Hilbert function of powers of ideals of low codimension (Q1297987)

We study the relationship between the Hilbert function of an ideal \(I\) and that of its powers \(I^k\) where \(I\) is either Cohen-Macaulay of codimension 2 or Gorenstein of codimension 3. We show that if \(I\) is an ideal in one of these two classes and it is of linear type then the Hilbert function of \(I^k\) depends only on the Hilbert function of \(I\). In other words, if we assume that \(I\) and \(J\) are either Cohen-Macaulay of codimension 2 or Gorenstein of codimension 3 ideals, and that they are both of linear type and they have the same Hilbert function then also the powers \(I^k\) and \(J^k\) have the same Hilbert function for all \(k\).