sagemathgh-39579: some details in schemes, typing annotation for gens

    
adding a little bit of typing annotation for `gens` methods in schemes

There remains several such methods in `elliptic_curves` that return
lists and should be changed later to return tuples.

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#39579
Reported by: Frédéric Chapoton
Reviewer(s): Vincent Macri

src/sage/schemes/curves/plane_curve_arrangement.py

@@ -43,10 +43,11 @@
 # it under the terms of the GNU General Public License as published by
 # the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
-#                  http://www.gnu.org/licenses/
+#                  https://www.gnu.org/licenses/
 # *****************************************************************************
-
 from itertools import combinations
+from typing import Any
+
 from sage.categories.sets_cat import Sets
 from sage.groups.free_group import FreeGroup
 from sage.misc.abstract_method import abstract_method
@@ -71,7 +72,7 @@ class PlaneCurveArrangementElement(Element):
     """
     An ordered plane curve arrangement.
     """
-    def __init__(self, parent, curves, check=True):
+    def __init__(self, parent, curves, check=True) -> None:
         """
         Construct a plane curve arrangement.
 
@@ -120,7 +121,7 @@ def __getitem__(self, i):
         """
         return self._curves[i]
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         r"""
         TESTS::
 
@@ -130,7 +131,7 @@ def __hash__(self):
         """
         return hash(self.curves())
 
-    def ncurves(self):
+    def ncurves(self) -> int:
         r"""
         Return the number of curves in the arrangement.
 
@@ -149,7 +150,7 @@ def ncurves(self):
 
     __len__ = ncurves
 
-    def curves(self):
+    def curves(self) -> tuple:
         r"""
         Return the curves in the arrangement as a tuple.
 
@@ -170,7 +171,7 @@ def curves(self):
         """
         return self._curves
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         String representation for a curve arrangement.
 
@@ -195,11 +196,11 @@ def _repr_(self):
             curves = ', '.join(h.defining_polynomial()._repr_()
                                for h in self._curves)
             return 'Arrangement ({}) in {}'.format(curves,
-                                                     self.parent().ambient_space())
+                                                   self.parent().ambient_space())
         return 'Arrangement of {} curves in {}'.format(len(self),
-                                                         self.parent().ambient_space())
+                                                       self.parent().ambient_space())
 
-    def _richcmp_(self, other, op):
+    def _richcmp_(self, other, op) -> bool:
         """
         Compare two curve arrangements.
 
@@ -337,7 +338,7 @@ def coordinate_ring(self):
         """
         return self._curves[0].defining_polynomial().parent()
 
-    def defining_polynomials(self):
+    def defining_polynomials(self) -> tuple:
         r"""
         Return the defining polynomials of the elements of ``self``.
 
@@ -428,7 +429,7 @@ class AffinePlaneCurveArrangementElement(PlaneCurveArrangementElement):
     """
     An ordered affine plane curve arrangement.
     """
-    def __init__(self, parent, curves, check=True):
+    def __init__(self, parent, curves, check=True) -> None:
         """
         Construct an ordered affine plane curve arrangement.
 
@@ -584,7 +585,7 @@ def fundamental_group(self, simplified=True, vertical=True,
             self._meridians_nonsimpl_nonvertical = dic
         return G
 
-    def meridians(self, simplified=True, vertical=True):
+    def meridians(self, simplified=True, vertical=True) -> dict:
         r"""
         Return the meridians of each irreducible component.
 
@@ -921,7 +922,7 @@ def fundamental_group(self, simplified=True):
             self._meridians_nonsimpl = dic1
         return G
 
-    def meridians(self, simplified=True):
+    def meridians(self, simplified=True) -> dict:
         r"""
         Return the meridians of each irreducible component.
 
@@ -991,7 +992,7 @@ class PlaneCurveArrangements(UniqueRepresentation, Parent):
     Element = PlaneCurveArrangementElement
 
     @staticmethod
-    def __classcall__(cls, base, names=()):
+    def __classcall__(cls, base, names: tuple[str, ...] = ()):
         """
         Normalize the inputs to ensure a unique representation.
 
@@ -1005,7 +1006,7 @@ def __classcall__(cls, base, names=()):
         names = normalize_names(len(names), names)
         return super().__classcall__(cls, base, names)
 
-    def __init__(self, base_ring, names=()):
+    def __init__(self, base_ring, names: tuple[str, ...] = ()):
         """
         Initialize ``self``.
 
