small cleanup in tropical files

Author: fchapoton

src/sage/rings/semirings/tropical_mpolynomial.py

@@ -33,9 +33,9 @@
 # ****************************************************************************
 
 from sage.misc.cachefunc import cached_method
+from sage.rings.polynomial.multi_polynomial_element import MPolynomial_polydict
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.rings.polynomial.multi_polynomial_element import MPolynomial_polydict
 
 
 class TropicalMPolynomial(MPolynomial_polydict):
@@ -286,8 +286,9 @@ def plot3d(self, color='random'):
             multivariate polynomial in two variables
         """
         from random import random
-        from sage.plot.graphics import Graphics
+
         from sage.geometry.polyhedron.constructor import Polyhedron
+        from sage.plot.graphics import Graphics
         from sage.sets.real_set import RealSet
         from sage.symbolic.relation import solve
 
@@ -399,7 +400,11 @@ def tropical_variety(self):
             sage: p1.tropical_variety()
             Tropical surface of 1*x*y + (-1/2)*x*z + 4*z^2
         """
-        from sage.rings.semirings.tropical_variety import TropicalCurve, TropicalSurface, TropicalVariety
+        from sage.rings.semirings.tropical_variety import (
+            TropicalCurve,
+            TropicalSurface,
+            TropicalVariety,
+        )
 
         if self.parent().ngens() == 2:
             return TropicalCurve(self)
@@ -617,8 +622,8 @@ def dual_subdivision(self):
              A vertex at (1, 0, 1, 0),
              A vertex at (1, 1, 0, 0))]
         """
-        from sage.geometry.polyhedron.constructor import Polyhedron
         from sage.geometry.polyhedral_complex import PolyhedralComplex
+        from sage.geometry.polyhedron.constructor import Polyhedron
 
         TV = self.tropical_variety()
         cycles = []
@@ -627,18 +632,14 @@ def dual_subdivision(self):
             for indices in TV._vertices_components().values():
                 cycle = []
                 for index in indices:
-                    vertices = TV._keys[index[0]]
-                    for v in vertices:
-                        cycle.append(v)
+                    cycle.extend(TV._keys[index[0]])
                 cycles.append(cycle)
         else:
             line_comps = TV.weight_vectors()[1]
             for indices in line_comps.values():
                 cycle = []
                 for index in indices:
-                    vertices = TV._keys[index]
-                    for v in vertices:
-                        cycle.append(v)
+                    cycle.extend(TV._keys[index])
                 cycles.append(cycle)
 
         polyhedron_lst = []
@@ -729,8 +730,8 @@ def __init__(self, base_semiring, n, names, order):
             sage: R = PolynomialRing(T, 5, 'x')
             sage: TestSuite(R).run()
         """
-        from sage.rings.semirings.tropical_semiring import TropicalSemiring
         from sage.categories.semirings import Semirings
+        from sage.rings.semirings.tropical_semiring import TropicalSemiring
         if not isinstance(base_semiring, TropicalSemiring):
             raise ValueError(f"{base_semiring} is not a tropical semiring")
         Parent.__init__(self, base=base_semiring, names=names, category=Semirings())

src/sage/rings/semirings/tropical_polynomial.py

@@ -35,9 +35,9 @@
 # ****************************************************************************
 
 from sage.misc.cachefunc import cached_method
-from sage.structure.unique_representation import UniqueRepresentation
-from sage.structure.parent import Parent
 from sage.rings.polynomial.polynomial_element_generic import Polynomial_generic_sparse
+from sage.structure.parent import Parent
+from sage.structure.unique_representation import UniqueRepresentation
 
 
 class TropicalPolynomial(Polynomial_generic_sparse):
@@ -371,9 +371,9 @@ def piecewise_function(self):
             sage: p3.piecewise_function()
             3*x + 1
         """
-        from sage.symbolic.ring import SR
         from sage.functions.piecewise import piecewise
         from sage.sets.real_set import RealSet
+        from sage.symbolic.ring import SR
 
         x = SR.var('x')
         data = self.monomial_coefficients()

src/sage/rings/semirings/tropical_variety.py

@@ -26,12 +26,11 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+from sage.rings.infinity import infinity
+from sage.rings.rational_field import QQ
 from sage.structure.sage_object import SageObject
 from sage.structure.unique_representation import UniqueRepresentation
 
