The Hilbert Function is a mathematical tool used in algebraic geometry to describe the growth of the dimensions of graded components of a polynomial ring. It provides a way to understand how the number of independent polynomial equations increases as the degree of the polynomials increases. This function is particularly useful in studying the properties of algebraic varieties, which are geometric objects defined by polynomial equations. In more technical terms, the Hilbert Function is defined for a graded ring and gives the dimension of the vector space of homogeneous polynomials of a given degree. It is closely related to the Hilbert Polynomial, which provides a polynomial approximation of the Hilbert Function for large degrees. Together, these concepts help mathematicians analyze the structure and characteristics of algebraic varieties, such as those studied by David Hilbert, after whom the function is named.