silently wrong result in coercion between InfinitePolynomialRing and LazyPowerSeriesRing

[mantepse]

Steps To Reproduce

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: R.<a> = InfinitePolynomialRing(QQ)
sage: o = L(0); r = a[0]
sage: o + r
y
sage: _.parent()
Infinite polynomial ring in a over Multivariate Lazy Taylor Series Ring in x, y over Rational Field

Expected Behavior

The result should either be a[0] (ideally with parent LazyPowerSeriesRing(InfinitePolynomialRing(QQ))) or an error should be raised.

Actual Behavior

A nonsense result,

Additional Information

The parent is explained as follows:

sage: cm.explain(o, r, op="add")
Coercion on left operand via
    Generic morphism:
      From: Multivariate Lazy Taylor Series Ring in x, y over Rational Field
      To:   Infinite polynomial ring in a over Multivariate Lazy Taylor Series Ring in x, y over Rational Field
Coercion on right operand via
    Coercion map:
      From: Infinite polynomial ring in a over Rational Field
      To:   Infinite polynomial ring in a over Multivariate Lazy Taylor Series Ring in x, y over Rational Field
Arithmetic performed after coercions.
Result lives in Infinite polynomial ring in a over Multivariate Lazy Taylor Series Ring in x, y over Rational Field
Infinite polynomial ring in a over Multivariate Lazy Taylor Series Ring in x, y over Rational Field

It is unclear to me, why we would want to have

sage: L.construction()[0]
Completion[('x', 'y'), prec=+Infinity]
sage: _.rank
4
sage: R.construction()[0]
InfPoly{[a], "lex", "dense"}
sage: _.rank
9.5

[mantepse]

In the end, the bug boils down to:

sage: L(a[0])
y

and within L._element_constructor_, this happens because

sage: P.<x,y> = QQ[]
sage: FF = P.fraction_field()
sage: FF(a[0])
y

[mantepse]

@nbruin, help greatly appreciated, of course.

[mantepse]

Making InfinitePolynomial_sparse.numerator() and denominator return elements of the InfinitePolynomialRing helps, but is not quite enough.

The InfinitePolynomialRing_sparse._element_constructor tries conversion to self._base (which is L below) of x._p (which is a_0 as an element of the multivariate polynomial ring), and this succeeds. To illustrate:

    sage: L.<x, y> = LazyPowerSeriesRing(QQ)
    sage: R.<a> = InfinitePolynomialRing(QQ)
    sage: M = InfinitePolynomialRing(L, names=["a"])
    sage: M(a[0])
    y

I'm not sure what to do about that. Note that this is exactly the coercion from R to M discovered.

[mantepse]

I forgot to realize (:-) that this has little to do with the LazyPowerSeriesRing:

sage: L.<x, y> = QQ[]
sage: R.<a> = InfinitePolynomialRing(QQ)
sage: M = InfinitePolynomialRing(L, names=["a"])
sage: c = a[0]
sage: M(c)
y

[mantepse]

I think the patch below might do the trick. I now get:

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: R.<a> = InfinitePolynomialRing(QQ)
sage: o = L(0); r = a[0]
sage: o + r
a_0
sage: _.parent()
Infinite polynomial ring in a over Multivariate Lazy Taylor Series Ring in x, y over Rational Field

However, I get an interesting error message here:

sage: sage: R.<a> = InfinitePolynomialRing(QQ)
sage: P.<x,y> = QQ[]
sage: FF = P.fraction_field()
sage: FF(a[0])  # correctly fails after providing numerator and denominator
---------------------------------------------------------------------------
...
RuntimeError: There is a bug in the coercion code in Sage.
Both x (=1) and y (=1) are supposed to have identical parents but they don't.
In fact, x has parent 'Infinite polynomial ring in a over Multivariate Polynomial Ring in x, y over Rational Field'
whereas y has parent 'Integer Ring'
Original elements 1 (parent Multivariate Polynomial Ring in x, y over Rational Field) and 1 (parent Infinite polynomial ring in a over Rational Field) and maps
<class 'sage.categories.morphism.SetMorphism'> (map internal to coercion system -- copy before use)
Generic morphism:
  From: Multivariate Polynomial Ring in x, y over Rational Field
  To:   Infinite polynomial ring in a over Multivariate Polynomial Ring in x, y over Rational Field
<class 'sage.structure.coerce_maps.DefaultConvertMap_unique'> (map internal to coercion system -- copy before use)
Coercion map:
  From: Infinite polynomial ring in a over Rational Field
  To:   Infinite polynomial ring in a over Multivariate Polynomial Ring in x, y over Rational Field

Any ideas what's going on?

diff --git a/src/sage/rings/polynomial/infinite_polynomial_element.py b/src/sage/rings/polynomial/infinite_polynomial_element.py
index 76bb926c17..87225b4804 100644
--- a/src/sage/rings/polynomial/infinite_polynomial_element.py
+++ b/src/sage/rings/polynomial/infinite_polynomial_element.py
@@ -560,6 +560,14 @@ class InfinitePolynomial(CommutativePolynomial, metaclass=InheritComparisonClass
         """
         return self._p.is_nilpotent()
 
+    def numerator(self):
+        P = self.parent()
+        return InfinitePolynomial(P, self._p.numerator())
+
+    def denominator(self):
+        P = self.parent()
+        return InfinitePolynomial(P, self._p.denominator())
+
     @cached_method
     def variables(self):
         """
diff --git a/src/sage/rings/polynomial/infinite_polynomial_ring.py b/src/sage/rings/polynomial/infinite_polynomial_ring.py
index a23b038731..169978acf9 100644
--- a/src/sage/rings/polynomial/infinite_polynomial_ring.py
+++ b/src/sage/rings/polynomial/infinite_polynomial_ring.py
@@ -922,7 +922,7 @@ class InfinitePolynomialRing_sparse(CommutativeRing):
             if isinstance(self._base, MPolynomialRing_polydict):
                 x = sage_eval(repr(), next(self.gens_dict()))
             else:
-                x = self._base(x)
+                x = self._base.coerce(x)
             # remark: Conversion to self._P (if applicable)
             # is done in InfinitePolynomial()
             return InfinitePolynomial(self, x)