Hi Oliver,
On 30 Jan., 01:04, Oliver Block <[email protected]> wrote:
> Hello,
>
> I would like to define a function d from polynomial ring to itself (in
> the easiest case) that is linear over the base ring
That's to say: You don't simply want a function, but you want a
morphism of modules over the base ring, isn't it?
> How can this be achieved?
>
> What I found is the "hom" method of the Polynomial ring, but using this
> method, ring homomorphisms are created.
Let
sage: P.<x> = QQ[]
Indeed, the method P.hom(P) is not what you want, since it creates
ring homomorphisms.
Since you want an element of a homset in the category of modules over
QQ, let's see first how to create the homset:
sage: H = P.Hom(P, category=Modules(QQ))
sage: H
Set of Morphisms from Univariate Polynomial Ring in x over Rational
Field to Univariate Polynomial Ring in x over Rational Field in
Category of vector spaces over Rational Field
As you can see, P.Hom(...) (in contrast to P.hom) accepts an optional
argument that allows for considering a super category.
Do you know how to obtain the documentation? If not: Simply type
P.Hom? and hit return. If you want to see the source code, type
P.Hom?? and hit return.
> Is there already such a functionality in SAGE? If not, my idea would be
> to create a method "der", similar to "hom", but with multiplication
> defined differently. Would this be a good approach? If so, could someone
> give me some hints where to look for in the source?
I am not aware of general derivations in Sage, but I could be
mistaken. So, let's do a search:
sage: search_src("derivation")
schemes/elliptic_curves/heegner.py:6911: # period lattice.
See the following paper for a derivation:
symbolic/function_factory.py:150: - ``derivative_func`` - method
to be used for (partial) derivation
symbolic/function_factory.py:224: Defining custom methods for
automatic or numeric evaluation, derivation,
symbolic/pynac.pyx:87: Used to pass return values of custom
python evaluation, derivation
symbolic/units.py:883:unit_derivations = {'acceleration':'length/
time^2',
symbolic/units.py:938:def unit_derivations_expr(v):
symbolic/units.py:954: sage:
sage.symbolic.units.unit_derivations_expr('volume')
symbolic/units.py:956: sage:
sage.symbolic.units.unit_derivations_expr('electric_potential')
symbolic/units.py:961: sage:
sage.symbolic.units.unit_derivations_expr('invalid')
symbolic/units.py:967: Z = unit_derivations[v]
symbolic/units.py:972: unit_derivations[v] = Z
symbolic/units.py:1354: if unit_derivations.has_key(str(v)):
symbolic/units.py:1355: base = unit_derivations_expr(v)
tensor/differential_form_element.py:807: The exterior
differential operator is a derivation of degree one
categories/coxeter_groups.py:1116: ``self`` as subword.
See Stembridge, A short derivation of
So, it doesn't look like what you want.
I think a method P.der(images_of_generators) returning a derivation
given by the images of generators would be useful. Of course, such
method needs to return a derivation, which is a morphism. So, I am
showing you how to define a morphism:
sage: from sage.categories.morphism import Morphism
sage: Morphism.__call__?
Apply this map to ``x``.
IMPLEMENTATION:
- To implement the call method in a subclass of Map,
implement
:meth:`_call_` and possibly also :meth:`_call_with_args`
and
:meth:`pushforward`.
...
So, that is an important information: Do NOT override __call__ (double
underscore), but provide a single underscore _call_!
Here is a toy implementation, namely for univariate polynomial rings
only, and without the possibility to map the generator to anything but
1.
sage: class Derivative(Morphism):
....: def __init__(self, P, gen_images):
....: if len(gen_images)!=1:
....: raise ValueError
....: B = P.base_ring()
....: if P not in Modules(B):
....: raise ValueError
....: if gen_images[0]!=B.one_element():
....: raise NotImplementedError
....: Morphism.__init__(self,P.Hom(P, category=Modules(B)))
....: # here, add stuff for
....: # general derivations.
....: # Store the generator images, e.g.
....: def _call_(self, p):
....: # this is just a toy example,
....: # implement more fancy stuff here
....: return derivative(p)
....: def _repr_type(self):
....: # This is to make the derivative print like a derivative
....: return "Derivative"
....:
sage: f = Derivative(P,[1])
sage: f
Derivative endomorphism of Univariate Polynomial Ring in x over
Rational Field
# you may think of a nicer string representation - look at the source
code
# of other morphisms
# f is not a morphism of rings, but of modules:
sage: f.category_for()
Category of vector spaces over Rational Field
# and it maps polynomials as we wanted it:
sage: p = P.random_element()
sage: p
-x^2 + 7/4
sage: f(p)
-2*x
Best regards,
Simon
> PS: There was a similar question on this list posed by Scott on Feb. 7, 2011,
> but there is no answer until now.
Sorry, I missed that.
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