The Method of Multiplicities

Polynomials have played a fundamental role in the construction of objects with inter-esting combinatorial properties, such as error correcting codes, pseudorandom gener-ators 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 ex-tractors, 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 thesis we improve on several of the results mentioned above by a more pow-erful application of polynomials that takes into account the information contained in the derivatives of the polynomials. We call this technique the method of multiplicities