The Method of Multiplicities
Polynomials have played a fundamental role in the construction of objects
with interesting combinatorial properties, such as error correcting codes,
pseudorandom generators and randomness extractors. 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 improved bounds of Kakeya sets over finite
fields, using the method of multiplicities [SS'13].
Time permitting, I will also give a proof of generalized Schwartz Zipplel