20848: merge with ticket 20839

src/sage/schemes/curves/affine_curve.py

@@ -417,6 +417,58 @@ def plot(self, *args, **kwds):
         I = self.defining_ideal()
         return I.plot(*args, **kwds)
 
+    def is_transverse(self, C, P):
+        r"""
+        Return whether the intersection of this curve with the curve ``C`` at the point ``P`` is transverse.
+
+        The intersection at ``P`` is transverse if ``P`` is a nonsingular point of both curves, and if the
+        tangents of the curves at ``P`` are distinct.
+
+        INPUT:
+
+        - ``C`` -- a curve in the ambient space of this curve.
+
+        - ``P`` -- a point in the intersection of both curves.
+
+        OUPUT: Boolean.
+
+        EXAMPLES::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: C = Curve([x^2 + y^2 - 1], A)
+            sage: D = Curve([x - 1], A)
+            sage: Q = A([1,0])
+            sage: C.is_transverse(D, Q)
+            False
+
+        ::
+
+            sage: R.<a> = QQ[]
+            sage: K.<b> = NumberField(a^3 + 2)
+            sage: A.<x,y> = AffineSpace(K, 2)
+            sage: C = A.curve([x*y])
+            sage: D = A.curve([y - b*x])
+            sage: Q = A([0,0])
+            sage: C.is_transverse(D, Q)
+            False
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: C = Curve([y - x^3], A)
+            sage: D = Curve([y + x], A)
+            sage: Q = A([0,0])
+            sage: C.is_transverse(D, Q)
+            True
+        """
+        if not self.intersects_at(C, P):
+            raise TypeError("(=%s) must be a point in the intersection of (=%s) and this curve"%(P,C))
+        if self.is_singular(P) or C.is_singular(P):
+            return False
+
+        # there is only one tangent at a nonsingular point of a plane curve
+        return not self.tangents(P)[0] == C.tangents(P)[0]
+
     def multiplicity(self, P):
         r"""
         Return the multiplicity of this affine plane curve at the point ``P``.

src/sage/schemes/curves/curve.py

@@ -2,10 +2,10 @@
 Generic curves.
 """
 
+from sage.categories.finite_fields import FiniteFields
 from sage.categories.fields import Fields
 from sage.misc.all import latex
 
-
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
 
 from sage.schemes.generic.divisor_group import DivisorGroup
@@ -309,3 +309,115 @@ def is_singular(self, P=None):
             True
         """
         return not self.is_smooth(P)
+
+    def intersects_at(self, C, P):
+        r"""
+        Return whether the point ``P`` is or is not in the intersection of this curve with the curve ``C``.
+
+        INPUT:
+
+        - ``C`` -- a curve in the same ambient space as this curve.
+
+        - ``P`` -- a point in the ambient space of this curve.
+
+        OUTPUT:
+
+        Boolean.
+
+        EXAMPLES::
+
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: C = Curve([x^2 - z^2, y^3 - w*x^2], P)
+            sage: D = Curve([w^2 - 2*x*y + z^2, y^2 - w^2], P)
+            sage: Q1 = P([1,1,-1,1])
+            sage: C.intersects_at(D, Q1)
+            True
+            sage: Q2 = P([0,0,1,-1])
+            sage: C.intersects_at(D, Q2)
+            False
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(GF(13), 2)
+            sage: C = Curve([y + 12*x^5 + 3*x^3 + 7], A)
+            sage: D = Curve([y^2 + 7*x^2 + 8], A)
+            sage: Q1 = A([9,6])
+            sage: C.intersects_at(D, Q1)
+            True
+            sage: Q2 = A([3,7])
+            sage: C.intersects_at(D, Q2)
+            False
+        """
+        if C.ambient_space() != self.ambient_space():
+            raise TypeError("(=%s) must be a curve in the same ambient space as (=%s)"%(C,self))
+        if not isinstance(C, Curve_generic):
+            raise TypeError("(=%s) must be a curve"%C)
+        try:
+            P = self.ambient_space()(P)
+        except TypeError:
+            raise TypeError("(=%s) must be a point in the ambient space of this curve"%P)
+        try:
+            P = self(P)
+            P = C(P)
+        except TypeError:
+            return False
+        return True
+
+    def intersection_points(self, C, F=None):
+        r"""
+        Return the points in the intersection of this curve and the curve ``C``.
+
+        If the intersection of these two curves has dimension greater than zero, and if
+        the base ring of this curve is not a finite field, then an error is returned.
+
+        INPUT:
+
+        - ``C`` -- a curve in the same ambient space as this curve.
+
+        - ``F`` -- (default: None). Field over which to compute the intersection points. If not specified,
+          the base ring of this curve is used.
+
+        OUTPUT:
+
+        - a list of points in the ambient space of this curve.
+
+        EXAMPLES::
+
+            sage: R.<a> = QQ[]
+            sage: K.<b> = NumberField(a^2 + a + 1)
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: C = Curve([y^2 - w*z, w^3 - y^3], P)
+            sage: D = Curve([x*y - w*z, z^3 - y^3], P)
+            sage: C.intersection_points(D, F=K)
+            [(-b - 1 : -b - 1 : b : 1), (b : b : -b - 1 : 1), (1 : 0 : 0 : 0), (1 : 1 : 1 : 1)]
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(GF(7), 2)
+            sage: C = Curve([y^3 - x^3], A)
+            sage: D = Curve([-x*y^3 + y^4 - 2*x^3 + 2*x^2*y], A)
+            sage: C.intersection_points(D)
+            [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 3), (5, 5), (5, 6), (6, 6)]
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: C = Curve([y^3 - x^3], A)
+            sage: D = Curve([-x*y^3 + y^4 - 2*x^3 + 2*x^2*y], A)
+            sage: C.intersection_points(D)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: the intersection must have dimension zero or
+            (=Rational Field) must be a finite field
+        """
+        if C.ambient_space() != self.ambient_space():
+            raise TypeError("(=%s) must be a curve in the same ambient space as (=%s)"%(C,self))
+        if not isinstance(C, Curve_generic):
+            raise TypeError("(=%s) must be a curve"%C)
+        X = self.intersection(C)
+        if F is None:
+            F = self.base_ring()
+        if X.dimension() == 0 or F in FiniteFields():
+            return X.rational_points(F=F)
+        else:
+            raise NotImplementedError("the intersection must have dimension zero or (=%s) must be a finite field"%F)

