Dimension of Ideals

Let I be an ideal of the polynomial ring P = K[x_1, ..., x_n], where K is a field. Let X be the set { x_1, ..., x_n } of variables of P. A subset U of X is called independent modulo I if I intersect K[U] = emptyset. A subset U of X is called maximally independent modulo I if U is independent modulo I, and no proper superset of U is independent modulo I. The dimension of I is defined to be the maximum of the cardinalities of all the independent sets modulo I.

Note that the definition given above of zero-dimensionality (as the case when the quotient of P by I has finite dimension as a vector space over the coefficient field) coincides with the definition of zero-dimensionality as dimension 0.

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

Given an ideal I of a polynomial ring P 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 polynomial ring P, 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].