improve conversions from polynomial rings to the base ring

[mantepse]

This ticket is a follow up on #26929. As pointed out there, conversions from univariate quotient rings to subrings of the base ring should be harmonized with those of multivariate quotient rings.

For example, we currently have the following:

sage:  R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2);
sage: ZZ.coerce_map_from(S) is None
True
sage: ZZ.convert_map_from(S)
Conversion via _integer_ method map:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
  To:   Integer Ring

In the univariate case another map gets picked up; one that apparently always fails.

sage: R.<x>=QQ[]; S.<a>=R.quo(x^2+1)
sage: m=ZZ.convert_map_from(S); m
Conversion map:
  From: Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^2 + 1
  To:   Integer Ring

This map simply ends up calling ZZ._element_constructor_(<argument>), which fails.

So the difference seems to be whether an _integer_ method is available on the elements.

(not sure whether this belongs to coercion or categories, or neither.)

[mantepse]

Maybe the following is part of the problem:

sage: R.<x,y> = PolynomialRing(QQ); S = R.quotient(x^2+y^2)
sage: R1.<x> = PolynomialRing(QQ); S1 = R1.quotient(x^2+x)
sage: S1 in Algebras(QQ)
True
sage: S in Algebras(QQ)
False

I don't know yet how (and where) this should be fixed.

[embray]

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[embray]

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[embray]

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