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A.1.3 Rings associated to monomial orderings
In SINGULAR we may implement localizations of the polynomial ring
by choosing an appropriate monomial ordering (when defining the ring by
the At this point, we restrict ourselves to describing the relation between a monomial ordering and the ring (as mathematical object) which is implemented by the ordering. This is most easily done by describing the set of units: if > is a monomial ordering then precisely those elements which have leading monomial 1 are considered as units (in all computations performed with respect to this ordering).
In mathematical terms: choosing a monomial ordering If > is global (that is, 1 is the smallest monomial), the implemented ring is just the polynomial ring. If > is local (that is, if 1 is the largest monomial), the implemented ring is the localization of the polynomial ring w.r.t. the homogeneous maximal ideal. For a mixed ordering, we obtain "something in between these two rings":
Note, that even if we implictly compute over the localization of
the polynomial ring, most computations are explicitly performed with
polynomial data only.
In particular, See division for division with remainder in the localization and invunit for a procedure returning a truncated power series expansion of the inverse of a unit. |
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