示例#1
0
    def rational_points(self, algorithm="enum"):
        r"""
        Return sorted list of all rational points on this curve.

        INPUT:

        -  ``algorithm`` - string:

           +  ``'enum'`` - straightforward enumeration

           +  ``'bn'`` - via Singular's Brill-Noether package.

           +  ``'all'`` - use all implemented algorithms and
              verify that they give the same answer, then return it

        .. note::

           The Brill-Noether package does not always work. When it
           fails a RuntimeError exception is raised.

        EXAMPLE::

            sage: x, y = (GF(5)['x,y']).gens()
            sage: f = y^2 - x^9 - x
            sage: C = Curve(f); C
            Affine Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
            sage: C.rational_points(algorithm='bn')
            [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
            sage: C = Curve(x - y + 1)
            sage: C.rational_points()
            [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]

        We compare Brill-Noether and enumeration::

            sage: x, y = (GF(17)['x,y']).gens()
            sage: C = Curve(x^2 + y^5 + x*y - 19)
            sage: v = C.rational_points(algorithm='bn')
            sage: w = C.rational_points(algorithm='enum')
            sage: len(v)
            20
            sage: v == w
            True
        """
        if algorithm == "enum":

            return AffineCurve_finite_field.rational_points(self,
                                                            algorithm="enum")

        elif algorithm == "bn":
            f = self.defining_polynomial()._singular_()
            singular = f.parent()
            singular.lib('brnoeth')
            try:
                X1 = f.Adj_div()
            except (TypeError, RuntimeError), s:
                raise RuntimeError, str(
                    s
                ) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."

            X2 = singular.NSplaces(1, X1)
            R = X2[5][1][1]
            singular.set_ring(R)

            # We use sage_flattened_str_list since iterating through
            # the entire list through the sage/singular interface directly
            # would involve hundreds of calls to singular, and timing issues
            # with the expect interface could crop up.  Also, this is vastly
            # faster (and more robust).
            v = singular('POINTS').sage_flattened_str_list()
            pnts = [
                self(int(v[3 * i]), int(v[3 * i + 1]))
                for i in range(len(v) / 3) if int(v[3 * i + 2]) != 0
            ]
            # remove multiple points
            pnts = list(set(pnts))
            pnts.sort()
            return pnts
示例#2
0
    def riemann_roch_basis(self, D):
        r"""
        Return a basis for the Riemann-Roch space corresponding to
        `D`.

        This uses Singular's Brill-Noether implementation.

        INPUT:

        -  ``D`` - a divisor

        OUTPUT:

        A list of function field elements that form a basis of the Riemann-Roch space

        EXAMPLE::

            sage: R.<x,y,z> = GF(2)[]
            sage: f = x^3*y + y^3*z + x*z^3
            sage: C = Curve(f); pts = C.rational_points()
            sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])
            sage: C.riemann_roch_basis(D)
            [x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]

        ::

            sage: R.<x,y,z> = GF(5)[]
            sage: f = x^7 + y^7 + z^7
            sage: C = Curve(f); pts = C.rational_points()
            sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
            sage: C.riemann_roch_basis(D)
            [(-2*x + y)/(x + y), (-x + z)/(x + y)]


        .. NOTE::
            Currently this only works over prime field and divisors supported on rational points.
        """
        f = self.defining_polynomial()._singular_()
        singular = f.parent()
        singular.lib('brnoeth')
        try:
            X1 = f.Adj_div()
        except (TypeError, RuntimeError) as s:
            raise RuntimeError(str(s) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above).")
        X2 = singular.NSplaces(1, X1)
        # retrieve list of all computed closed points (possibly of degree >1)
        v = X2[3].sage_flattened_str_list()    # We use sage_flattened_str_list since iterating through
                                               # the entire list through the sage/singular interface directly
                                               # would involve hundreds of calls to singular, and timing issues with
                                               # the expect interface could crop up.  Also, this is vastly
                                               # faster (and more robust).
        v = [ v[i].partition(',') for i in range(len(v)) ]
        pnts = [ ( int(v[i][0]), int(v[i][2])-1 ) for i in range(len(v))]
        # retrieve coordinates of rational points
        R = X2[5][1][1]
        singular.set_ring(R)
        v = singular('POINTS').sage_flattened_str_list()
        coords = [self(int(v[3*i]), int(v[3*i+1]), int(v[3*i+2])) for i in range(len(v)//3)]
        # build correct representation of D for singular
        Dsupport = D.support()
        Dcoeffs = []
        for x in pnts:
            if x[0] == 1:
                Dcoeffs.append(D.coefficient(coords[x[1]]))
            else:
                Dcoeffs.append(0)
        Dstr = str(tuple(Dcoeffs))
        G = singular(','.join([str(x) for x in Dcoeffs]), type='intvec')
        # call singular's brill noether routine and return
        T = X2[1][2]
        T.set_ring()
        LG = G.BrillNoether(X2)
        LG = [X.split(',\n') for X in LG.sage_structured_str_list()]
        x,y,z = self.ambient_space().coordinate_ring().gens()
        vars = {'x':x, 'y':y, 'z':z}
        V = [(sage_eval(a, vars)/sage_eval(b, vars)) for a, b in LG]
        return V
示例#3
0
    def rational_points(self, algorithm="enum"):
        r"""
        Return sorted list of all rational points on this curve.

