Hilbert series (also called Poincare series) are important tools for studying algebra. They give a measure of size for certain vector spaces which have extra structure called a grading. In this video, we introduce the notion of graded vector spaces and their Hilbert series. We show how to compute this Hilbert series for polynomial rings with generators which are not necessarily in degree one. We also look at the ring of symmetric polynomials. We look at some basic properties of HIlbert series such as additivity and shifting which allows us to compute many Hilbert series. As an application, we look at the ordinary double point and show how Hilbert series can be used to identify its coordinate ring as a quotient of a polynomial ring.
