coefficients() function more consistent between Expressions and polynomial rings

[sagetrac-JoalHeagney]

The different behaviour between the two rings consists of

1. the coefficients(sparse=True) (which is default) method returns a list of pairs in SR, and a list in PolynomialRing,
2. Expression.dict() does not exist.
   Example:

y = 3*x^3 + 2*x^2 - 4*x
print(y)
type(y)

Gives:

3*x^3 + 2*x^2 - 4*x
<type 'sage.symbolic.expression.Expression'>

And

M = matrix(SR,[[1,2],[0,-2]])
ch = M.charpoly()
print(ch)
type(ch)

gives

x^2 + x - 2
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'>

But:

y.coeffs()

returns

[[−4,1],[2,2],[3,3]]

and

ch.coeffs()

returns

[−2,1,1]

I'd prefer if these two functions returned the same format, preferably the Expression format, as having access to the index allows list comprehension tastiness.

Component: algebra

Keywords: coeffs, rings, polynomials, expression, symbolic

Issue created by migration from https://trac.sagemath.org/ticket/17922

[sagetrac-JoalHeagney]

Changed keywords from coeffs to coeffs, rings, polynomials, expression, symbolic

[rwst]

comment:2

Have list comprehension tastiness with dict:

sage: ch.dict().values()
[-2, 1, 1]

In #17518 we started being more consistent by deprecating coeffs.

[sagetrac-JoalHeagney]

comment:3

That works for polynomial rings, are there plans to add that to sage.symbolic.expression.Expressions?

[rwst]

comment:4

Clarified the ticket description.

[rwst]

Description changed:

--- 
+++ 
@@ -1,3 +1,7 @@
+The different behaviour between the two rings consists of
+1. the `coefficients(sparse=True)` (which is default) method returns a list of pairs in `SR`, and a list in `PolynomialRing`,
+2. `Expression.dict()` does not exist.
+Example:
 
 ```
 y = 3*x^3 + 2*x^2 - 4*x

[rwst]

comment:5

Replying to @sagetrac-JoalHeagney:

That works for polynomial rings, are there plans to add that to sage.symbolic.expression.Expressions?

That would not be difficult (in comparison). If you expect it then to behave identically you will be disappointed however, because symbolics have no default generator (although Expression.coefficients() has the lexically first occuring var hard-wired when no varname is given).

sage: var('a,b,c')
(a, b, c)
sage: (a+2*a^2+3*b).list()
[3*b, 1, 2]
sage: (3*b+a+2*a^2).list()
[3*b, 1, 2]
sage: (3*b+c+2*c^2).list()
[2*c^2 + c, 3]

so you could never have true polymorphism.