gh-38729: provide monomial_coefficients for polynomials and allow sin…

…gle argument for MPolynomialRing_base.monomial

    
The goal of this pull request is to allow generic programming with
instances of `CombinatorialFreeModule` and polynomial rings.

In particular, we provide an alias `monomial_coefficients` for the
existing method `dict` of polynomial rings, and we allow to specify the
exponents in `monomial` of multivariate polynomial rings as a single
tuple.
    
URL: #38729
Reported by: Martin Rubey
Reviewer(s): Frédéric Chapoton, Martin Rubey, Travis Scrimshaw

src/sage/algebras/cluster_algebra.py

@@ -462,7 +462,7 @@ def d_vector(self) -> tuple:
             sage: x.d_vector()
             (1, 1, 2, -2)
         """
-        monomials = self.lift().dict()
+        monomials = self.lift().monomial_coefficients()
         minimal = map(min, zip(*monomials))
         return tuple(-vector(minimal))[:self.parent().rank()]
 
@@ -615,8 +615,9 @@ def theta_basis_decomposition(self):
             f_poly = components[g_vect].F_polynomial()
             g_vect = vector(g_vect)
             while f_poly != zero_U:
-                y_exp = min(f_poly.dict())
-                coeff = f_poly.dict()[y_exp]
+                coeffs = f_poly.monomial_coefficients()
+                y_exp = min(coeffs)
+                coeff = coeffs[y_exp]
                 g_theta = tuple(g_vect + B * vector(y_exp))
                 out[g_theta] = out.get(g_theta, zero_A) + A({zero_t + tuple(y_exp): coeff})
                 f_poly -= U({y_exp: coeff}) * A.theta_basis_F_polynomial(g_theta)

src/sage/algebras/commutative_dga.py

@@ -222,7 +222,7 @@ def image_monomial(exponent):
             return A.zero()
 
         for g in I.gens():
-            d = g.dict()
+            d = g.monomial_coefficients()
             res = A.sum(d[ex] * image_monomial(ex) for ex in d)
             if not res.is_zero():
                 raise ValueError("the differential does not preserve the ideal")
@@ -303,7 +303,7 @@ def _call_(self, x):
         if x.is_zero():
             return self.codomain().zero()
         res = self.codomain().zero()
-        dic = x.dict()
+        dic = x.monomial_coefficients()
         for key in dic:
             keyl = list(key)
             coef = dic[key]
@@ -392,11 +392,11 @@ def differential_matrix(self, n):
         A = self.domain()
         dom = A.basis(n)
         cod = A.basis(n + 1)
-        cokeys = [next(iter(a.lift().dict().keys())) for a in cod]
+        cokeys = [next(iter(a.lift().monomial_coefficients().keys())) for a in cod]
         m = matrix(A.base_ring(), len(dom), len(cod))
         for i, domi in enumerate(dom):
             im = self(domi)
-            dic = im.lift().dict()
+            dic = im.lift().monomial_coefficients()
             for j in dic.keys():
                 k = cokeys.index(j)
                 m[i, k] = dic[j]
@@ -670,11 +670,11 @@ def differential_matrix_multigraded(self, n, total=False):
         n = G(vector(n))
         dom = A.basis(n)
         cod = A.basis(n + self._degree_of_differential)
-        cokeys = [next(iter(a.lift().dict().keys())) for a in cod]
+        cokeys = [next(iter(a.lift().monomial_coefficients().keys())) for a in cod]
         m = matrix(self.base_ring(), len(dom), len(cod))
         for i, domi in enumerate(dom):
             im = self(domi)
-            dic = im.lift().dict()
+            dic = im.lift().monomial_coefficients()
             for j in dic.keys():
                 k = cokeys.index(j)
                 m[i, k] = dic[j]
@@ -1182,7 +1182,7 @@ def basis(self, n):
         basis = []
         for v in free_basis:
             el = prod([self.gen(i)**v[i] for i in range(len(v))])
-            di = el.dict()
+            di = el.monomial_coefficients()
             if len(di) == 1:
                 k, = di.keys()
                 if tuple(k) == v:
@@ -1477,7 +1477,7 @@ def degree(self, total=False):
             """
             if self.is_zero():
                 raise ValueError("the zero element does not have a well-defined degree")
-            exps = self.lift().dict().keys()
+            exps = self.monomial_coefficients().keys()
             degrees = self.parent()._degrees
             n = self.parent().ngens()
             l = [sum(e[i] * degrees[i] for i in range(n)) for e in exps]
@@ -1548,7 +1548,7 @@ def homogeneous_parts(self):
                 sage: a.homogeneous_parts()
                 {1: -2*e3 + e5, 2: e1*e2, 3: e1*e3*e5 - 3*e2*e3*e5}
             """
-            dic = self.dict()
+            dic = self.monomial_coefficients()
             terms = [self.parent()({t: dic[t]}) for t in dic.keys()]
             res = {}
             for term in terms:
@@ -1559,7 +1559,7 @@ def homogeneous_parts(self):
                     res[deg] = term
             return {i: res[i] for i in sorted(res.keys())}
 
-        def dict(self):
+        def monomial_coefficients(self):
             r"""
             A dictionary that determines the element.
 