-from sage.rings.rational_field import QQ
-from sage.rings.infinity import infinity
-
 
 class TropicalVariety(UniqueRepresentation, SageObject):
     r"""
@@ -188,10 +187,11 @@ def __init__(self, poly):
         """
         import operator
         from itertools import combinations
-        from sage.symbolic.ring import SR
-        from sage.symbolic.relation import solve
+
         from sage.arith.misc import gcd
         from sage.rings.semirings.tropical_mpolynomial import TropicalMPolynomial
+        from sage.symbolic.relation import solve
+        from sage.symbolic.ring import SR
 
         if not isinstance(poly, TropicalMPolynomial):
             raise ValueError(f"{poly} is not a multivariate tropical polynomial")
@@ -448,9 +448,10 @@ def _components_intersection(self):
             5: [((0, t2, 0), {0 <= t2}), ((1/2*t2, t2, t2), {t2 <= 0})]}
         """
         import operator
+
         from sage.functions.min_max import max_symbolic, min_symbolic
-        from sage.symbolic.relation import solve
         from sage.sets.set import Set
+        from sage.symbolic.relation import solve
 
         def update_result(result):
             sol_param = solve(new_expr, vars)
@@ -592,19 +593,20 @@ def weight_vectors(self):
             sage: all(a == vector([0,0,0,0]) for a in [sum(lst) for lst in vec])
             True
         """
-        from sage.symbolic.ring import SR
-        from sage.symbolic.relation import solve
-        from sage.calculus.functional import diff
+        from itertools import combinations
+
         from sage.arith.misc import gcd
+        from sage.calculus.functional import diff
         from sage.matrix.constructor import matrix
         from sage.modules.free_module_element import vector, zero_vector
-        from itertools import combinations
+        from sage.symbolic.relation import solve
+        from sage.symbolic.ring import SR
 
         dim = self.dimension()
         t = SR.var('t')
-        t_vars = [SR.var('t{}'.format(i)) for i in range(dim)]
-        u_vars = [SR.var('u{}'.format(i)) for i in range(dim)]
-        convert_tu = {ti: ui for ti, ui in zip(t_vars, u_vars)}
+        t_vars = [SR.var(f't{i}') for i in range(dim)]
+        u_vars = [SR.var(f'u{i}') for i in range(dim)]
+        convert_tu = dict(zip(t_vars, u_vars))
         CI = self._components_intersection()
         unique_line = set()
         index_line = {}
@@ -657,9 +659,9 @@ def weight_vectors(self):
                 if is_unique:
                     new_eqn = tuple([eq.subs(convert_tu) for eq in eqn])
                     cdns = line[1]
-                    new_cdn = set([cdn.subs(convert_tu) for cdn in cdns])
+                    new_cdn = {cdn.subs(convert_tu) for cdn in cdns}
                     unique_line.add(new_eqn)
-                    index_line[index] = tuple([new_eqn, new_cdn])
+                    index_line[index] = (new_eqn, new_cdn)
                     line_comps[index] = [i]
                     index += 1
                 else:
@@ -676,9 +678,7 @@ def weight_vectors(self):
                 for v in l.variables():
                     all_var.add(v)
             for vpar in all_var:
-                par_drv = []
-                for l in line:
-                    par_drv.append(QQ(diff(l, vpar)))
+                par_drv = [QQ(diff(l, vpar)) for l in line]
                 par_drv = vector(par_drv)
                 dir_vecs.append(par_drv)
 
@@ -688,32 +688,24 @@ def weight_vectors(self):
                 surface = self._hypersurface[i][0]
                 drv_vectors = []
                 for vpar in self._vars:
-                    temp_vec = []
-                    for s in surface:
-                        temp_vec.append(QQ(diff(s, vpar)))
+                    temp_vec = [QQ(diff(s, vpar)) for s in surface]
                     temp_vec = vector(temp_vec)
                     drv_vectors.append(temp_vec)
                 temp = [t_vars]
-                for vec in drv_vectors:
-                    temp.append(vec)
+                temp.extend(drv_vectors)
                 vec_matrix = matrix(SR, temp)
                 normal_vec = vec_matrix.det()
-                temp_nor = []
-                for tvar in t_vars:
-                    temp_nor.append(QQ(diff(normal_vec, tvar)))
+                temp_nor = [QQ(diff(normal_vec, tvar)) for tvar in t_vars]
                 normal_vec = vector(temp_nor)
                 normal_vec *= 1/gcd(normal_vec)
 
