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# hilbertFunction(ZZ,DualSpace)

- Function: hilbertFunction
- Usage:
    `k = hilbertFunction(d,S)`
    `T = hilbertFunction(S)`
    `K = hilbertFunction(D,S)`
- Inputs:
  - d, an integer, an integer, or a list of arguments
  - S, an instance of the type DualSpace, a dual space
- Outputs:
  - T, a tally, of dimensions for each degree
  - k, an integer, dimensions
  - K, a list, of dimensions

## Description

This function counts the dimension of the dual space in each degree. A single degree can be specified, or a list of degrees. If no degree is specified, then a Tally is returned pairing each degree with its dimension.

    i1 : R = CC[x,y];

    i2 : I = (ideal {x,y})^4;

    o2 : Ideal of R

    i3 : D = zeroDimensionalDual(origin R, I)

    o3 = | 1 y x .5y2 xy .5x2 .166667y3 .5xy2 .5x2y .166667x3 |

    o3 : DualSpace

    i4 : hilbertFunction D

    o4 = Tally{0 => 1}
               1 => 2
               2 => 3
               3 => 4

    o4 : Tally

## Ways to use this method:

- `hilbertFunction(DualSpace)`
- `hilbertFunction(List,DualSpace)`
- hilbertFunction(ZZ,DualSpace)

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NoetherianOperators.m2:2352:0.