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Let I be an ideal of the polynomial ring
P = K[x1, ..., xn], where K is a field. Let X be the set
{ x1, ..., xn } of variables of P.
A subset U of X is called independent modulo I if
I ∩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. It is not too hard to see in this case that this coincides
with the more abstract commutative algebra definition of the Krull
dimension of the quotient algebra P/I as the maximal length of a
chain of prime ideals.
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].
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