Martin Rubey wrote:
>
> Martin Rubey <[EMAIL PROTECTED]> writes:
>
> > > > + leadingCoefficient s ==
> > > > + leadingCoefficient(coefficient(s, order(s pretend PS)$PS))
> > > > +
> > >
> > > That looks wrong. In fact, I do not think we can implement this: since
> > > we use lexicographic order to know that say x is a leading term we
> > > would have to examine infinitely many terms and check that each of
> > > them contains x in power at least 1.
>
> Actually, I don't understand. For me, for power series the leading
> coefficient
> is the term of minimal total degree. In POLY, MPOLY, etc. we define leading
> coefficient to be coefficient with maximal degree, minimum in lexicographic
> order. I think it makes sense to generalize this.
>
We have:
++ \spadtype{PowerSeriesCategory} is the most general power series
++ category with exponents in an ordered abelian monoid.
PowerSeriesCategory(Coef,Expon,Var): Category == Definition where
...
leadingMonomial: % -> %
++ leadingMonomial(f) returns the monomial of \spad{f} of lowest order.
leadingCoefficient: % -> Coef
++ leadingCoefficient(f) returns the coefficient of the lowest order
++ term of \spad{f}
the "lowest order" can be only the order in Expon (since we have
no other order at our disposal).
Later we have:
MultivariateTaylorSeriesCategory(Coef,Var): Category == Definition where
Coef : Ring
Var : OrderedSet
L ==> List
NNI ==> NonNegativeInteger
Definition ==> Join(PartialDifferentialRing Var,_
PowerSeriesCategory(Coef,IndexedExponents Var,Var),_
InnerEvalable(Var,%),Evalable %) with
so, the monoid is IndexedExponents Var, which inherits from
IndexedDirectProductOrderedAbelianMonoidSup which in turn interits
from IndexedDirectProductOrderedAbelianMonoid where finally we see
that order is lexicographic.
In POLY, MPOLY we have similar definitions, only using terms which are
maximal with respect to given order. Now, wording may be a bit misleading,
because sometimes degree means exponent (element of monoid), sometimes
we want total degree:
AbelianMonoidRing(R:Ring, E:OrderedAbelianMonoid): Category ==
Join(Ring,BiModule(R,R)) with
leadingCoefficient: % -> R
++ leadingCoefficient(p) returns the coefficient highest degree term of p.
however, here again only interpretation as element of monoid makes
sense.
> > > AFAICS your implementation corresponds to "total degree, reverse
> > > lexicographic" order -- this is a reasonable order,
> >
> > Yes
> >
> > > but using it would imply giving SMTS different category.
> >
> > Why? I thought it specialises well to the univariate case.
>
> Actually, "total degree, reverse lexicographic" is not completely true. The
> definition above is rather "total degree, underlying order":
>
As explained above MultivariateTaylorSeriesCategory fixes underlying
order to be lexicographic.
--
Waldek Hebisch
[EMAIL PROTECTED]
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