Implement dual subdivision and weight vectors for tropical variety

src/doc/en/reference/references/index.rst

@@ -4651,6 +4651,9 @@ REFERENCES:
              University Press, New York, 1995, With contributions
              by A. Zelevinsky, Oxford Science Publications.
 
+.. [Mac2015] Diane Maclagan and Bernd Sturmfels, *Introduction to
+             Tropical Geometry*, American Mathematical Society, 2015.
+
 .. [MagmaHGM] *Hypergeometric motives* in Magma,
    http://magma.maths.usyd.edu.au/~watkins/papers/HGM-chapter.pdf
 

src/sage/rings/semirings/tropical_mpolynomial.py

@@ -11,8 +11,6 @@
     sage: R.<x,y,z> = PolynomialRing(T)
     sage: z.parent()
     Multivariate Tropical Polynomial Semiring in x, y, z over Rational Field
-    sage: R(2)*x + R(-1)*x + R(5)*y + R(-3)
-    (-1)*x + 5*y + (-3)
     sage: (x+y+z)^2
     0*x^2 + 0*x*y + 0*y^2 + 0*x*z + 0*y*z + 0*z^2
 
@@ -151,6 +149,23 @@ class TropicalMPolynomial(MPolynomial_polydict):
         p1 = R(3)*a*b + a + R(-1)*b
         sphinx_plot(p1.plot3d())
 
+    Another way to represent tropical curve is through dual subdivision,
+    which is a subdivision of Newton polytope of tropical polynomial::
+
+        sage: p1.Newton_polytope()
+        A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
+        sage: p1.dual_subdivision()
+        Polyhedral complex with 1 maximal cell
+
+    .. PLOT::
+        :width: 300 px
+
+        T = TropicalSemiring(QQ, use_min=False)
+        R = PolynomialRing(T, ('a,b'))
+        a, b = R.gen(), R.gen(1)
+        p1 = R(3)*a*b + a + R(-1)*b
+        sphinx_plot(p1.dual_subdivision().plot())
+
     TESTS:
 
     There is no subtraction defined for tropical polynomials::
@@ -285,7 +300,7 @@ def plot3d(self, color='random'):
         T = self.parent().base()
         R = self.base_ring().base_ring()
 
-        # Finding the point of curve that touch the edge of the axes
+        # Find the point of curve that touch the edge of the axes
         for comp in tv.components():
             if len(comp[1]) == 1:
                 valid_int = RealSet(comp[1][0])
@@ -359,7 +374,7 @@ def tropical_variety(self):
         curve. For dimensions higher than two, it is referred to as a
         tropical hypersurface.
 
-        OUTPUT: a :class:`sage.rings.semirings.tropical_variety.TropicalVariety`
+        OUTPUT: :class:`sage.rings.semirings.tropical_variety.TropicalVariety`
 
         EXAMPLES:
 
@@ -390,6 +405,180 @@ def tropical_variety(self):
             return TropicalSurface(self)
         return TropicalVariety(self)
 
