A Hilbert polynomial is a mathematical tool used in algebraic geometry to describe the growth of the dimensions of the graded components of a graded ring. It provides a way to understand how the number of independent sections of a line bundle changes as one considers higher and higher degrees. This polynomial is particularly useful in studying projective varieties and their properties. The concept is named after the mathematician David Hilbert, who made significant contributions to various areas of mathematics. Hilbert polynomials are closely related to the Hilbert function, which counts the dimensions of the graded components, and they play a crucial role in the study of schemes and algebraic varieties.