gh-37761: rings/infinite polynomial ring fixes

    
Fix #37756
    
URL: #37761
Reported by: Martin Rubey
Reviewer(s): Martin Rubey, Travis Scrimshaw

src/sage/rings/polynomial/infinite_polynomial_element.py

@@ -552,6 +552,41 @@ def is_nilpotent(self):
         """
         return self._p.is_nilpotent()
 
+    def numerator(self):
+        r"""
+        Return a numerator of ``self``, computed as ``self * self.denominator()``.
+
+        .. WARNING::
+
+           This is not the numerator of the rational function
+           defined by ``self``, which would always be ``self`` since it is a
+           polynomial.
+
+        EXAMPLES::
+
+            sage: X.<x> = InfinitePolynomialRing(QQ)
+            sage: p = 2/3*x[1] + 4/9*x[2] - 2*x[1]*x[3]
+            sage: num = p.numerator(); num
+            -18*x_3*x_1 + 4*x_2 + 6*x_1
+
+        TESTS::
+
+            sage: num.parent()
+            Infinite polynomial ring in x over Rational Field
+
+        Check that :issue:`37756` is fixed::
+
+            sage: R.<a> = InfinitePolynomialRing(QQ)
+            sage: P.<x,y> = QQ[]
+            sage: FF = P.fraction_field()
+            sage: FF(a[0])
+            Traceback (most recent call last):
+            ...
+            TypeError: Could not find a mapping of the passed element to this ring.
+        """
+        P = self.parent()
+        return InfinitePolynomial(P, self._p.numerator())
+
     @cached_method
     def variables(self):
         """
@@ -569,9 +604,69 @@ def variables(self):
             ()
         """
         if hasattr(self._p, 'variables'):
-            return tuple(self._p.variables())
+            P = self.parent()
+            return tuple(InfinitePolynomial(P, v) for v in self._p.variables())
         return ()
 
+    def monomials(self):
+        """
+        Return the list of monomials in ``self``.
+
+        The returned list is decreasingly ordered by the term ordering of
+        ``self.parent()``.
+
+        EXAMPLES::
+
+            sage: X.<x> = InfinitePolynomialRing(QQ)
+            sage: p = x[1]^3 + x[2] - 2*x[1]*x[3]
+            sage: p.monomials()
+            [x_3*x_1, x_2, x_1^3]
+
+            sage: X.<x> = InfinitePolynomialRing(QQ, order='deglex')
+            sage: p = x[1]^3 + x[2] - 2*x[1]*x[3]
+            sage: p.monomials()
+            [x_1^3, x_3*x_1, x_2]
+        """
+        P = self.parent()
+        return [InfinitePolynomial(P, m) for m in self._p.monomials()]
+
+    def monomial_coefficient(self, mon):
+        """
+        Return the base ring element that is the coefficient of ``mon``
+        in ``self``.
+
+        This function contrasts with the function :meth:`coefficient`,
+        which returns the coefficient of a monomial viewing this
+        polynomial in a polynomial ring over a base ring having fewer
+        variables.
+
+        INPUT:
+
+        - ``mon`` -- a monomial in the parent of ``self``
+
+        OUTPUT: coefficient in base ring
+
+        .. SEEALSO::
+
+            For coefficients in a base ring of fewer variables,
+            look at :meth:`coefficient`.
+
+        EXAMPLES::
+
+            sage: X.<x> = InfinitePolynomialRing(QQ)
+            sage: f = 2*x[0]*x[2] + 3*x[1]^2
+            sage: c = f.monomial_coefficient(x[1]^2); c
+            3
+            sage: c.parent()
+            Rational Field
+
+            sage: c = f.coefficient(x[2]); c
+            2*x_0
+            sage: c.parent()
+            Infinite polynomial ring in x over Rational Field
+        """
+        return self._p.monomial_coefficient(mon._p)
+
     @cached_method
     def max_index(self):
         r"""
@@ -997,42 +1092,42 @@ def coefficient(self, monomial):
             sage: a.coefficient({x[0]:1, x[1]:1})
             2
         """
+        P = self.parent()
         if self._p == 0:
-            res = 0
-        elif isinstance(monomial, self.__class__):
-            if not (self.parent().has_coerce_map_from(monomial.parent())):
-                res = 0
+            return P.zero()
+        if isinstance(monomial, self.__class__):
+            if not P.has_coerce_map_from(monomial.parent()):
+                return P.zero()
+            if hasattr(self._p, 'variables'):
+                VarList = [str(X) for X in self._p.variables()]
             else:
-                if hasattr(self._p, 'variables'):
-                    VarList = [str(X) for X in self._p.variables()]
-                else:
-                    VarList = []
-                if hasattr(monomial._p, 'variables'):
-                    VarList.extend([str(X) for X in monomial._p.variables()])
-                VarList = list(set(VarList))
-                VarList.sort(key=self.parent().varname_key, reverse=True)
-                from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-                if len(VarList) == 1:
-                    # 'xx' is guaranteed to be no variable
-                    # name of monomial, since coercions
-                    # were tested before
-                    R = PolynomialRing(self._p.base_ring(), VarList + ['xx'], order=self.parent()._order)
-
-                    res = PolynomialRing(self._p.base_ring(), VarList, order=self.parent()._order)(R(self._p).coefficient(R(monomial._p)))
-                else:
-                    R = PolynomialRing(self._p.base_ring(), VarList, order=self.parent()._order)
-                    res = R(self._p).coefficient(R(monomial._p))
-        elif isinstance(monomial, dict):
+                VarList = []
+            if hasattr(monomial._p, 'variables'):
+                VarList.extend([str(X) for X in monomial._p.variables()])
+            VarList = list(set(VarList))
+            VarList.sort(key=P.varname_key, reverse=True)
+            from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+            if len(VarList) == 1:
+                # 'xx' is guaranteed to be no variable
+                # name of monomial, since coercions
+                # were tested before
+                R = PolynomialRing(self._p.base_ring(), VarList + ['xx'], order=P._order)
+                S = PolynomialRing(self._p.base_ring(), VarList, order=P._order)
+                res = S(R(self._p).coefficient(R(monomial._p)))
+                return InfinitePolynomial(P, res)
+
+            R = PolynomialRing(self._p.base_ring(), VarList, order=P._order)
+            res = R(self._p).coefficient(R(monomial._p))
+            return InfinitePolynomial(P, res)
+
+        if isinstance(monomial, dict):
             if monomial:
                 I = iter(monomial)
                 K = next(I)
                 del monomial[K]
-                res = self.coefficient(K).coefficient(monomial)
-            else:
-                return self
-        else:
-            raise TypeError("Objects of type %s have no coefficients in InfinitePolynomials" % (type(monomial)))
-        return self.parent()(res)
+                return self.coefficient(K).coefficient(monomial)
+            return self
+        raise TypeError("Objects of type %s have no coefficients in InfinitePolynomials" % (type(monomial)))
 