                 # Calculate the weight vector
                 temp_final = [t_vars]
-                for v in dir_vecs:
-                    temp_final.append(v)
+                temp_final.extend(dir_vecs)
                 temp_final.append(normal_vec)
                 vec_matrix = matrix(SR, temp_final)
                 weight_vec = vec_matrix.det()
-                temp_weight = []
-                for tvar in t_vars:
-                    temp_weight.append(QQ(diff(weight_vec, tvar)))
+                temp_weight = [QQ(diff(weight_vec, tvar)) for tvar in t_vars]
                 weight_vec = vector(temp_weight)
                 order = self._hypersurface[i][2]
                 weight_vec *= order
@@ -722,7 +714,7 @@ def weight_vectors(self):
             balance = False
             for i in range(1, len(WV[k])+1):
                 for j in combinations(range(len(WV[k])), i):
-                    test_vectors = [v for v in WV[k]]
+                    test_vectors = list(WV[k])
                     for idx in j:
                         test_vectors[idx] = -test_vectors[idx]
                     if sum(test_vectors) == zero_vector(QQ, dim):
@@ -791,8 +783,8 @@ def _axes(self):
             sage: p2.tropical_variety()._axes()
             [[-1, 2], [-1, 2], [-1, 2]]
         """
-        from sage.symbolic.relation import solve
         from sage.arith.srange import srange
+        from sage.symbolic.relation import solve
 
         if not self._hypersurface:
             return [[-1, 1], [-1, 1], [-1, 1]]
@@ -911,8 +903,8 @@ def _polygon_vertices(self):
             7: {(-1/2, -1, -1), (-1/2, 2, -1), (0, 0, 0), (0, 2, 0)},
             8: {(1, 1, 1), (1, 2, 1), (2, 1, 1), (2, 2, 1)}}
         """
-        from sage.symbolic.relation import solve
         from sage.sets.real_set import RealSet
+        from sage.symbolic.relation import solve
 
         poly_verts = {i: set() for i in range(self.number_of_components())}
         axes = self._axes()
@@ -1057,8 +1049,9 @@ def plot(self, color='random'):
             sphinx_plot(p2.tropical_variety().plot())
         """
         from random import random
-        from sage.plot.graphics import Graphics
+
         from sage.geometry.polyhedron.constructor import Polyhedron
+        from sage.plot.graphics import Graphics
 
         if color == 'random':
             colors = []
@@ -1283,17 +1276,17 @@ def _vertices_components(self):
                 if lower != -infinity:
                     x = parametric_function[0].subs(**{str(v): lower})
                     y = parametric_function[1].subs(**{str(v): lower})
-                    if (x,y) not in comp_vert:
-                        comp_vert[(x,y)] = [(i, 1)]
+                    if (x, y) not in comp_vert:
+                        comp_vert[(x, y)] = [(i, 1)]
                     else:
-                        comp_vert[(x,y)].append((i, 1))
+                        comp_vert[(x, y)].append((i, 1))
                 if upper != infinity:
                     x = parametric_function[0].subs(**{str(v): upper})
                     y = parametric_function[1].subs(**{str(v): upper})
-                    if (x,y) not in comp_vert:
-                        comp_vert[(x,y)] = [(i, -1)]
+                    if (x, y) not in comp_vert:
+                        comp_vert[(x, y)] = [(i, -1)]
                     else:
-                        comp_vert[(x,y)].append((i, -1))
+                        comp_vert[(x, y)].append((i, -1))
         return comp_vert
 
     def weight_vectors(self):
@@ -1334,9 +1327,9 @@ def weight_vectors(self):
              (-1, 0): [(-1, 0), (0, -1), (1, 1)],
              (3, 4): [(-1, -1), (0, 1), (1, 0)]}
         """
+        from sage.arith.misc import gcd
         from sage.calculus.functional import diff
         from sage.modules.free_module_element import vector
-        from sage.arith.misc import gcd
 
         if not self._vertices_components():
             return {}
@@ -1636,10 +1629,9 @@ def plot(self):
                  + R(4)*x**3*y + x**2*y**2 + R(2)*x*y**3 + y**4)
             sphinx_plot(p3.tropical_variety().plot())
         """
-        from sage.plot.plot import plot
-        from sage.plot.text import text
         from sage.plot.graphics import Graphics
-        from sage.plot.plot import parametric_plot
+        from sage.plot.plot import parametric_plot, plot
+        from sage.plot.text import text
 
         if not self._hypersurface:
             return plot(lambda x: float('nan'), {-1, 1})