Thanks for the discussion about this topic. I send this mail to re-iterate
and summarize. It seems there are two things that you might want:
1) Get the coefficient of a specific monomial in the multivariate polynomial
ring.
2) Get the coefficient of the polynomial in a tower of (two) polynomial
extensions
I suggest that MPolynomial.coefficient method perform function #1 above. It
currently does some strange things over the base rings I've tested. It does
vary on base ring due to implementation differences.
I suggest that there be a new method which performs function #2 and I suggest
it be called "polynomial_coefficient", but I really don't like that name. I
almost prefer "coefficient_polynomial" since it will sort beside coefficient
in documentation and help people realize that if they are not happy
with "coefficient" that they have an alternative. Of course, each method's
documentation should probably mention the other.
I'm wondering if we could have a vote on preferred syntax. I'm not going to
describe the parameters because if they are not clear enough from context, it
probably isn't a good parameter choice :) :
sage: P.<v,w,x,y,z>=ZZ[]
sage: f=(1-v)*(1-2*w)*(1-3*x); f
-6*v*w*x + 2*v*w + 3*v*x + 6*w*x - v - 2*w - 3*x + 1
##############
# Alternative Number 1
##############
sage: f.polynomial_coefficient({w:0,v:1})
3*x - 1
##############
# Alternative Number 2
##############
sage: f.polynomial_coefficient([x,y,z],v)
3*x - 1
##############
# Alternative Number 3
# Overloading of coefficient to handle both functionalities
##############
sage: f.coefficient(v,base=ZZ['x,y,z'])
3*x - 1
sage: f.coefficient(v) # here's an example of functionality #1 above
-1
Those are listed in the order of my preference for the interface. It feels to
me that "Alternative Number 3" would not be much fun to implement (and
involve coercions that some view as suspect). I basically
implemented "Alternative Number 1" as a function in my own program and it
feels like a straightforward implementation just iterating through the
polydict. I'd love to have some other voters.
NOTE: It is NOT sufficient to simply provide a monomial expression to
the "polynomial_coefficient" method because it doesn't provide for the case
when you explicitly want an exponent to be 0 -- this point seems to me to
have caused a great deal of confusion in the current implementation.
--
Joel
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