@@ -1569,11 +1569,18 @@ def dict(self):
             EXAMPLES::
 
                 sage: A.<x,y,z,t> = GradedCommutativeAlgebra(QQ, degrees=(1, 2, 2, 3))
-                sage: dic = (x*y - 5*y*z + 7*x*y^2*z^3*t).dict()
-                sage: sorted(dic.items())
+                sage: elt = x*y - 5*y*z + 7*x*y^2*z^3*t
+                sage: sorted(elt.monomial_coefficients().items())
+                [((0, 1, 1, 0), -5), ((1, 1, 0, 0), 1), ((1, 2, 3, 1), 7)]
+
+            ``dict`` is an alias::
+
+                sage: sorted(elt.dict().items())
                 [((0, 1, 1, 0), -5), ((1, 1, 0, 0), 1), ((1, 2, 3, 1), 7)]
             """
-            return self.lift().dict()
+            return self.lift().monomial_coefficients()
+
+        dict = monomial_coefficients
 
         def __call__(self, *values, **kwargs):
             r"""
@@ -1645,8 +1652,8 @@ def __call__(self, *values, **kwargs):
                     if gstr in kwargs:
                         images[i] = kwargs[gstr]
             res = 0
-            for (m, c) in self.dict().items():
-                term = prod((gen**y for (y, gen) in zip(m, images)), c)
+            for m, c in self.monomial_coefficients().items():
+                term = prod((gen ** y for y, gen in zip(m, images)), c)
                 res += term
             return res
 
@@ -2000,7 +2007,7 @@ def degree(self, total=False):
                 raise ValueError("the zero element does not have a well-defined degree")
             degrees = self.parent()._degrees_multi
             n = self.parent().ngens()
-            exps = self.lift().dict().keys()
+            exps = self.monomial_coefficients().keys()
             l = [sum(exp[i] * degrees[i] for i in range(n)) for exp in exps]
             if len(set(l)) == 1:
                 return l[0]
@@ -3861,7 +3868,7 @@ def _call_(self, x):
         """
         codomain = self.codomain()
         result = codomain.zero()
-        for mono, coeff in x.dict().items():
+        for mono, coeff in x.monomial_coefficients().items():
             term = prod([gen**y for (y, gen) in zip(mono, self.im_gens())],
                         codomain.one())
             result += coeff * term

src/sage/algebras/free_algebra.py

@@ -651,7 +651,7 @@ def exp_to_monomial(T):
                     return M([(i % ngens, Ti) for i, Ti in enumerate(T) if Ti])
 
                 return self.element_class(self, {exp_to_monomial(T): c
-                                                 for T, c in x.letterplace_polynomial().dict().items()})
+                                                 for T, c in x.letterplace_polynomial().monomial_coefficients().items()})
         # ok, not a free algebra element (or should not be viewed as one).
         if isinstance(x, str):
             from sage.misc.sage_eval import sage_eval

src/sage/algebras/fusion_rings/fast_parallel_fmats_methods.pyx

@@ -273,7 +273,7 @@ cdef get_reduced_hexagons(factory, tuple mp_params):
         if i % n_proc == child_id:
             he = req_cy(basis, r_matrix, fvars, _Nk_ij, id_anyon, sextuple)
             if he:
-                red = reduce_poly_dict(he.dict(), _nnz, _ks, one)
+                red = reduce_poly_dict(he.monomial_coefficients(), _nnz, _ks, one)
 
                 # Avoid pickling cyclotomic coefficients
                 red = _flatten_coeffs(red)
@@ -341,7 +341,7 @@ cdef get_reduced_pentagons(factory, tuple mp_params):
         if i % n_proc == child_id:
             pe = feq_cy(basis, fvars, _Nk_ij, id_anyon, zero, nonuple, prune=True)
             if pe:
-                red = reduce_poly_dict(pe.dict(), _nnz, _ks, one)
+                red = reduce_poly_dict(pe.monomial_coefficients(), _nnz, _ks, one)
 
