Title: The Method of Multiplicities
Speaker: Shubhangi Saraf, Institute for Advanced Study
Date: Wednesday, March 28, 2012 11:00-12:00pm
Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ
Polynomials have played a fundamental role in the construction of objects with interesting combinatorial properties, such as error correcting codes, pseudorandom generators, randomness extractors etc. Somewhat strikingly, polynomials have also been found to be a powerful tool in the analysis of combinatorial parameters of objects that have some algebraic structure. This method of analysis has found applications in works on list-decoding of error correcting codes, constructions of randomness extractors, and in obtaining strong bounds for the size of Kakeya Sets. Remarkably, all these applications have relied on very simple and elementary properties of polynomials such as the sparsity of the zero sets of low degree polynomials.
In this talk we will discuss improvements on several of the results mentioned above by a more powerful application of polynomials that takes into account the information contained in the *derivatives* of the polynomials. We call this technique the ``method of multiplicities". The information about higher multiplicity vanishings of polynomials, which is encoded in the derivative polynomials, enables us to meaningfully reason about the zero sets of polynomials of degree much higher than the underlying field size.
We will discuss some of these applications of the method of multiplicities, to obtain improved constructions of error correcting codes, and qualitatively tighter analyses of Kakeya sets, and randomness extractors.
(Based on joint works with Zeev Dvir, Swastik Kopparty, Madhu Sudan, Sergey Yekhanin)