src/sage/schemes/curves/projective_curve.py

@@ -233,6 +233,36 @@ def multiplicity(self, P):
         Q = [1/t*Q[j] for j in range(self.ambient_space().dimension_relative())]
         return C.multiplicity(C.ambient_space()(Q))
 
+    def is_complete_intersection(self):
+        r"""
+        Return whether this projective curve is or is not a complete intersection.
+
+        OUTPUT: Boolean.
+
+        EXAMPLES::
+
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: C = Curve([x*y - z*w, x^2 - y*w, y^2*w - x*z*w], P)
+            sage: C.is_complete_intersection()
+            False
+
+        ::
+
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: C = Curve([y*w - x^2, z*w^2 - x^3], P)
+            sage: C.is_complete_intersection()
+            True
+
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: C = Curve([z^2 - y*w, y*z - x*w, y^2 - x*z], P)
+            sage: C.is_complete_intersection()
+            False
+        """
+        singular.lib("sing.lib")
+        I = singular.simplify(self.defining_ideal(), 10)
+        L = singular.is_ci(I).sage()
+        return len(self.ambient_space().gens()) - len(I.sage().gens()) == L[-1]
+
 class ProjectivePlaneCurve(ProjectiveCurve):
     def __init__(self, A, f):
         r"""
@@ -691,6 +721,57 @@ def is_ordinary_singularity(self, P):
         # otherwise they are distinct
         return True
 
+    def is_transverse(self, C, P):
+        r"""
+        Return whether the intersection of this curve with the curve ``C`` at the point ``P`` is transverse.
+
+        The intersection at ``P`` is transverse if ``P`` is a nonsingular point of both curves, and if the
+        tangents of the curves at ``P`` are distinct.
+
+        INPUT:
+
+        - ``C`` -- a curve in the ambient space of this curve.
+
+        - ``P`` -- a point in the intersection of both curves.
+
+        OUPUT: Boolean.
+
+        EXAMPLES::
+
+            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
+            sage: C = Curve([x^2 - y^2], P)
+            sage: D = Curve([x - y], P)
+            sage: Q = P([1,1,0])
+            sage: C.is_transverse(D, Q)
+            False
+
+        ::
+
+            sage: K = QuadraticField(-1)
+            sage: P.<x,y,z> = ProjectiveSpace(K, 2)
+            sage: C = Curve([y^2*z - K.0*x^3], P)
+            sage: D = Curve([z*x + y^2], P)
+            sage: Q = P([0,0,1])
+            sage: C.is_transverse(D, Q)
+            False
+
+        ::
+
+            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
+            sage: C = Curve([x^2 - 2*y^2 - 2*z^2], P)
+            sage: D = Curve([y - z], P)
+            sage: Q = P([2,1,1])
+            sage: C.is_transverse(D, Q)
+            True
+        """
+        if not self.intersects_at(C, P):
+            raise TypeError("(=%s) must be a point in the intersection of (=%s) and this curve"%(P,C))
+        if self.is_singular(P) or C.is_singular(P):
+            return False
+
+        # there is only one tangent at a nonsingular point of a plane curve
+        return not self.tangents(P)[0] == C.tangents(P)[0]
+
 class ProjectivePlaneCurve_finite_field(ProjectivePlaneCurve):
     def rational_points_iterator(self):
         r"""