24.4 Commands
The commands and algorithms were done by Eric Rowell with help from Dell Kronewitter and Bill
Helton.
24.4.1 NCHilbertCoefficient[integer1, aListOfExpressions, integer2, anOption]
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- Aliases: none
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- Description: NCHilbertCoefficient[integer1,aListOfExpressions,integer2, anOption] attempts to
compute the first integer1 coefficients of the Hilbert series for the algebra generated by the
relations in aListOfExpressions. There are four possible calls to this function, here
expressed in order of longest time used to least.
- The default (no fifth arguement) is for algebras that are nonhomogeneous. This
will compute a (possibly partial) Groebner Basis out to integer2 iterations of
Mora’s algorithm with respect to the ambient order, convert this basis to rules,
and procede to compute the specified number of Hilbert series coefficients. Unless
the partial Groebner basis computed contains all the polynomials that will ever
appear in the (possibly infinite) full Groebner basis up to degree integer1, the
dimensions of Tt computed will only be upper bounds.
- ExpressionForm → Homogeneous. This is only valid for homogeneous ideals.
This does as above, only the resulting dimensions are for the standard gradation,
and takes much less time. In theory, there should be no problem with the
dimensions being inaccurate provided enough iterations are used. There is an
algorithm for homogeneous problems that will return all the polynomials of a
specified degree and less.
- ExpressionForm → partialGBHomogeneous. This is an option that will
avoid the Groebner basis computation and simply convert the relations in
aListOfExpressions to rules and use them to compute the Hilbert coefficients.
This is useful particularly when one has already gone to the trouble of computing a
(partial) Groebner basis. This is only coded for homogeneous ideals. The iteration
number integer2 is ignored (although it must be there) so one may as well set it
to 0.
- ExpressionForm → HomogeneousBinomial. This is a very specific option for
ideals whose generators are the difference of two monic monomials (i.e. of the
form: xyxz - yxzx). This is essentially the same as the homogeneous version
above, only faster.
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- Arguments: integer1, aListOfExpressions, integer2, anOption
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- Comments / Limitations: The order is alway the ambient order. Make certain that your
order is only length lexicographic as this will save time. There is no reason to use
any order other than length lexicographic for Hilbert series computations that the
author of this code can think of. Currently the with the default version of this function
the ambient order will be cleared during the computation, as there is a new variable
introduced that is later removed. For now, just remember to reset the order before
proceding.
24.4.2 NCX1VectorDimension[alist]
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- Aliases: none
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- Description: NCX1VectorDimension computes the dimension of the span of a set of
polynomials as a vector space over the ground field.
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- Arguments: alist
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- Comments / Limitations: none