                 # Avoid pickling cyclotomic coefficients
                 red = _flatten_coeffs(red)

src/sage/algebras/fusion_rings/poly_tup_engine.pyx

@@ -26,7 +26,7 @@ cpdef inline tuple poly_to_tup(MPolynomial_libsingular poly):
         sage: poly_to_tup(x**2*y**4 - 4/5*x*y**2 + 1/3 * y)
         (((2, 4), 1), ((1, 2), -4/5), ((0, 1), 1/3))
     """
-    return tuple(poly.dict().items())
+    return tuple(poly.monomial_coefficients().items())
 
 cpdef inline MPolynomial_libsingular _tup_to_poly(tuple eq_tup, MPolynomialRing_libsingular parent):
     r"""

src/sage/algebras/hecke_algebras/cubic_hecke_base_ring.py

@@ -148,7 +148,7 @@ def _act_(self, perm, pol):
         if not self.is_left():
             perm, pol = pol, perm
         pol_dict = {}
-        for key, value in pol.dict().items():
+        for key, value in pol.monomial_coefficients().items():
             newkey = [0] * len(key)
             for pos, k in enumerate(key):
                 newkey[perm(pos + 1) - 1] = k

src/sage/algebras/iwahori_hecke_algebra.py

@@ -83,7 +83,7 @@ def normalized_laurent_polynomial(R, p):
         u + v^-1 + u^-1
     """
     try:
-        return R({k: R._base(c) for k, c in p.dict().items()})
+        return R({k: R._base(c) for k, c in p.monomial_coefficients().items()})
     except (AttributeError, TypeError):
         return R(p)
 

src/sage/algebras/letterplace/free_algebra_element_letterplace.pyx

@@ -139,8 +139,8 @@ cdef class FreeAlgebraElement_letterplace(AlgebraElement):
             sage: sorted(p)   # indirect doctest
             [((0, 0, 0, 1, 0, 0, 0, 1), 2), ((0, 1, 0, 0, 0, 0, 1, 0), 1)]
         """
-        cdef dict d = self._poly.dict()
-        yield from d.iteritems()
+        cdef dict d = self._poly.monomial_coefficients()
+        yield from d.items()
 
     def _repr_(self):
         """

src/sage/algebras/quantum_clifford.py

@@ -533,7 +533,7 @@ def product_on_basis(self, m1, m2):
         poly *= self._w_poly.monomial(*v)
         poly = poly.reduce([vp[i]**(4*k) - (1 + q**(-2*k)) * vp[i]**(2*k) + q**(-2*k)
                             for i in range(self._n)])
-        pdict = poly.dict()
+        pdict = poly.monomial_coefficients()
         ret = {(self._psi(p), tuple(e)): pdict[e] * q**q_power * sign
                for e in pdict}
 
@@ -610,7 +610,7 @@ def inverse(self):
                               for wi in wp})
             poly = poly.reduce([wi**(4*k) - (1 + q**(-2*k)) * wi**(2*k) + q**(-2*k)
                                 for wi in wp])
-            pdict = poly.dict()
+            pdict = poly.monomial_coefficients()
             coeff = coeff.inverse_of_unit()
             ret = {(p, tuple(e)): coeff * c for e, c in pdict.items()}
             return Cl.element_class(Cl, ret)

src/sage/algebras/rational_cherednik_algebra.py

@@ -328,11 +328,11 @@ def product_on_basis(self, left, right):
         def commute_w_hd(w, al): # al is given as a dictionary
             ret = P.one()
             for k in al:
-                x = sum(c * gens_dict[i] for i,c in alpha[k].weyl_action(w))
+                x = sum(c * gens_dict[i] for i, c in alpha[k].weyl_action(w))
                 ret *= x**al[k]
-            ret = ret.dict()
+            ret = ret.monomial_coefficients()
             for k in ret:
-                yield (self._hd({I[i]: e for i,e in enumerate(k) if e != 0}), ret[k])
+                yield (self._hd({I[i]: e for i, e in enumerate(k) if e != 0}), ret[k])
 
         # Do Lac Ra if they are both non-trivial
         if dl and dr:
@@ -374,7 +374,7 @@ def commute_w_hd(w, al): # al is given as a dictionary
                         for i,c in alphacheck[k].weyl_action(right[1].reduced_word(),
                                                              inverse=True))
                 ret *= x**dl[k]
-            ret = ret.dict()
+            ret = ret.monomial_coefficients()
             w = left[1]*right[1]
             return self._from_dict({(left[0], w,
                                       self._h({I[i]: e for i,e in enumerate(k)

src/sage/algebras/splitting_algebra.py

@@ -102,19 +102,26 @@ def is_unit(self):
 
         return super().is_unit()
 
-    def dict(self):
+    def monomial_coefficients(self):
         r"""
         Return the dictionary of ``self`` according to its lift to the cover.
 
