Let I be an ideal of the local polynomial ring K[x1, ..., xn]< x1, ..., xn >, 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).
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 [BW93, p. 449].