Retrieve more information of fundamental groups of plane curves

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

@@ -7,6 +7,7 @@ Curves
    sage/schemes/curves/constructor
    sage/schemes/curves/curve
    sage/schemes/curves/affine_curve
+   sage/schemes/curves/plane_curve_arrangement
    sage/schemes/curves/projective_curve
    sage/schemes/curves/point
    sage/schemes/curves/closed_point

src/sage/geometry/all.py

@@ -16,4 +16,5 @@
 lazy_import('sage.geometry.voronoi_diagram', 'VoronoiDiagram')
 lazy_import('sage.geometry.ribbon_graph', 'RibbonGraph')
 lazy_import('sage.geometry.hyperplane_arrangement.arrangement', 'HyperplaneArrangements')
+lazy_import('sage.geometry.hyperplane_arrangement.arrangement', 'OrderedHyperplaneArrangements')
 lazy_import('sage.geometry.hyperplane_arrangement.library', 'hyperplane_arrangements')

src/sage/geometry/hyperplane_arrangement/library.py

@@ -156,7 +156,7 @@ def bigraphical(self, G, A=None, K=QQ, names=None):
         for u, v in G.edge_iterator(labels=False, sort_vertices=False):
             i = vertex_to_int[u]
             j = vertex_to_int[v]
-            hyperplanes.append( x[i] - x[j] - A[i][j])
+            hyperplanes.append(x[i] - x[j] - A[i][j])
             hyperplanes.append(-x[i] + x[j] - A[j][i])
         return H(*hyperplanes)
 
@@ -794,7 +794,7 @@ def Shi(self, data, K=QQ, names=None, m=1):
         hyperplanes = []
 
         for a in PR:
-            for const in range(-m+1,m+1):
+            for const in range(-m + 1, m + 1):
                 hyperplanes.append(sum(a[j]*x[j] for j in range(d))-const)
         A = H(*hyperplanes)
         x = polygen(QQ, 'x')

src/sage/schemes/curves/affine_curve.py

@@ -1725,7 +1725,6 @@ def tangent_line(self, p):
              defined by: -2*y + z + 1, x + y + z
             sage: _ == C.tangent_line(p)
             True
-
         """
         A = self.ambient_space()
         R = A.coordinate_ring()
@@ -1752,8 +1751,48 @@ class AffinePlaneCurve_field(AffinePlaneCurve, AffineCurve_field):
     """
     _point = AffinePlaneCurvePoint_field
 
+    def has_vertical_asymptote(self):
+        """
+        Check if the curve is not a line and has vertical asymptotes.
+
+        EXAMPLES::
+
+            sage: A2.<x,y> = AffineSpace(2, QQ)
+            sage: Curve(x).has_vertical_asymptote()
+            False
+            sage: Curve(y^2 * x + x + y).has_vertical_asymptote()
+            True
+        """
+        A = self.ambient_space()
+        R = A.coordinate_ring()
+        x, y = R.gens()
+        f = self.defining_polynomial().radical()
+        dy = f.degree(y)
+        dxy = f.coefficient({y: dy}).degree()
+        return dxy > 0 and f.degree() > 1
+
+    def is_vertical_line(self):
+        """
+        Check if the curve is a vertical line.
+
+        EXAMPLES::
+
+            sage: A2.<x, y> = AffineSpace(2, QQ)
+            sage: Curve(x - 1).is_vertical_line()
+            True
+            sage: Curve(x - y).is_vertical_line()
+            False
+            sage: Curve(y^2 * x + x + y).is_vertical_line()
+            False
+        """
+        A = self.ambient_space()
+        R = A.coordinate_ring()
+        x, y = R.gens()
+        f = self.defining_polynomial().radical()
+        return f.degree(y) == 0 and f.degree() == 1
+
     @cached_method
-    def fundamental_group(self, simplified=True, puiseux=False):
+    def fundamental_group(self, simplified=True, puiseux=True):
         r"""
         Return a presentation of the fundamental group of the complement
         of ``self``.
@@ -1762,7 +1801,7 @@ def fundamental_group(self, simplified=True, puiseux=False):
 
         - ``simplified`` -- (default: ``True``) boolean to simplify the presentation.
 
