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# minimumDimension -- computes the minimum dimension of $I$.

- Usage:
    `minimumDimension I`
- Inputs:
  - I, an ideal, a homogeneous ideal of the polynomial ring $S=K[x_1,\ldots,x_n]$, with $K$ a field
- Outputs:
  - k, an integer, the minimum dimension of $I$

## Description

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring. Given a homogeneous ideal $I\subset S$, it returns the minimum dimension, defined as the minimum integer $k$ such that the filter ideal $I^{<k>}$ differs from $I$.

    i1 : S = QQ[x_1..x_10,y_1..y_10];

    i2 : E = {{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{6,7},{8,9},{8,10},{9,10}};

    i3 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0));

    o3 : Ideal of S

    i4 : minimumDimension J

    o4 = 11

## See also

- filterIdeal -- computes the $i$th filter ideal of $I$.
- isSCM -- checks whether a module or an ideal is sequentially Cohen-Macaulay

## Ways to use minimumDimension:

- `minimumDimension(Ideal)`

## For the programmer

The object minimumDimension is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SCMAlgebras.m2:481:0.