# macaulayBound -- the bound on the growth of a Hilbert function from Macaulay's Theorem

## Synopsis

• Usage:
h=macaulayBound(a,d)
• Inputs:
• Outputs:
• h, an integer, the Macaulay upper bound for the Hilbert function in degree d+1 given that it is a in degree d.

## Description

Given a Hilbert function of a in degree d, macaulayBound yields the upper bound from Macaulay's Theorem for the Hilbert function in degree d+1.

 i1 : macaulayBound(3,1) o1 = 6 i2 : macaulayBound(15,5) o2 = 18

• macaulayRep -- the Macaulay representation of an integer
• macaulayLowerOperator -- the a_<d> operator used in Green's proof of Macaulay's Theorem
• isHF -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal

## Ways to use macaulayBound :

• "macaulayBound(ZZ,ZZ)"

## For the programmer

The object macaulayBound is .