Python PolynomialRing.variable_names方法代码示例

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import variable_names [as 别名]
    def dual(self):
        r"""
        Return the projective dual of the given subscheme of projective space.

        INPUT:

        - ``X`` -- A subscheme of projective space. At present, ``X`` is
          required to be an irreducible and reduced hypersurface defined
          over `\QQ` or a finite field.

        OUTPUT:

        - The dual of ``X`` as a subscheme of the dual projective space.

        EXAMPLES:

        The dual of a smooth conic in the plane is also a smooth conic::

            sage: R.<x, y, z> = QQ[]
            sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
            sage: I = R.ideal(x^2 + y^2 + z^2)
            sage: X = P.subscheme(I)
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
              y0^2 + y1^2 + y2^2

        The dual of the twisted cubic curve in projective 3-space is a singular
        quartic surface. In the following example, we compute the dual of this
        surface, which by double duality is equal to the twisted cubic itself.
        The output is the twisted cubic as an intersection of three quadrics::

            sage: R.<x, y, z, w> = QQ[]
            sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
            sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
            sage: X = P.subscheme(I)
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 3 over
            Rational Field defined by:
              y2^2 - y1*y3,
              y1*y2 - y0*y3,
              y1^2 - y0*y2

        The singular locus of the quartic surface in the last example
        is itself supported on a twisted cubic::

            sage: X.Jacobian().radical()
            Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
            Polynomial Ring in x, y, z, w over Rational Field

        An example over a finite field::

            sage: R = PolynomialRing(GF(61), 'a,b,c')
            sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
            sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 2 over
            Finite Field of size 61 defined by:
            y0^2 - 30*y1^2 - 20*y2^2

        TESTS::

            sage: R = PolynomialRing(Qp(3), 'a,b,c')
            sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
            sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
            sage: X.dual()
            Traceback (most recent call last):
            ...
            NotImplementedError: base ring must be QQ or a finite field
        """
        from sage.libs.singular.function_factory import ff

        K = self.base_ring()
        if not(is_RationalField(K) or is_FiniteField(K)):
            raise NotImplementedError("base ring must be QQ or a finite field")
        I = self.defining_ideal()
        m = I.ngens()
        n = I.ring().ngens() - 1
        if (m != 1 or (n < 1) or I.is_zero()
            or I.is_trivial() or not I.is_prime()):
            raise NotImplementedError("At the present, the method is only"
                                      " implemented for irreducible and"
                                      " reduced hypersurfaces and the given"
                                      " list of generators for the ideal must"
                                      " have exactly one element.")
        R = PolynomialRing(K, 'x', n + 1)
        from sage.schemes.projective.projective_space import ProjectiveSpace
        Pd = ProjectiveSpace(n, K, 'y')
        Rd = Pd.coordinate_ring()
        x = R.variable_names()
        y = Rd.variable_names()
        S = PolynomialRing(K, x + y + ('t',))
        if S.has_coerce_map_from(I.ring()):
            T = PolynomialRing(K, 'w', n + 1)
            I_S = (I.change_ring(T)).change_ring(S)
        else:
            I_S = I.change_ring(S)
        f_S = I_S.gens()[0]
        z = S.gens()
        J = I_S
        for i in range(n + 1):
#.........这里部分代码省略.........

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import variable_names [as 别名]
    def Chow_form(self):
        r"""
        Returns the Chow form associated to this subscheme.

        For a `k`-dimensional subvariety of `\mathbb{P}^N` of degree `D`.
        The `(N-k-1)`-dimensional projective linear subspaces of `\mathbb{P}^N`
        meeting `X` form a hypersurface in the Grassmannian `G(N-k-1,N)`.
        The homogeneous form of degree `D` defining this hypersurface in Plucker
        coordinates is called the Chow form of `X`.

        The base ring needs to be a number field, finite field, or `\QQbar`.

        ALGORITHM:

        For a `k`-dimension subscheme `X` consider the `k+1` linear forms
        `l_i = u_{i0}x_0 + \cdots + u_{in}x_n`. Let `J` be the ideal in the
        polynomial ring `K[x_i,u_{ij}]` defined by the equations of `X` and the `l_i`.
        Let `J'` be the saturation of `J` with respect to the irrelevant ideal of
        the ambient projective space of `X`. The elimination ideal `I = J' \cap K[u_{ij}]`
        is a principal ideal, let `R` be its generator. The Chow form is obtained by
        writing `R` as a polynomial in Plucker coordinates (i.e. bracket polynomials).
        [DalbecSturmfels]_.