@@ -1061,6 +1062,8 @@ def change_ring(self, base_ring):
             sage: L.change_ring(QQ) is L
             True
         """
+        # return self.__class__(base_ring, names=self.variable_names())
+        # line below is ugly but line above does not work
         return self.__reduce__()[1][0](base_ring, names=self.variable_names())
 
     @abstract_method
@@ -1124,7 +1127,7 @@ def _element_constructor_(self, *args, **kwds):
             arg = tuple(args)
         ambient_space = self.ambient_space()
         R = ambient_space.coordinate_ring()
-        curves = ()
+        curves: tuple[Any, ...] = ()
         for h in arg:
             try:
                 ambient = h.ambient_space()
@@ -1173,7 +1176,7 @@ def ngens(self):
         """
         return len(self.variable_names())
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the coordinates.
 

src/sage/schemes/generic/ambient_space.py

@@ -315,7 +315,7 @@ def gen(self, n=0):
         """
         return self.coordinate_ring().gen(n)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of the coordinate ring of the scheme
         ``self``.

src/sage/schemes/hyperelliptic_curves/monsky_washnitzer.py

@@ -570,15 +570,10 @@ def poly_ring(self):
         """
         return self._poly_ring
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
-        Return a tuple (x, T) where x and T are the generators of the ring
-        (as element *of this ring*).
-
-        .. NOTE::
-
-             I have no idea if this is compatible with the usual Sage
-             'gens' interface.
+        Return (x, T) where x and T are the generators of the ring
+        (as elements *of this ring*).
 
         EXAMPLES::
 
@@ -2539,7 +2534,7 @@ def zero(self):
         """
         return self.element_class(self, self._poly_ring.zero(), check=False)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self``.
 
@@ -2680,7 +2675,7 @@ def monomial_diff_coeffs_matrices(self):
             mat_2[i] = self._precomputed_diff_coeffs[i][2]
         return mat_1.transpose(), mat_2.transpose()
 
-    def _precompute_monomial_diffs(self):
+    def _precompute_monomial_diffs(self) -> list:
         r"""
         Precompute coefficients of the basis representation of `d(x^iy^j)`
         for small `i`, `j`.
@@ -3804,6 +3799,7 @@ def frob_basis_elements(self, prec, p):
             F.append(F_i)
         return F
 
+    @cached_method
     def helper_matrix(self):
         r"""
         We use this to solve for the linear combination of
@@ -3821,22 +3817,16 @@ def helper_matrix(self):
             [-100/2101 -125/2101 -625/8404  -64/2101  -80/2101]
             [ -80/2101 -100/2101 -125/2101 -625/8404  -64/2101]
         """
-        try:
-            return self._helper_matrix
-        except AttributeError:
-            pass
-
         # The smallest y term of (1/j) d(x^i y^j) is constant for all j.
         x, y = self.base_ring().gens()
         n = self.degree()
         L = [(y * x**i).diff().extract_pow_y(0) for i in range(n)]
         A = matrix(L).transpose()
         if A.base_ring() not in IntegralDomains():
-            # must be using integer_mod or something to approximate
-            self._helper_matrix = (~A.change_ring(QQ)).change_ring(A.base_ring())
-        else:
-            self._helper_matrix = ~A
-        return self._helper_matrix
+            # must be using integer_mod or something to approximate ?
+            return (~A.change_ring(QQ)).change_ring(A.base_ring())
+
+        return ~A
 
     def _element_constructor_(self, val=0, offset=0):
         r"""

src/sage/schemes/plane_conics/con_field.py

@@ -377,7 +377,7 @@ def diagonalization(self, names=None):
         con = Conic(D, names=names)
         return con, con.hom(T, self), self.hom(T.inverse(), con)
 
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the generators of the coordinate ring of ``self``.
 