     # Essentials for Buchberger
     def reduce(self, I, tailreduce=False, report=None):

src/sage/rings/polynomial/infinite_polynomial_ring.py

@@ -279,8 +279,8 @@
 class InfinitePolynomialRingFactory(UniqueFactory):
     """
     A factory for creating infinite polynomial ring elements.  It
-    handles making sure that they are unique as well as handling
-    pickling.  For more details, see
+    makes sure that they are unique as well as handling pickling.
+    For more details, see
     :class:`~sage.structure.factory.UniqueFactory` and
     :mod:`~sage.rings.polynomial.infinite_polynomial_ring`.
 
@@ -564,7 +564,7 @@ def __init__(self, parent, start):
 
     def __next__(self):
         """
-        Return a dictionary that can be used to interprete strings in the base ring of ``self``.
+        Return a dictionary that can be used to interpret strings in the base ring of ``self``.
 
         EXAMPLES::
 
@@ -701,7 +701,7 @@ def __init__(self, R, names, order):
             names = ['x']
         for n in names:
             if not (isinstance(n, str) and n.isalnum() and (not n[0].isdigit())):
-                raise ValueError("generator names must be alpha-numeric strings not starting with a  digit, but %s is not" % n)
+                raise ValueError("generator names must be alphanumeric strings not starting with a digit, but %s is not" % n)
         if len(names) != len(set(names)):
             raise ValueError("generator names must be pairwise different")
         self._names = tuple(names)
@@ -884,6 +884,17 @@ def _element_constructor_(self, x):
             Traceback (most recent call last):
             ...
             ValueError: cannot convert 1/3 into an element of Infinite polynomial ring in x over Integer Ring
+
+        .. WARNING::
+
+            The :issue:`37756` is not yet fixed::
+
+                sage: L.<x, y> = QQ[]
+                sage: R.<a> = InfinitePolynomialRing(QQ)
+                sage: M = InfinitePolynomialRing(L, names=["a"])
+                sage: c = a[0]
+                sage: M(c)  # known bug
+                a_0
         """
         from sage.rings.polynomial.infinite_polynomial_element import InfinitePolynomial
         # In many cases, the easiest solution is to "simply" evaluate