        INPUT:


        -  ``algorithm`` - string:

           +  ``'enum'`` - straightforward enumeration

           +  ``'bn'`` - via Singular's Brill-Noether package.

           +  ``'all'`` - use all implemented algorithms and
              verify that they give the same answer, then return it


        .. note::

           The Brill-Noether package does not always work. When it
           fails a RuntimeError exception is raised.

        EXAMPLE::

            sage: x, y = (GF(5)['x,y']).gens()
            sage: f = y^2 - x^9 - x
            sage: C = Curve(f); C
            Affine Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
            sage: C.rational_points(algorithm='bn')
            [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
            sage: C = Curve(x - y + 1)
            sage: C.rational_points()
            [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]

        We compare Brill-Noether and enumeration::

            sage: x, y = (GF(17)['x,y']).gens()
            sage: C = Curve(x^2 + y^5 + x*y - 19)
            sage: v = C.rational_points(algorithm='bn')
            sage: w = C.rational_points(algorithm='enum')
            sage: len(v)
            20
            sage: v == w
            True
        """
        if algorithm == "enum":

            return AffineCurve_finite_field.rational_points(self, algorithm="enum")

        elif algorithm == "bn":
            f = self.defining_polynomial()._singular_()
            singular = f.parent()
            singular.lib('brnoeth')
            try:
                X1 = f.Adj_div()
            except (TypeError, RuntimeError) as s:
                raise RuntimeError(str(s) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above).")

            X2 = singular.NSplaces(1, X1)
            R = X2[5][1][1]
            singular.set_ring(R)

            # We use sage_flattened_str_list since iterating through
            # the entire list through the sage/singular interface directly
            # would involve hundreds of calls to singular, and timing issues
            # with the expect interface could crop up.  Also, this is vastly
            # faster (and more robust).
            v = singular('POINTS').sage_flattened_str_list()
            pnts = [self(int(v[3*i]), int(v[3*i+1])) for i in range(len(v)/3) if int(v[3*i+2])!=0]
            # remove multiple points
            pnts = sorted(set(pnts))
            return pnts

        elif algorithm == "all":

            S_enum = self.rational_points(algorithm = "enum")
            S_bn = self.rational_points(algorithm = "bn")
            if S_enum != S_bn:
                raise RuntimeError("Bug in rational_points -- different algorithms give different answers for curve %s!"%self)
            return S_enum

        else:
            raise ValueError("No algorithm '%s' known"%algorithm)
示例#4
0
    def _points_via_singular(self, sort=True):
        r"""
        Return all rational points on this curve, computed using Singular's
        Brill-Noether implementation.

        INPUT:


        -  ``sort`` - bool (default: True), if True return the
           point list sorted. If False, returns the points in the order
           computed by Singular.


        EXAMPLE::

            sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
            sage: f = y^2*z^7 - x^9 - x*z^8
            sage: C = Curve(f); C
            Projective Curve over Finite Field of size 5 defined by
            -x^9 + y^2*z^7 - x*z^8
            sage: C._points_via_singular()
            [(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),
             (3 : 1 : 1), (3 : 4 : 1)]
            sage: C._points_via_singular(sort=False)     #random
            [(0 : 1 : 0), (3 : 1 : 1), (3 : 4 : 1), (2 : 2 : 1),
             (0 : 0 : 1), (2 : 3 : 1)]


        .. note::

           The Brill-Noether package does not always work (i.e., the
           'bn' algorithm. When it fails a RuntimeError exception is
           raised.
        """
        f = self.defining_polynomial()._singular_()
        singular = f.parent()
        singular.lib('brnoeth')
        try:
            X1 = f.Adj_div()
        except (TypeError, RuntimeError) as s:
            raise RuntimeError(str(s) + "\n\n ** Unable to use the\
                                          Brill-Noether Singular package to\
                                          compute all points (see above).")

