Dear all,
in https://github.com/sagemath/sage/pull/38108, which provides a solver for
functional equations in lazy completions of graded algebras with basis, I
(want to) use generic methods, so that I do not have to write special code
for every other algebra. Here is an example which I find a bit depressing:
sage: s = SymmetricFunctions(QQ).s()
sage: f = (s[1,1] + s[2])^2
sage: f.monomial_coefficients()
{[2, 1, 1]: 3, [2, 2]: 2, [1, 1, 1, 1]: 1, [3, 1]: 3, [4]: 1}
sage: list(f)
[([2, 1, 1], 3), ([2, 2], 2), ([1, 1, 1, 1], 1), ([3, 1], 3), ([4], 1)]
sage: s.monomial(Partition([2,1,1]))
s[2, 1, 1]
This works for algebras subclassing CombinatorialFreeModule, for example
symmetric functions and the free algebra.
However, it will not work for polynomials:
sage: R.<x,y> = QQ[]
sage: f = (x+2*y)^2
sage: list(f)
[(1, x^2), (4, x*y), (4, y^2)]
sage: R.monomial(0,2)
y^2
sage: T.<x> = QQ[]
sage: t = (x+1)^2
sage: list(t)
[1, 2, 1]
Note that
* coefficient_monomials does not exist
* list gives pairs in the multivariate case, but the coefficient comes
first, and a flat list in the univariate case
* monomial takes one argument for each variable, and does not accept tuples.
My question is: do we want to keep this discrepancy forever, or should we
work on resolving it.
I see the following first steps:
* slightly generalize MPolynomialRing_base.monomial and to accept also a
single tuple of the correct length.
* introduce a method `monomial_coefficients` which produces a dict from
exponents to coefficients.
Any opinions?
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