Dimension of Ideals

Let \(I\) be an ideal of the local polynomial ring \(K[x_1, \ldots, x_n]_{\langle x_1, \ldots, x_n\rangle}\), where \(K\) is a field. As for polynomial rings, the dimension of the ideal \(I\) can be defined as the the maximum of the cardinalities of all the independent sets modulo \(I\) (see Section Dimension of Ideals for details).

Dimension(I): RngMPolLoc -> RngIntElt, [ RngIntElt ]

Given an ideal \(I\) of a local polynomial ring \(R\) defined over a field, return the dimension \(d\) of \(I\), together with a (sorted) sequence \(U\) of integers of length \(d\) such that the variables of \(P\) corresponding to the integers of \(U\) constitute a maximally independent set modulo \(I\). If \(I\) is the full local polynomial ring \(R\), the dimension is defined to be \(-1\), and the second return value is not set. The algorithm implemented is that given in [Becker and Weispfenning, 1993, p. 449].