Hilbert function

Definition
Let $$R$$ be a graded algebra over a field and $$M$$ a graded module over $$R$$. The Hilbert function of $$M$$, sometimes denoted $$h_M$$, is a function that sends any integer $$n$$ to the dimension of the $$n^{th}$$ graded component of $$M$$, as a vector space over the underlying field.

We usually consider the Hilbert function for a graded algebra that occurs as a quotient of a multivariate polynomial ring over a field, by a graded ideal. In other words, we study the Hilbert function for a graded algebra over a field that is generated by its degree one terms, and where the degree one component is finite-dimensional as a vector space.

For sufficiently large values, the Hilbert function equals a polynomial, termed the Hilbert polynomial.

Related notions

 * Hilbert-Samuel function and Hilbert-Samuel polynomial