        OUTPUT: a homogeneous polynomial.

        REFERENCES:

        .. [DalbecSturmfels] J. Dalbec and B. Sturmfels. Invariant methods in discrete and computational geometry,
           chapter Introduction to Chow forms, pages 37-58. Springer Netherlands, 1994.

        EXAMPLES::

            sage: P.<x0,x1,x2,x3> = ProjectiveSpace(GF(17), 3)
            sage: X = P.subscheme([x3+x1,x2-x0,x2-x3])
            sage: X.Chow_form()
            t0 - t1 + t2 + t3

        ::

            sage: P.<x0,x1,x2,x3> = ProjectiveSpace(QQ,3)
            sage: X = P.subscheme([x3^2 -101*x1^2 - 3*x2*x0])
            sage: X.Chow_form()
            t0^2 - 101*t2^2 - 3*t1*t3

        ::

            sage: P.<x0,x1,x2,x3>=ProjectiveSpace(QQ,3)
            sage: X = P.subscheme([x0*x2-x1^2, x0*x3-x1*x2, x1*x3-x2^2])
            sage: Ch = X.Chow_form(); Ch
            t2^3 + 2*t2^2*t3 + t2*t3^2 - 3*t1*t2*t4 - t1*t3*t4 + t0*t4^2 + t1^2*t5
            sage: Y = P.subscheme_from_Chow_form(Ch, 1); Y
            Closed subscheme of Projective Space of dimension 3 over Rational Field
            defined by:
              x2^2*x3 - x1*x3^2,
              -x2^3 + x0*x3^2,
              -x2^2*x3 + x1*x3^2,
              x1*x2*x3 - x0*x3^2,
              3*x1*x2^2 - 3*x0*x2*x3,
              -2*x1^2*x3 + 2*x0*x2*x3,
              -3*x1^2*x2 + 3*x0*x1*x3,
              x1^3 - x0^2*x3,
              x2^3 - x1*x2*x3,
              -3*x1*x2^2 + 2*x1^2*x3 + x0*x2*x3,
              2*x0*x2^2 - 2*x0*x1*x3,
              3*x1^2*x2 - 2*x0*x2^2 - x0*x1*x3,
              -x0*x1*x2 + x0^2*x3,
              -x0*x1^2 + x0^2*x2,
              -x1^3 + x0*x1*x2,
              x0*x1^2 - x0^2*x2
            sage: I = Y.defining_ideal()
            sage: I.saturation(I.ring().ideal(list(I.ring().gens())))[0]
            Ideal (x2^2 - x1*x3, x1*x2 - x0*x3, x1^2 - x0*x2) of Multivariate
            Polynomial Ring in x0, x1, x2, x3 over Rational Field
        """
        I = self.defining_ideal()
        P = self.ambient_space()
        R = P.coordinate_ring()
        N = P.dimension()+1
        d = self.dimension()
        #create the ring for the generic linear hyperplanes
        # u0x0 + u1x1 + ...
        SS = PolynomialRing(R.base_ring(), 'u', N*(d+1), order='lex')
        vars = SS.variable_names() + R.variable_names()
        S = PolynomialRing(R.base_ring(), vars, order='lex')
        n = S.ngens()
        newcoords = [S.gen(n-N+t) for t in range(N)]
        #map the generators of the subscheme into the ring with the hyperplane variables
        phi = R.hom(newcoords,S)
        phi(self.defining_polynomials()[0])
        #create the dim(X)+1 linear hyperplanes
        l = []
        for i in range(d+1):
            t = 0
            for j in range(N):
                t += S.gen(N*i + j)*newcoords[j]
            l.append(t)
        #intersect the hyperplanes with X
        J = phi(I) + S.ideal(l)
        #saturate the ideal with respect to the irrelevant ideal
        J2 = J.saturation(S.ideal([phi(t) for t in R.gens()]))[0]
#.........这里部分代码省略.........