Macaulay's theorem

Macaulay's theorem provides a way to describe the possible sequences of numbers called Hilbert functions, which measure the size of algebraic structures like polynomial rings and their quotients. Essentially, it offers criteria to determine whether a given sequence can represent the growth of these structures in successive degrees. The theorem states that such sequences must follow specific numerical patterns and inequalities, capturing how the complexity of algebraic objects can develop. This helps mathematicians understand the constraints and potential structures within algebraic geometry and commutative algebra.