 Comments

Alon-Furedi, Schwartz-Zippel, DeMillo-Lipton 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 , , , | 4 Comments

A timeline of the polynomial method up-to combinatorial nullstellensatz

Over the past 30-40 years, the so-called 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 , , , | 3 Comments

On Zeros of a Polynomial in a Finite Grid: the Alon-Furedi 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 Chevalley-Warning theorem using something known as … Continue reading

Posted in Combinatorics, Finite Geometry, Polynomial Method | Tagged , , , | 5 Comments

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 are sets of points in an affine geometry (aka affine space) that intersect every hyperplane. … Continue reading

Posted in Polynomial Method | Tagged , , | Leave a comment

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 . But what about a polynomial in -variables, , where … Continue reading

Posted in Finite Geometry, Polynomial Method | Tagged , , | 8 Comments