+    def Newton_polytope(self):
+        """
+        Return the Newton polytope of ``self``.
+
+        The Newton polytope is the convex hull of all the points
+        corresponding to the exponents of the monomials of tropical
+        polynomial.
+
+        OUTPUT: :class:`sage.geometry.polyhedron.constructor.Polyhedron`
+
+        EXAMPLES:
+
+        Newton polytope for a two-variable tropical polynomial::
+
+            sage: T = TropicalSemiring(QQ)
+            sage: R.<x,y> = PolynomialRing(T)
+            sage: p1 = x + y
+            sage: p1.Newton_polytope()
+            A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices
+
+        .. PLOT::
+            :width: 300 px
+
+            T = TropicalSemiring(QQ)
+            R = PolynomialRing(T, ('x,y'))
+            x, y = R.gen(), R.gen(1)
+            p1 = x + y
+            sphinx_plot(p1.Newton_polytope().plot())
+
+        Newton polytope in three dimension::
+
+            sage: T = TropicalSemiring(QQ)
+            sage: R.<x,y,z> = PolynomialRing(T)
+            sage: p1 = x^2 + x*y*z + x + y + z + R(0)
+            sage: p1.Newton_polytope()
+            A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 5 vertices
+
+        .. PLOT::
+            :width: 300 px
+
+            T = TropicalSemiring(QQ)
+            R = PolynomialRing(T, ('x,y,z'))
+            x, y, z = R.gen(), R.gen(1), R.gen(2)
+            p1 = x**2 + x*y*z + x + y + z + R(0)
+            sphinx_plot(p1.Newton_polytope().plot())
+        """
+        from sage.geometry.polyhedron.constructor import Polyhedron
+
+        exponents = self.exponents()
+        return Polyhedron(exponents)
+
+    def dual_subdivision(self):
+        """
+        Return the dual subdivision of ``self``.
+
+        Dual subdivision refers to a specific decomposition of the
+        Newton polytope of a tropical polynomial. The term "dual" is
+        used in the sense that the combinatorial structure of the
+        tropical variety is reflected in the dual subdivision.
+        Specifically, vertices of the dual subdivision correspond to
+        the intersection of multiple components. Edges of the dual
+        subdivision correspond to the individual components.
+
+        OUTPUT: :class:`sage.geometry.polyhedral_complex.PolyhedralComplex`
+
+        EXAMPLES:
+
+        Dual subdivision of a tropical curve::
+
+            sage: T = TropicalSemiring(QQ, use_min=False)
+            sage: R.<x,y> = PolynomialRing(T)
+            sage: p1 = R(3) + R(2)*x + R(2)*y + R(3)*x*y + x^2 + y^2
+            sage: p1.dual_subdivision()
+            Polyhedral complex with 4 maximal cells
+
+        .. PLOT::
+            :width: 300 px
+
+            T = TropicalSemiring(QQ,  use_min=False)
+            R = PolynomialRing(T, ('x,y'))
+            x, y = R.gen(), R.gen(1)
+            p1 = R(3) + R(2)*x + R(2)*y + R(3)*x*y + x**2 + y**2
+            sphinx_plot(p1.dual_subdivision().plot())
+
+        A subdivision of a pentagonal Newton polytope::
+
+            sage: p2 = R(3) + x^2 + R(-2)*y + R(1/2)*x^2*y + R(2)*x*y^3 + R(-1)*x^3*y^4
+            sage: p2.dual_subdivision()
+            Polyhedral complex with 5 maximal cells
+
+        .. PLOT::
+            :width: 300 px
+
+            T = TropicalSemiring(QQ,  use_min=False)
+            R = PolynomialRing(T, ('x,y'))
+            x, y = R.gen(), R.gen(1)
+            p2 = R(3) + x**2 + R(-2)*y + R(1/2)*x**2*y + R(2)*x*y**3 + R(-1)*x**3*y**4
+            sphinx_plot(p2.dual_subdivision().plot())
+
+        A subdivision with many faces, not all of which are triangles::
+
+            sage: p3 = R(8) + R(4)*x + R(2)*y + R(1)*x^2 + x*y + R(1)*y^2 \
+            ....:     + R(2)*x^3 + x^2*y + x*y^2 + R(4)*y^3 + R(8)*x^4 \
+            ....:     + R(4)*x^3*y + x^2*y^2 + R(2)*x*y^3 + y^4
+            sage: p3.dual_subdivision().plot()
+            Graphics object consisting of 10 graphics primitives
+
+        .. PLOT::
+            :width: 300 px
+
+            T = TropicalSemiring(QQ,  use_min=False)
+            R = PolynomialRing(T, ('x,y'))
+            x, y = R.gen(), R.gen(1)
+            p3 = R(8) + R(4)*x + R(2)*y + R(1)*x**2 + x*y + R(1)*y**2 \
+                 + R(2)*x**3 + x**2*y + x*y**2 + R(4)*y**3 + R(8)*x**4 \
+                 + R(4)*x**3*y + x**2*y**2 + R(2)*x*y**3 + y**4
+            sphinx_plot(p3.dual_subdivision().plot())
+
+        Dual subdivision of a tropical surface::
+
+            sage: T = TropicalSemiring(QQ)
+            sage: R.<x,y,z> = PolynomialRing(T)
+            sage: p1 = x + y + z + x^2 + R(1)
+            sage: p1.dual_subdivision()
+            Polyhedral complex with 7 maximal cells
+
+        .. PLOT::
+            :width: 300 px
+
+            T = TropicalSemiring(QQ,  use_min=False)
+            R = PolynomialRing(T, ('x,y,z'))
+            x, y, z = R.gen(), R.gen(1), R.gen(2)
+            p1 = x + y + z + x**2 + R(1)
+            sphinx_plot(p1.dual_subdivision().plot())
+
+        Dual subdivision of a tropical hypersurface::
+
+            sage: T = TropicalSemiring(QQ)
+            sage: R.<a,b,c,d> = PolynomialRing(T)
+            sage: p1 = R(2)*a*b + R(3)*a*c + R(-1)*c^2 + R(-1/3)*a*d
+            sage: p1.dual_subdivision()
+            Polyhedral complex with 4 maximal cells
+        """
+        from sage.geometry.polyhedron.constructor import Polyhedron
+        from sage.geometry.polyhedral_complex import PolyhedralComplex
+
+        TV = self.tropical_variety()
+        cycles = []
+
+        if TV.dimension() == 2:
+            for indices in TV._vertices_components().values():
+                cycle = []
+                for index in indices:
+                    vertices = TV._keys[index[0]]
+                    for v in vertices:
+                        cycle.append(v)
+                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)
+                cycles.append(cycle)
+
+        polyhedron_lst = []
+        for cycle in cycles:
+            polyhedron = Polyhedron(vertices=cycle)
+            polyhedron_lst.append(polyhedron)
+        pc = PolyhedralComplex(polyhedron_lst)
+        return pc
+
     def _repr_(self):
         r"""
         Return string representation of ``self``.
@@ -595,7 +784,7 @@ def random_element(self, degree=2, terms=None, choose_degree=False,
         r"""
         Return a random multivariate tropical polynomial from ``self``.
 