        X2 = singular.NSplaces(1, X1)
        R = X2[5][1][1]
        singular.set_ring(R)

        # We use sage_flattened_str_list since iterating through
        # the entire list through the sage/singular interface directly
        # would involve hundreds of calls to singular, and timing issues with
        # the expect interface could crop up.  Also, this is vastly
        # faster (and more robust).
        v = singular('POINTS').sage_flattened_str_list()
        pnts = [self(int(v[3*i]), int(v[3*i+1]), int(v[3*i+2]))
                for i in range(len(v)//3)]
        # singular always dehomogenizes with respect to the last variable
        # so if this variable divides the curve equation, we need to add
        # points at infinity
        F = self.defining_polynomial()
        z = F.parent().gens()[-1]
        if z.divides(F):
            pnts += [self(1,a,0) for a in self.base_ring()]
            pnts += [self(0,1,0)]
        # remove multiple points
        pnts = list(set(pnts))
        if sort:
            pnts.sort()
        return pnts
示例#5
0
    def riemann_roch_basis(self, D):
        r"""
        Return a basis for the Riemann-Roch space corresponding to
        `D`.

        This uses Singular's Brill-Noether implementation.

        INPUT:

        -  ``D`` - a divisor

        OUTPUT:

        A list of function field elements that form a basis of the Riemann-Roch space

        EXAMPLE::

            sage: R.<x,y,z> = GF(2)[]
            sage: f = x^3*y + y^3*z + x*z^3
            sage: C = Curve(f); pts = C.rational_points()
            sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])
            sage: C.riemann_roch_basis(D)
            [x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]

        ::

            sage: R.<x,y,z> = GF(5)[]
            sage: f = x^7 + y^7 + z^7
            sage: C = Curve(f); pts = C.rational_points()
            sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
            sage: C.riemann_roch_basis(D)
            [(-2*x + y)/(x + y), (-x + z)/(x + y)]


        .. NOTE::
            Currently this only works over prime field and divisors supported on rational points.
        """
        f = self.defining_polynomial()._singular_()
        singular = f.parent()
        singular.lib('brnoeth')
        try:
            X1 = f.Adj_div()
        except (TypeError, RuntimeError) as s:
            raise RuntimeError(
                str(s) +
                "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."
            )
        X2 = singular.NSplaces(1, X1)
        # retrieve list of all computed closed points (possibly of degree >1)
        v = X2[3].sage_flattened_str_list(
        )  # We use sage_flattened_str_list since iterating through
        # the entire list through the sage/singular interface directly
        # would involve hundreds of calls to singular, and timing issues with
        # the expect interface could crop up.  Also, this is vastly
        # faster (and more robust).
        v = [v[i].partition(',') for i in range(len(v))]
        pnts = [(int(v[i][0]), int(v[i][2]) - 1) for i in range(len(v))]
        # retrieve coordinates of rational points
        R = X2[5][1][1]
        singular.set_ring(R)
        v = singular('POINTS').sage_flattened_str_list()
        coords = [
            self(int(v[3 * i]), int(v[3 * i + 1]), int(v[3 * i + 2]))
            for i in range(len(v) // 3)
        ]
        # build correct representation of D for singular
        Dsupport = D.support()
        Dcoeffs = []
        for x in pnts:
            if x[0] == 1:
                Dcoeffs.append(D.coefficient(coords[x[1]]))
            else:
                Dcoeffs.append(0)
        Dstr = str(tuple(Dcoeffs))
        G = singular(','.join([str(x) for x in Dcoeffs]), type='intvec')
        # call singular's brill noether routine and return
        T = X2[1][2]
        T.set_ring()
        LG = G.BrillNoether(X2)
        LG = [X.split(',\n') for X in LG.sage_structured_str_list()]
        x, y, z = self.ambient_space().coordinate_ring().gens()
        vars = {'x': x, 'y': y, 'z': z}
        V = [(sage_eval(a, vars) / sage_eval(b, vars)) for a, b in LG]
        return V
示例#6
0
    def _points_via_singular(self, sort=True):
        r"""
        Return all rational points on this curve, computed using Singular's
        Brill-Noether implementation.

        INPUT:


        -  ``sort`` - bool (default: True), if True return the
           point list sorted. If False, returns the points in the order
           computed by Singular.


