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