We have domains for linear ordinary differential operators, but
no for linear partial differential operators. I have
implemented an experimental domain for LPDO-s (easy, since
we have almost all ingredients), but the question remain
how to integrate it nicely into existing categories.
I think that LPDO-s can be treated as polynomials in variables
that commute between themselves, but do not commute with
coefficients. So we can introduce a new category,
currently I call it 'VariablesCommuteWithCoefficients'.
Then in AMR we have:
if % has VariablesCommuteWithCoefficients and R has CommutativeRing then
CommutativeRing
that is both conditions have to be satisfied to get CommutativeRing.
In AMR and FAMR we need few similar changes.
Then we can split PolynomialCategory into two. One category
which I call MaybeSkewPolynomialCategory is for polynomials
without assuming that variables commute with coefficients.
Then PolynomialCategory inherits from MaybeSkewPolynomialCategory
and VariablesCommuteWithCoefficients and defines functions
which make sense only for ordinary polynomials.
For technical reasons we have to move functions from IntegralDomain
into EntireRing (all functions defined in IntegralDomain make
sense in noncommutative case) and in few places we have to work
around compiler limitations.
However, I am not entirely satisfied with this. One thing is that
new functions in EntireRing are defined in asymetric way. One
could make dual definitions replacing left by right. However,
at least for 'unitNormal' the intent is to make a choice
which breaks symmetry, so provinding only one version may
be appropriate. Introducing something like LeftEntireRing
and RightEntireRing does not look nice, particulary because
mathematically ring is entire iff ring with opposite
multiplication is entire.
Another problem is that I would like to consider linear
differential operators as a particular case of Ore algebra,
and for general Ore algebra variables do not commute with
themselves, so we need new category (hmm, something
between indexed product and AMR ???). Additionally
it is a bit fuzzy which variables in Ore algebra are
"Ore variables" and which are coefficients. For
purpose of noncommutative Groebner bases we want
everything invertible to be a coefficient. But in other
situations we may need fractions with respect to Ore
variables.
--
Waldek Hebisch
[email protected]
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