-        OUTPUT: a :class:`TropicalMPolynomial`
+        OUTPUT: :class:`TropicalMPolynomial`
 
         .. SEEALSO::
 

src/sage/rings/semirings/tropical_polynomial.py

@@ -11,10 +11,8 @@
     sage: R.<x> = PolynomialRing(T)
     sage: x.parent()
     Univariate Tropical Polynomial Semiring in x over Rational Field
-    sage: (x + R(3)*x) * (x^2 + x)
-    3*x^3 + 3*x^2
-    sage: (x^2 + R(1)*x + R(-1))^2
-    0*x^4 + 1*x^3 + 2*x^2 + 0*x + (-2)
+    sage: (x^2 + x + R(0))^2
+    0*x^4 + 0*x^3 + 0*x^2 + 0*x + 0
 
 REFERENCES:
 
@@ -56,7 +54,7 @@ class TropicalPolynomial(Polynomial_generic_sparse):
 
     EXAMPLES:
 
-    First, we construct a tropical polynomial semiring by defining a base
+    We construct a tropical polynomial semiring by defining a base
     tropical semiring and then inputting it to :class:`PolynomialRing`::
 
         sage: T = TropicalSemiring(QQ, use_min=False)
@@ -168,7 +166,7 @@ def roots(self):
         Return the list of all tropical roots of ``self``, counted with
         multiplicity.
 
-        OUTPUT: a list of tropical numbers
+        OUTPUT: A list of tropical numbers
 
         ALGORITHM:
 
@@ -294,7 +292,7 @@ def factor(self):
         `x + x_0` is `x_0` and not `-x_0`. However, not every tropical
         polynomial can be factored.
 
-        OUTPUT: a :class:'Factorization'
+        OUTPUT: :class:'Factorization'
 
         EXAMPLES::
 
@@ -800,7 +798,7 @@ def random_element(self, degree=(-1, 2), monic=False, *args, **kwds):
         r"""
         Return a random tropical polynomial of given degrees (bounds).
 
-        OUTPUT: a :class:`TropicalPolynomial`
+        OUTPUT: :class:`TropicalPolynomial`
 
         .. SEEALSO::
 
@@ -875,7 +873,7 @@ def interpolation(self, points):
 
         - ``points`` -- a list of tuples ``(x, y)``
 
-        OUTPUT: a :class:`TropicalPolynomial`
+        OUTPUT: :class:`TropicalPolynomial`
 
         EXAMPLES::