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.
> > 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":
(25) -> sin(X+2*Y)
4 3 2 2 1 3
(25) (2y + x) + (- - y - 2x y - x y - - x ) + O(5)
3 6
Type: TaylorSeries(Fraction(Integer))
(26) -> leadingCoefficient reductum reductum sin(X+2*Y)
4
(26) - -
3
Type: Fraction(Integer)
(27) -> sin(R+2*S)
1 3 2 2 4 3
(27) (x + 2y) + (- - x - y x - 2y x - - y ) + O(5)
6 3
Type:
SparseMultivariateTaylorSeries(Fraction(Integer),OrderedVariableList([x,y]),MultivariatePolynomial([x,y],Fraction(Integer)))
(28) -> leadingCoefficient reductum reductum sin(R+2*S)
1
(28) - -
6
Type: Fraction(Integer)
Martin
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