-        - ``puiseux`` -- (default: ``False``) boolean to decide if the
+        - ``puiseux`` -- (default: ``True``) boolean to decide if the
           presentation is constructed in the classical way or using Puiseux
           shortcut. If ``True``, ``simplified`` is set to ``False``.
 
@@ -1788,10 +1827,10 @@ def fundamental_group(self, simplified=True, puiseux=False):
             sage: # optional - sirocco
             sage: A.<x,y> = AffineSpace(QQ, 2)
             sage: C = A.curve(y^2 - x^3 - x^2)
-            sage: C.fundamental_group()
+            sage: C.fundamental_group(puiseux=False)
             Finitely presented group < x0 |  >
             sage: bm = C.braid_monodromy()
-            sage: g = C.fundamental_group(puiseux=True)
+            sage: g = C.fundamental_group(simplified=False)
             sage: g.sorted_presentation()
             Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0, x1^-1*x0 >
             sage: g.simplified()
@@ -1800,22 +1839,41 @@ def fundamental_group(self, simplified=True, puiseux=False):
         In the case of number fields, they need to have an embedding
         to the algebraic field::
 
-            sage: # needs sage.rings.number_field
-            sage: a = QQ[x](x^2 + 5).roots(QQbar)[0][0]
+            sage: # needs sage.rings.number_field sirocco
+            sage: T.<t> = QQ[]
+            sage: a = (t^2 + 5).roots(QQbar)[0][0]
             sage: F = NumberField(a.minpoly(), 'a', embedding=a)
             sage: F.inject_variables()
             Defining a
             sage: A.<x,y> = AffineSpace(F, 2)
             sage: C = A.curve(y^2 - a*x^3 - x^2)
-            sage: C.fundamental_group()         # optional - sirocco
+            sage: C.fundamental_group()
             Finitely presented group < x0 |  >
+            sage: C = A.curve(x * (x - 1))
+            sage: C.fundamental_group()
+            Finitely presented group < x0, x1 |  >
 
         .. WARNING::
 
             This functionality requires the sirocco package to be installed.
         """
         from sage.schemes.curves.zariski_vankampen import fundamental_group_from_braid_mon
-        return fundamental_group_from_braid_mon(self.braid_monodromy(), simplified=simplified, puiseux=puiseux)
+        bm = self.braid_monodromy()
+        if bm == []:
+            f = self.defining_polynomial()
+            x, y = f.parent().gens()
+            d0 = f.degree(y)
+            f0 = f.coefficient({y: d0})
+            d = d0 + f0.degree(x)
+        else:
+            d = bm[0].parent().strands()
+        G = fundamental_group_from_braid_mon(self.braid_monodromy(),
+                                             degree=d,
+                                             simplified=simplified,
+                                             puiseux=puiseux)
+        if simplified:
+            G = G.simplified()
+        return G
 
     @cached_method
     def braid_monodromy(self):

src/sage/schemes/curves/all.py

@@ -2,7 +2,7 @@
 Plane curves
 """
 
-#*****************************************************************************
+# *****************************************************************************
 #
 #   Sage: Open Source Mathematical Software
 #
@@ -18,8 +18,12 @@
 #  The full text of the GPL is available at:
 #
 #                  http://www.gnu.org/licenses/
-#*****************************************************************************
+# *****************************************************************************
 
 from .constructor import Curve
 
 from .projective_curve import Hasse_bounds
+
+from .plane_curve_arrangement import AffinePlaneCurveArrangements
+
+from .plane_curve_arrangement import ProjectivePlaneCurveArrangements