         EXAMPLES::
 
             sage: from sage.algebras.splitting_algebra import SplittingAlgebra
             sage: CR3.<e3> = SplittingAlgebra(cyclotomic_polynomial(3))
-            sage: (e3 + 42).dict()
+            sage: f = e3 + 42
+            sage: f.monomial_coefficients()
+            {0: 42, 1: 1}
+
+        ``dict`` is an alias::
+
+            sage: f.dict()
             {0: 42, 1: 1}
         """
-        return self.lift().dict()
+        return self.lift().monomial_coefficients()
 
+    dict = monomial_coefficients
 
 # ------------------------------------------------------------------------------------------------------------------
 # Parent class of the splitting algebra
@@ -282,7 +289,7 @@ def __init__(self, monic_polynomial, names='X', iterate=True, warning=True):
             root_names_reduces.remove(root_name)
 
             P = base_ring_step[root_names_reduces[0]]
-            p = P(monic_polynomial.dict())
+            p = P(monic_polynomial.monomial_coefficients())
             q, _ = p.quo_rem(P.gen() - first_root)
 
             verbose("Invoking recursion with: %s" % (q,))

src/sage/algebras/weyl_algebra.py

@@ -786,7 +786,7 @@ def _element_constructor_(self, x):
             return self.element_class(self, {i: R(c) for i, c in x if R(c) != zero})
         x = self._poly_ring(x)
         return self.element_class(self, {(tuple(m), t): c
-                                         for m, c in x.dict().items()})
+                                         for m, c in x.monomial_coefficients().items()})
 
     def _coerce_map_from_(self, R):
         """

src/sage/combinat/ncsf_qsym/qsym.py

@@ -697,7 +697,7 @@ def from_polynomial(self, f, check=True):
             -F[1, 1] + F[2]
         """
         assert self.base_ring() == f.base_ring()
-        exponent_coefficient = f.dict()
+        exponent_coefficient = f.monomial_coefficients()
         z = {}
         for e, c in exponent_coefficient.items():
             I = Compositions()([ei for ei in e if ei])

src/sage/combinat/root_system/root_lattice_realization_algebras.py

@@ -121,7 +121,7 @@ def from_polynomial(self, p):
             """
             L = self.basis().keys()
             return self.sum_of_terms((L.from_vector(vector(t)), c)
-                                     for (t,c) in p.dict().items())
+                                     for t, c in p.monomial_coefficients().items())
 
         @cached_method
         def divided_difference_on_basis(self, weight, i):

src/sage/combinat/sf/monomial.py

@@ -135,21 +135,25 @@ def product(self, left, right):
         return self._from_dict(z_elt)
 
     def from_polynomial(self, f, check=True):
-        """
-        Return the symmetric function in the monomial basis corresponding to the polynomial ``f``.
+        r"""
+        Return the symmetric function in the monomial basis corresponding
+        to the polynomial ``f``.
 
         INPUT:
 
         - ``self`` -- a monomial symmetric function basis
-        - ``f`` -- a polynomial in finitely many variables over the same base ring as ``self``;
-          it is assumed that this polynomial is symmetric
-        - ``check`` -- boolean (default: ``True``); checks whether the polynomial is indeed symmetric
+        - ``f`` -- a polynomial in finitely many variables over the
+          same base ring as ``self``; it is assumed that this
+          polynomial is symmetric
+        - ``check`` -- boolean (default: ``True``); checks whether
+          the polynomial is indeed symmetric
 
         OUTPUT:
 
-        - This function converts a symmetric polynomial `f` in a polynomial ring in finitely
-          many variables to a symmetric function in the monomial
-          basis of the ring of symmetric functions over the same base ring.
+        - This function converts a symmetric polynomial `f` in a
+          polynomial ring in finitely many variables to a symmetric
+          function in the monomial basis of the ring of symmetric
+          functions over the same base ring.
 
         EXAMPLES::
 
@@ -173,12 +177,13 @@ def from_polynomial(self, f, check=True):
             sage: f = (2*m[2,1]+m[1,1]+3*m[3]).expand(3)
             sage: m.from_polynomial(f)
             m[1, 1] + 2*m[2, 1] + 3*m[3]
+
         """
         assert self.base_ring() == f.base_ring()
         if check and not f.is_symmetric():
             raise ValueError("%s is not a symmetric polynomial" % f)
         out = self._from_dict({_Partitions.element_class(_Partitions, list(e)): c
-                               for (e,c) in f.dict().items()
+                               for e, c in f.monomial_coefficients().items()
                                if all(e[i+1] <= e[i] for i in range(len(e)-1))},
                               remove_zeros=False)
         return out

src/sage/combinat/sf/sfa.py

@@ -6900,7 +6900,7 @@ def _from_polynomial(p, f):
     n = p.parent().ngens()
     if n == 1:
         d = {_Partitions.from_exp([e]): c
-             for e, c in p.dict().items()}
+             for e, c in p.monomial_coefficients().items()}
     else:
         d = {_Partitions.from_exp(e): c
              for e, c in p.iterator_exp_coeff(False)}