Ralf Hemmecke wrote:
>
> Can someone convince me why it is a good idea (apart from convenience)
> that degree(0) returns the unit from the exponent domain?
>
> According to the usage in FriCAS' sources it must be common knowledge,
> but if I consider a graded ring, then 0 lives in every grade, so it
> should be undefined what "degree(0)" returns or at least we should have
> a note in the documentation that degree(0)=1 to make it unquestionable.
>
> http://fricas.github.io/api/AbelianMonoidRing.html#l-abelian-monoid-ring-degree
>
> http://fricas.github.io/api/GradedModule.html#l-graded-module-degree
Strictly speaking both descriptions of 'degree' are incorrect.
In GradedModule what is meant is "elements of degree <= n
form an R-module". Once you take this into account definition
of 'degree' from AbelianMonoidRing applies with little correction:
degree of general element x is least upper bound of set of degrees
of (nonzero) homogeneous components of x. List of homogeneous components
of 0 is empty, so it is mininal element of set of exponents.
In case of polynomials it is unit in set of exponents, in case of
Laurent polynomials it is minus infinity (undefined if we do not
take ordered completition).
In AbelianMonoidRing 'degree' really is really more general than
what the description says: the fundamental notion is filtration
and AbelianMonoidRing is supposed to support both increasing
and decreasing filtrations. 'degree' should give matched
term of filtration. For good increasing filtrations F_i this means
taking greatest lower bound of indices i such that element x
belong to F_i. Of course, it may happen that greatest lower bound d
does not exist or x does not belong to F_d where d. Or we
may be unable to compute approproate value -- it goes without
mentioning this in every documentation string that FriCAS
functions may fail (singnal error).
There is some fuzzines in definition of degree of 0, namely it
depends on ordered set used for exponents: if you consider
exponents of univariate polynomials to be nonnegative integers,
then degree(0) = 0. If you take integers as set of exponents,
then degree is undefined (should be -infinity, but this is outside
of integers). If you take integers startiong from -1 as
exponents, then degree(0) = -1. However, in case of AbelianMonoidRing
set of exponents is specified, so no problem here.
--
Waldek Hebisch
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