Hilbert Series, h-vectors, and the Fibonacci Sequence

The Hilbert series of a module over a commutative ring is a generating function that captures many invariants of the module. In the case of zero dimensional standard graded rings, the Hilbert series is a polynomial and we call the coefficients the h-vector. As it turns out, the number of such h-vectors of length n is bounded above by the nth Fibonacci number. In this talk, we will define the Hilbert series and give some basic examples. We will also discuss the sequence defined by the number of h-vectors of a given length and its relation to the Fibonacci numbers and partitions. This talk will be accessible to anyone who has taken (or is currently taking) linear algebra.