        EXAMPLE::

            sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
            sage: f = y^2*z^7 - x^9 - x*z^8
            sage: C = Curve(f); C
            Projective Curve over Finite Field of size 5 defined by
            -x^9 + y^2*z^7 - x*z^8
            sage: C._points_via_singular()
            [(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),
             (3 : 1 : 1), (3 : 4 : 1)]
            sage: C._points_via_singular(sort=False)     #random
            [(0 : 1 : 0), (3 : 1 : 1), (3 : 4 : 1), (2 : 2 : 1),
             (0 : 0 : 1), (2 : 3 : 1)]


        .. note::

           The Brill-Noether package does not always work (i.e., the
           'bn' algorithm. When it fails a RuntimeError exception is
           raised.
        """
        f = self.defining_polynomial()._singular_()
        singular = f.parent()
        singular.lib('brnoeth')
        try:
            X1 = f.Adj_div()
        except (TypeError, RuntimeError) as s:
            raise RuntimeError(
                str(s) + "\n\n ** Unable to use the\
                                          Brill-Noether Singular package to\
                                          compute all points (see above).")

        X2 = singular.NSplaces(1, X1)
        R = X2[5][1][1]
        singular.set_ring(R)

        # We use sage_flattened_str_list since iterating through
        # the entire list through the sage/singular interface directly
        # would involve hundreds of calls to singular, and timing issues with
        # the expect interface could crop up.  Also, this is vastly
        # faster (and more robust).
        v = singular('POINTS').sage_flattened_str_list()
        pnts = [
            self(int(v[3 * i]), int(v[3 * i + 1]), int(v[3 * i + 2]))
            for i in range(len(v) // 3)
        ]
        # singular always dehomogenizes with respect to the last variable
        # so if this variable divides the curve equation, we need to add
        # points at infinity
        F = self.defining_polynomial()
        z = F.parent().gens()[-1]
        if z.divides(F):
            pnts += [self(1, a, 0) for a in self.base_ring()]
            pnts += [self(0, 1, 0)]
        # remove multiple points
        pnts = list(set(pnts))
        if sort:
            pnts.sort()
        return pnts
示例#7
0
 try:
     X1 = f.Adj_div()
 except (TypeError, RuntimeError), s:
     raise RuntimeError, str(s) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."
 X2 = singular.NSplaces(1, X1)
 # retrieve list of all computed closed points (possibly of degree >1)
 v = X2[3].sage_flattened_str_list()    # We use sage_flattened_str_list since iterating through
                                        # the entire list through the sage/singular interface directly
                                        # would involve hundreds of calls to singular, and timing issues with
                                        # the expect interface could crop up.  Also, this is vastly
                                        # faster (and more robust).
 v = [ v[i].partition(',') for i in range(len(v)) ]
 pnts = [ ( int(v[i][0]), int(v[i][2])-1 ) for i in range(len(v))]
 # retrieve coordinates of rational points
 R = X2[5][1][1]
 singular.set_ring(R)
 v = singular('POINTS').sage_flattened_str_list()
 coords = [self(int(v[3*i]), int(v[3*i+1]), int(v[3*i+2])) for i in range(len(v)/3)]
 # build correct representation of D for singular
 Dsupport = D.support()
 Dcoeffs = []
 for x in pnts:
     if x[0] == 1:
         Dcoeffs.append(D.coefficient(coords[x[1]]))
     else:
         Dcoeffs.append(0)
 Dstr = str(tuple(Dcoeffs))
 G = singular(','.join([str(x) for x in Dcoeffs]), type='intvec')
 # call singular's brill noether routine and return
 T = X2[1][2]
 T.set_ring()
示例#8
0
     raise RuntimeError, str(
         s
     ) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."
 X2 = singular.NSplaces(1, X1)
 # retrieve list of all computed closed points (possibly of degree >1)
 v = X2[3].sage_flattened_str_list(
 )  # We use sage_flattened_str_list since iterating through
 # the entire list through the sage/singular interface directly
 # would involve hundreds of calls to singular, and timing issues with
 # the expect interface could crop up.  Also, this is vastly
 # faster (and more robust).
 v = [v[i].partition(',') for i in range(len(v))]
 pnts = [(int(v[i][0]), int(v[i][2]) - 1) for i in range(len(v))]
 # retrieve coordinates of rational points
 R = X2[5][1][1]
 singular.set_ring(R)
 v = singular('POINTS').sage_flattened_str_list()
 coords = [
     self(int(v[3 * i]), int(v[3 * i + 1]), int(v[3 * i + 2]))
     for i in range(len(v) / 3)
 ]
 # build correct representation of D for singular
 Dsupport = D.support()
 Dcoeffs = []
 for x in pnts:
     if x[0] == 1:
         Dcoeffs.append(D.coefficient(coords[x[1]]))
     else:
         Dcoeffs.append(0)
 Dstr = str(tuple(Dcoeffs))
 G = singular(','.join([str(x) for x in Dcoeffs]), type='intvec')