coercion system for nested polynomial ring is not sane

[user202729]

Issue 1: sometimes coerce map is not callable

• done

sage: QQ["a"]["b,c"].coerce_map_from(QQ["a,c,b"])
Call morphism:
  From: Multivariate Polynomial Ring in a, c, b over Rational Field
  To:   Multivariate Polynomial Ring in b, c over Univariate Polynomial Ring in a over Rational Field
sage: QQ["a"]["b,c"].coerce_map_from(QQ["a,c,b"])(QQ["a,c,b"].1)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
...
TypeError: c is not a constant polynomial

I found this while working on #39089 .
The coerce map is reported to exist by Sage only because __call__ is (incorrectly)
overridden by multipolynomial ring implementation (parents are not supposed to override
__call__, they should override _element_constructor_ instead).

Issue 2: sometimes R ← S and S ← T has coerce map, but R ← T has no coerce map

• done

sage: R=QQ["a,b"]["c,d"]
....: S=QQ["a,b,c,d"]
....: T=QQ["a,c,b,d"]
....: U=QQ["a,c"]["b,d"]
sage: R.coerce_map_from(U) is None
True
sage: R.coerce_map_from(S) * S.coerce_map_from(T) * T.coerce_map_from(U)
Composite map:
  From: Multivariate Polynomial Ring in b, d over Multivariate Polynomial Ring in a, c over Rational Field
  To:   Multivariate Polynomial Ring in c, d over Multivariate Polynomial Ring in a, b over Rational Field
  Defn:   Coercion map:
          From: Multivariate Polynomial Ring in b, d over Multivariate Polynomial Ring in a, c over Rational Field
          To:   Multivariate Polynomial Ring in a, c, b, d over Rational Field
        then
          Coercion map:
          From: Multivariate Polynomial Ring in a, c, b, d over Rational Field
          To:   Multivariate Polynomial Ring in a, b, c, d over Rational Field
        then
          Call morphism:
          From: Multivariate Polynomial Ring in a, b, c, d over Rational Field
          To:   Multivariate Polynomial Ring in c, d over Multivariate Polynomial Ring in a, b over Rational Field

Issue 3: nested polynomial ring with coinciding variable name lead to non-commuting coerce map

• done

(
		QQ["a,b"]["a,c"].coerce_map_from(QQ["b"]["a,c"])
		* QQ["b"]["a,c"].coerce_map_from(QQ["a"])
		)(QQ["a"].0) == QQ["a,b"]["a,c"].coerce_map_from(QQ["a"])(QQ["a"].0)

(
		QQ["a,b"][["a,c"]].coerce_map_from(QQ["b"][["a,c"]])
		* QQ["b"][["a,c"]].coerce_map_from(QQ["a"])
		)(QQ["a"].0) == QQ["a,b"][["a,c"]].coerce_map_from(QQ["a"])(QQ["a"].0)

(
		QQ[["a,b"]][["a,c"]].coerce_map_from(QQ[["b"]][["a,c"]])
		* QQ[["b"]][["a,c"]].coerce_map_from(QQ[["a"]])
		)(QQ[["a"]].0) == QQ[["a,b"]][["a,c"]].coerce_map_from(QQ[["a"]])(QQ[["a"]].0)

It looks impossible to resolve this issue, unless we deprecate the construction of nested polynomial/power series/Laurent polynomial/series ring/etc. with same variable names.

It gets worse — what about quotient rings?

sage: R.<a,b>=QQ[]
sage: S=R.quotient(a^2+1)
sage: (S["a,c"].coerce_map_from(QQ["b"]["a,c"]) * QQ["b"]["a,c"].coerce_map_from(QQ["a"]))(QQ["a"].0)
a
sage: (S["a,c"].coerce_map_from(QQ["a"]))(QQ["a"].0)
abar

Issue 4: un-flattening morphism may or may not be coercion map depends on univariate/multivariate case

• done

sage: R = QQ["a"]["b"]
sage: S = R.flattening_morphism().codomain()
sage: R.coerce_map_from(S)
sage: S.coerce_map_from(R)
Coercion map:
  From: Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Rational Field
  To:   Multivariate Polynomial Ring in a, b over Rational Field
sage: R = QQ["a,b"]["c,d"]
sage: S = R.flattening_morphism().codomain()
sage: R.coerce_map_from(S)
Call morphism:
  From: Multivariate Polynomial Ring in a, b, c, d over Rational Field
  To:   Multivariate Polynomial Ring in c, d over Multivariate Polynomial Ring in a, b over Rational
 Field
sage: S.coerce_map_from(R)
Coercion map:
  From: Multivariate Polynomial Ring in c, d over Multivariate Polynomial Ring in a, b over Rational
 Field
  To:   Multivariate Polynomial Ring in a, b, c, d over Rational Field

Environment

• OS: Linux
• Sage Version: latest development source

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide