gh-38767: polynomials/fix category
    
We put polynomials (and friends) into the category of algebras with
basis.

Dependencies: #38729
    
URL: #38767
Reported by: Martin Rubey
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/categories/additive_magmas.py

@@ -971,8 +971,8 @@ def extra_super_categories(self):
                     [Category of unital magmas]
 
                     sage: C.super_categories()
-                    [Category of unital algebras with basis over Rational Field,
-                     Category of additive magma algebras over Rational Field]
+                    [Category of additive magma algebras over Rational Field,
+                     Category of unital algebras with basis over Rational Field]
                 """
                 from sage.categories.magmas import Magmas
                 return [Magmas().Unital()]

src/sage/categories/algebra_functor.py

@@ -128,7 +128,6 @@
      Category of semigroup algebras over Rational Field,
      ...
      Category of unital magma algebras over Rational Field,
-     ...
      Category of magma algebras over Rational Field,
      ...
      Category of set algebras over Rational Field,

src/sage/categories/map.pyx

@@ -655,9 +655,11 @@ cdef class Map(Element):
             sage: f = R.hom([x+y, x-y], R)
             sage: f.category_for()
             Join of Category of unique factorization domains
-            and Category of commutative algebras
-            over (number fields and quotient fields and metric spaces)
-            and Category of infinite sets
+             and Category of algebras with basis over
+              (number fields and quotient fields and metric spaces)
+             and Category of commutative algebras over
+              (number fields and quotient fields and metric spaces)
+             and Category of infinite sets
             sage: f.category()
             Category of endsets of unital magmas
              and right modules over (number fields and quotient fields and metric spaces)

src/sage/categories/modules.py

@@ -893,7 +893,7 @@ def __init_extra__(self):
                      (Vector space of dimension 2 over Rational Field,
                       Univariate Polynomial Ring in x over Rational Field)
                     sage: A.category()                                                  # needs sage.modules
-                    Category of Cartesian products of vector spaces
+                    Category of Cartesian products of vector spaces with basis
                      over (number fields and quotient fields and metric spaces)
                     sage: A.base_ring()                                                 # needs sage.modules
                     Rational Field

src/sage/interfaces/tab_completion.py

@@ -71,7 +71,7 @@ def completions(s, globs):
          sage: import sage.interfaces.tab_completion as s
          sage: p = x**2 + 1
          sage: s.completions('p.co',globals()) # indirect doctest
-         ['p.coefficients',...]
+         ['p.coefficient',...]
 
          sage: s.completions('dic',globals()) # indirect doctest
          ['dickman_rho', 'dict']

src/sage/rings/lazy_series_ring.py

@@ -1488,10 +1488,14 @@ def __init__(self, base_ring, names, sparse=True, category=None):
             sage: TestSuite(L).run()                                                    # needs sage.libs.singular
             sage: L.category()
             Category of infinite commutative no zero divisors algebras over
-             (unique factorization domains and commutative algebras over
+             (unique factorization domains and algebras with basis over
               (Dedekind domains and euclidean domains
-              and noetherian rings
-              and infinite enumerated sets and metric spaces)
+               and noetherian rings
+               and infinite enumerated sets and metric spaces)
+              and commutative algebras over
+               (Dedekind domains and euclidean domains
+                and noetherian rings
+                and infinite enumerated sets and metric spaces)
               and infinite sets)
 
             sage: L = LazyLaurentSeriesRing(GF(5), 't')

src/sage/rings/polynomial/laurent_polynomial_ring.py

@@ -445,6 +445,8 @@ def __init__(self, R):
         if R.ngens() != 1:
             raise ValueError("must be 1 generator")
         LaurentPolynomialRing_generic.__init__(self, R)
+        from sage.rings.integer_ring import IntegerRing
+        self._indices = IntegerRing()
 
     Element = LaurentPolynomial_univariate
 
@@ -561,6 +563,12 @@ def _element_constructor_(self, x):
 
         return self.element_class(self, x)
 
+    def monomial(self, arg):
+        r"""
+        Return the monomial with the given exponent.
+        """
+        return self.element_class(self, {arg: self.base_ring().one()})
+
     def __reduce__(self):
         """
         Used in pickling.
@@ -591,6 +599,9 @@ def __init__(self, R):
         if not R.base_ring().is_integral_domain():
             raise ValueError("base ring must be an integral domain")
         LaurentPolynomialRing_generic.__init__(self, R)
+        from sage.modules.free_module import FreeModule
+        from sage.rings.integer_ring import IntegerRing
+        self._indices = FreeModule(IntegerRing(), R.ngens())
 
     Element = LazyImport('sage.rings.polynomial.laurent_polynomial_mpair', 'LaurentPolynomial_mpair')
 
@@ -605,7 +616,7 @@ def _repr_(self):
         """
         return "Multivariate Laurent Polynomial Ring in %s over %s" % (", ".join(self._R.variable_names()), self._R.base_ring())
 
-    def monomial(self, *args):
+    def monomial(self, *exponents):
         r"""
         Return the monomial whose exponents are given in argument.
 
@@ -629,14 +640,27 @@ def monomial(self, *args):
             sage: L.monomial(1, 2, 3)                                                   # needs sage.modules
             Traceback (most recent call last):
             ...
-            TypeError: tuple key must have same length as ngens
-        """
-        if len(args) != self.ngens():
-            raise TypeError("tuple key must have same length as ngens")
+            TypeError: tuple key (1, 2, 3) must have same length as ngens (= 2)
+
+        We also allow to specify the exponents in a single tuple::
+
+            sage: L.monomial((-1, 2))                                                   # needs sage.modules
+            x0^-1*x1^2
 
+            sage: L.monomial((-1, 2, 3))                                                # needs sage.modules
+            Traceback (most recent call last):
+            ...
+            TypeError: tuple key (-1, 2, 3) must have same length as ngens (= 2)
+        """
         from sage.rings.polynomial.polydict import ETuple
-        m = ETuple(args, int(self.ngens()))
-        return self.element_class(self, self.polynomial_ring().one(), m)
+        if len(exponents) == 1 and isinstance((e := exponents[0]), (tuple, ETuple)):
+            exponents = e
+
+        if len(exponents) != self.ngens():
+            raise TypeError(f"tuple key {exponents} must have same length as ngens (= {self.ngens()})")
+
+        m = ETuple(exponents, int(self.ngens()))
+        return self.element_class(self, self.polynomial_ring().base_ring().one(), m)
 
     def _element_constructor_(self, x, mon=None):
         """

src/sage/rings/polynomial/laurent_polynomial_ring_base.py

@@ -42,6 +42,8 @@ class LaurentPolynomialRing_generic(CommutativeRing, Parent):
         sage: R.<x1,x2> = LaurentPolynomialRing(QQ)
         sage: R.category()
         Join of Category of unique factorization domains
+            and Category of algebras with basis
+                over (number fields and quotient fields and metric spaces)
             and Category of commutative algebras
                 over (number fields and quotient fields and metric spaces)
             and Category of infinite sets

src/sage/rings/polynomial/multi_polynomial_ring_base.pxd

@@ -4,6 +4,7 @@ from sage.structure.parent cimport Parent
 cdef class MPolynomialRing_base(CommutativeRing):
     cdef object _ngens
     cdef object _term_order
+    cdef public object _indices
     cdef public object _has_singular
     cdef public object _magma_gens
     cdef public dict _magma_cache

src/sage/rings/polynomial/multi_polynomial_ring_base.pyx

@@ -99,6 +99,8 @@ cdef class MPolynomialRing_base(CommutativeRing):
         else:
             category = polynomial_default_category(base_ring.category(), n)
         Ring.__init__(self, base_ring, names, category=category)
+        from sage.combinat.integer_vector import IntegerVectors
+        self._indices = IntegerVectors(self._ngens)
 
     def is_integral_domain(self, proof=True):
         """