Tag Archives: polynomials

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

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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

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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 , , , | 7 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

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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

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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

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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

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