@@ -945,28 +945,28 @@ def parametrization(self, point=None, morphism=True):
             point = Sequence(point)
             B = self.base_ring()
             Q = PolynomialRing(B, 'x,y')
-            [x, y] = Q.gens()
+            x, y = Q.gens()
             gens = self.ambient_space().gens()
             P = PolynomialRing(B, 4, ['X', 'Y', 'T0', 'T1'])
-            [X, Y, T0, T1] = P.gens()
-            c3 = [j for j in range(2,-1,-1) if point[j] != 0][0]
-            c1 = [j for j in range(3) if j != c3][0]
-            c2 = [j for j in range(3) if j != c3 and j != c1][0]
-            L = [0,0,0]
-            L[c1] = Y*T1*point[c1] + Y*T0
-            L[c2] = Y*T1*point[c2] + X*T0
-            L[c3] = Y*T1*point[c3]
+            X, Y, T0, T1 = P.gens()
+            c3 = next(j for j in range(2, -1, -1) if point[j] != 0)
+            c1 = next(j for j in range(3) if j != c3)
+            c2 = next(j for j in range(3) if j != c3 and j != c1)
+            L = [0, 0, 0]
+            L[c1] = Y * T1 * point[c1] + Y * T0
+            L[c2] = Y * T1 * point[c2] + X * T0
+            L[c3] = Y * T1 * point[c3]
             bezout = P(self.defining_polynomial()(L) / T0)
-            t = [bezout([x,y,0,-1]),bezout([x,y,1,0])]
-            par = (tuple([Q(p([x,y,t[0],t[1]])/y) for p in L]),
-                   tuple([gens[m]*point[c3]-gens[c3]*point[m]
-                       for m in [c2,c1]]))
+            t = [bezout([x, y, 0, -1]), bezout([x, y, 1, 0])]
+            par = (tuple([Q(p([x, y, t[0], t[1]]) / y) for p in L]),
+                   tuple([gens[m] * point[c3] - gens[c3] * point[m]
+                          for m in [c2, c1]]))
             if self._parametrization is None:
                 self._parametrization = par
         if not morphism:
             return par
         P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
-        return P1.hom(par[0],self), self.Hom(P1)(par[1], check=False)
+        return P1.hom(par[0], self), self.Hom(P1)(par[1], check=False)
 
     def point(self, v, check=True):
         r"""

src/sage/schemes/toric/chow_group.py

@@ -119,6 +119,7 @@
 # ****************************************************************************
 from __future__ import annotations
 
+from sage.misc.cachefunc import cached_method
 from sage.misc.flatten import flatten
 from sage.misc.fast_methods import WithEqualityById
 from sage.modules.fg_pid.fgp_module import FGP_Module_class
@@ -226,6 +227,7 @@ def _repr_(self) -> str:
         s += ')'
         return s
 
+    @cached_method
     def degree(self) -> int:
         r"""
         Return the degree of the Chow cycle.
@@ -245,18 +247,14 @@ def degree(self) -> int:
             sage: [ a.degree() for a in A.gens() ]
             [2, 1, 0]
         """
-        if '_dim' in self.__dict__:
-            return self._dim
-
         ambient_dim = self.parent()._variety.dimension()
         cone_dim = None
         for i, cone in enumerate(self.parent()._cones):
             if self.lift()[i] != 0:
                 if cone_dim not in [None, cone.dim()]:
                     raise ValueError('Chow cycle is not of definite degree')
                 cone_dim = cone.dim()
-        self._dim = ambient_dim - cone_dim
-        return self._dim
+        return ambient_dim - cone_dim
 
     def project_to_degree(self, degree):
         r"""
@@ -817,6 +815,7 @@ def _cone_to_V(self, cone):
         x[self._cones.index(cone)] = 1
         return self._V(x)
 
+    @cached_method
     def degree(self, k=None):
         r"""
         Return the degree-`k` Chow group.
@@ -911,15 +910,8 @@ def degree(self, k=None):
         """
         if k is not None:
             return self.degree()[k]
-
-        try:
-            return self._degree
-        except AttributeError:
-            pass
-
-        self._degree = tuple(ChowGroup_degree_class(self, d)
-                             for d in range(self._variety.dimension() + 1))
-        return self._degree
+        return tuple(ChowGroup_degree_class(self, d)
+                     for d in range(self._variety.dimension() + 1))
 
     def coordinate_vector(self, chow_cycle, degree=None, reduce=True):
         r"""
@@ -958,7 +950,7 @@ def coordinate_vector(self, chow_cycle, degree=None, reduce=True):
         a = chow_cycle.project_to_degree(degree)
         return self.degree(degree).module().coordinate_vector(a, reduce=reduce)
 
-    def gens(self, degree=None):
+    def gens(self, degree=None) -> tuple:
         r"""
         Return the generators of the Chow group.
 
@@ -1177,7 +1169,7 @@ def gen(self, i):
         """
         return self._gens[i]
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of the Chow group of fixed degree.
 

src/sage/schemes/toric/divisor.py

@@ -1742,7 +1742,7 @@ def ngens(self):
         return self.scheme().fan().nrays()
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the generators of the divisor group.
 

src/sage/schemes/toric/variety.py

@@ -3211,7 +3211,7 @@ def __call__(self, x, coerce=True):
         """
         return self._element_constructor_(x)
 
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the generators of the cohomology ring.
 
@@ -3235,7 +3235,7 @@ def gens(self):
 
     def gen(self, i):
         r"""
-        Return the generators of the cohomology ring.
+        Return a generator of the cohomology ring.
 
         INPUT: