Exemple #1
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    def __init__(self, polys_or_rat_fncts, domain):
        r"""
        The Python constructor.

        See :class:`DynamicalSystem` for details.

        EXAMPLES::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: DynamicalSystem_affine([3/5*x^2, y^2/(2*x^2)], domain=A)
            Dynamical System of Affine Space of dimension 2 over Rational Field
              Defn: Defined on coordinates by sending (x, y) to
                    (3/5*x^2, y^2/(2*x^2))
        """
        L = polys_or_rat_fncts
        # Next attribute needed for _fast_eval and _fastpolys
        self._is_prime_finite_field = is_PrimeFiniteField(L[0].base_ring())
        DynamicalSystem.__init__(self, L, domain)
Exemple #2
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def Conic(base_field, F=None, names=None, unique=True):
    r"""
    Return the plane projective conic curve defined by ``F``
    over ``base_field``.

    The input form ``Conic(F, names=None)`` is also accepted,
    in which case the fraction field of the base ring of ``F``
    is used as base field.

    INPUT:

    - ``base_field`` -- The base field of the conic.

    - ``names`` -- a list, tuple, or comma separated string
      of three variable names specifying the names
      of the coordinate functions of the ambient
      space `\Bold{P}^3`. If not specified or read
      off from ``F``, then this defaults to ``'x,y,z'``.

    - ``F`` -- a polynomial, list, matrix, ternary quadratic form,
      or list or tuple of 5 points in the plane.

                   If ``F`` is a polynomial or quadratic form,
                   then the output is the curve in the projective plane
                   defined by ``F = 0``.

                   If ``F`` is a polynomial, then it must be a polynomial
                   of degree at most 2 in 2 variables, or a homogeneous
                   polynomial in of degree 2 in 3 variables.

                   If ``F`` is a matrix, then the output is the zero locus
                   of `(x,y,z) F (x,y,z)^t`.

                   If ``F`` is a list of coefficients, then it has
                   length 3 or 6 and gives the coefficients of
                   the monomials `x^2, y^2, z^2` or all 6 monomials
                   `x^2, xy, xz, y^2, yz, z^2` in lexicographic order.

                   If ``F`` is a list of 5 points in the plane, then the output
                   is a conic through those points.

    - ``unique`` -- Used only if ``F`` is a list of points in the plane.
      If the conic through the points is not unique, then
      raise ``ValueError`` if and only if ``unique`` is True

    OUTPUT:

    A plane projective conic curve defined by ``F`` over a field.

    EXAMPLES:

    Conic curves given by polynomials ::

        sage: X,Y,Z = QQ['X,Y,Z'].gens()
        sage: Conic(X^2 - X*Y + Y^2 - Z^2)
        Projective Conic Curve over Rational Field defined by X^2 - X*Y + Y^2 - Z^2
        sage: x,y = GF(7)['x,y'].gens()
        sage: Conic(x^2 - x + 2*y^2 - 3, 'U,V,W')
        Projective Conic Curve over Finite Field of size 7 defined by U^2 + 2*V^2 - U*W - 3*W^2

    Conic curves given by matrices ::

        sage: Conic(matrix(QQ, [[1, 2, 0], [4, 0, 0], [7, 0, 9]]), 'x,y,z')
        Projective Conic Curve over Rational Field defined by x^2 + 6*x*y + 7*x*z + 9*z^2

        sage: x,y,z = GF(11)['x,y,z'].gens()
        sage: C = Conic(x^2+y^2-2*z^2); C
        Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
        sage: Conic(C.symmetric_matrix(), 'x,y,z')
        Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2

    Conics given by coefficients ::

        sage: Conic(QQ, [1,2,3])
        Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + 3*z^2
        sage: Conic(GF(7), [1,2,3,4,5,6], 'X')
        Projective Conic Curve over Finite Field of size 7 defined by X0^2 + 2*X0*X1 - 3*X1^2 + 3*X0*X2 - 2*X1*X2 - X2^2

    The conic through a set of points ::

        sage: C = Conic(QQ, [[10,2],[3,4],[-7,6],[7,8],[9,10]]); C
        Projective Conic Curve over Rational Field defined by x^2 + 13/4*x*y - 17/4*y^2 - 35/2*x*z + 91/4*y*z - 37/2*z^2
        sage: C.rational_point()
        (10 : 2 : 1)
        sage: C.point([3,4])
        (3 : 4 : 1)

        sage: a=AffineSpace(GF(13),2)
        sage: Conic([a([x,x^2]) for x in range(5)])
        Projective Conic Curve over Finite Field of size 13 defined by x^2 - y*z
    """
    if not (base_field is None or isinstance(base_field, IntegralDomain)):
        if names is None:
            names = F
        F = base_field
        base_field = None
    if isinstance(F, (list,tuple)):
        if len(F) == 1:
            return Conic(base_field, F[0], names)
        if names is None:
            names = 'x,y,z'
        if len(F) == 5:
            L=[]
            for f in F:
                if isinstance(f, SchemeMorphism_point_affine):
                    C = Sequence(f, universe = base_field)
                    if len(C) != 2:
                        raise TypeError("points in F (=%s) must be planar"%F)
                    C.append(1)
                elif isinstance(f, SchemeMorphism_point_projective_field):
                    C = Sequence(f, universe = base_field)
                elif isinstance(f, (list, tuple)):
                    C = Sequence(f, universe = base_field)
                    if len(C) == 2:
                        C.append(1)
                else:
                    raise TypeError("F (=%s) must be a sequence of planar " \
                                      "points" % F)
                if len(C) != 3:
                    raise TypeError("points in F (=%s) must be planar" % F)
                P = C.universe()
                if not isinstance(P, IntegralDomain):
                    raise TypeError("coordinates of points in F (=%s) must " \
                                     "be in an integral domain" % F)
                L.append(Sequence([C[0]**2, C[0]*C[1], C[0]*C[2], C[1]**2,
                                   C[1]*C[2], C[2]**2], P.fraction_field()))
            M=Matrix(L)
            if unique and M.rank() != 5:
                raise ValueError("points in F (=%s) do not define a unique " \
                                   "conic" % F)
            con = Conic(base_field, Sequence(M.right_kernel().gen()), names)
            con.point(F[0])
            return con
        F = Sequence(F, universe = base_field)
        base_field = F.universe().fraction_field()
        temp_ring = PolynomialRing(base_field, 3, names)
        (x,y,z) = temp_ring.gens()
        if len(F) == 3:
            return Conic(F[0]*x**2 + F[1]*y**2 + F[2]*z**2)
        if len(F) == 6:
            return Conic(F[0]*x**2 + F[1]*x*y + F[2]*x*z + F[3]*y**2 + \
                         F[4]*y*z + F[5]*z**2)
        raise TypeError("F (=%s) must be a sequence of 3 or 6" \
                         "coefficients" % F)
    if is_QuadraticForm(F):
        F = F.matrix()
    if is_Matrix(F) and F.is_square() and F.ncols() == 3:
        if names is None:
            names = 'x,y,z'
        temp_ring = PolynomialRing(F.base_ring(), 3, names)
        F = vector(temp_ring.gens()) * F * vector(temp_ring.gens())

    if not is_MPolynomial(F):
        raise TypeError("F (=%s) must be a three-variable polynomial or " \
                         "a sequence of points or coefficients" % F)

    if F.total_degree() != 2:
        raise TypeError("F (=%s) must have degree 2" % F)

    if base_field is None:
        base_field = F.base_ring()
    if not isinstance(base_field, IntegralDomain):
        raise ValueError("Base field (=%s) must be a field" % base_field)
    base_field = base_field.fraction_field()
    if names is None:
        names = F.parent().variable_names()
    pol_ring = PolynomialRing(base_field, 3, names)

    if F.parent().ngens() == 2:
        (x,y,z) = pol_ring.gens()
        F = pol_ring(F(x/z,y/z)*z**2)

    if F == 0:
        raise ValueError("F must be nonzero over base field %s" % base_field)

    if F.total_degree() != 2:
        raise TypeError("F (=%s) must have degree 2 over base field %s" % \
                          (F, base_field))

    if F.parent().ngens() == 3:
        P2 = ProjectiveSpace(2, base_field, names)
        if is_PrimeFiniteField(base_field):
            return ProjectiveConic_prime_finite_field(P2, F)
        if is_FiniteField(base_field):
            return ProjectiveConic_finite_field(P2, F)
        if is_RationalField(base_field):
            return ProjectiveConic_rational_field(P2, F)
        if is_NumberField(base_field):
            return ProjectiveConic_number_field(P2, F)
        if is_FractionField(base_field) and (is_PolynomialRing(base_field.ring()) or is_MPolynomialRing(base_field.ring())):
            return ProjectiveConic_rational_function_field(P2, F)
            
        return ProjectiveConic_field(P2, F)

    raise TypeError("Number of variables of F (=%s) must be 2 or 3" % F)
    def __init__(self, parent, polys, check=True):
        r"""
        The Python constructor.

        See :class:`SchemeMorphism_polynomial` for details.

        INPUT:

        - ``parent`` -- Hom.

        - ``polys`` -- list or tuple of polynomial or rational functions.

        - ``check`` -- Boolean.

        OUTPUT:

        - :class:`SchemeMorphism_polynomial_affine_space`.

        EXAMPLES::

            sage: A.<x,y> = AffineSpace(ZZ, 2)
            sage: H = Hom(A, A)
            sage: H([3/5*x^2, y^2/(2*x^2)])
            Scheme endomorphism of Affine Space of dimension 2 over Integer Ring
              Defn: Defined on coordinates by sending (x, y) to
                    (3*x^2/5, y^2/(2*x^2))

        ::

            sage: A.<x,y> = AffineSpace(ZZ, 2)
            sage: H = Hom(A, A)
            sage: H([3*x^2/(5*y), y^2/(2*x^2)])
            Scheme endomorphism of Affine Space of dimension 2 over Integer Ring
              Defn: Defined on coordinates by sending (x, y) to
                    (3*x^2/(5*y), y^2/(2*x^2))

        ::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: H = Hom(A, A)
            sage: H([3/2*x^2, y^2])
            Scheme endomorphism of Affine Space of dimension 2 over Rational Field
              Defn: Defined on coordinates by sending (x, y) to
                    (3/2*x^2, y^2)

        ::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: X = A.subscheme([x-y^2])
            sage: H = Hom(X, X)
            sage: H([9/4*x^2, 3/2*y])
            Scheme endomorphism of Closed subscheme of Affine Space of dimension 2
            over Rational Field defined by:
              -y^2 + x
              Defn: Defined on coordinates by sending (x, y) to
                    (9/4*x^2, 3/2*y)

            sage: P.<x,y,z> = ProjectiveSpace(ZZ, 2)
            sage: H = Hom(P, P)
            sage: f = H([5*x^3 + 3*x*y^2-y^3, 3*z^3 + y*x^2, x^3-z^3])
            sage: f.dehomogenize(2)
            Scheme endomorphism of Affine Space of dimension 2 over Integer Ring
              Defn: Defined on coordinates by sending (x0, x1) to
                    ((5*x0^3 + 3*x0*x1^2 - x1^3)/(x0^3 - 1), (x0^2*x1 + 3)/(x0^3 - 1))

            If you pass in quotient ring elements, they are reduced::

            sage: A.<x,y,z> = AffineSpace(QQ, 3)
            sage: X = A.subscheme([x-y])
            sage: H = Hom(X,X)
            sage: u,v,w = X.coordinate_ring().gens()
            sage: H([u, v, u+v])
            Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
            over Rational Field defined by:
              x - y
              Defn: Defined on coordinates by sending (x, y, z) to
                    (y, y, 2*y)

            You must use the ambient space variables to create rational functions::

            sage: A.<x,y,z> = AffineSpace(QQ, 3)
            sage: X = A.subscheme([x^2-y^2])
            sage: H = Hom(X,X)
            sage: u,v,w = X.coordinate_ring().gens()
            sage: H([u, v, (u+1)/v])
            Traceback (most recent call last):
            ...
            ArithmeticError: Division failed. The numerator is not a multiple of the denominator.
            sage: H([x, y, (x+1)/y])
            Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
            over Rational Field defined by:
              x^2 - y^2
              Defn: Defined on coordinates by sending (x, y, z) to
                    (x, y, (x + 1)/y)

            ::

            sage: R.<t> = PolynomialRing(QQ)
            sage: A.<x,y,z> = AffineSpace(R, 3)
            sage: X = A.subscheme(x^2-y^2)
            sage: H = End(X)
            sage: H([x^2/(t*y), t*y^2, x*z])
            Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
            over Univariate Polynomial Ring in t over Rational Field defined by:
              x^2 - y^2
              Defn: Defined on coordinates by sending (x, y, z) to
                    (x^2/(t*y), t*y^2, x*z)
        """
        if check:
            if not isinstance(polys, (list, tuple)):
                raise TypeError("polys (=%s) must be a list or tuple"%polys)
            source_ring = parent.domain().ambient_space().coordinate_ring()
            target = parent.codomain().ambient_space()
            if len(polys) != target.ngens():
                raise ValueError("there must be %s polynomials"%target.ngens())
            try:
                polys = [source_ring(poly) for poly in polys]
            except TypeError: #maybe given quotient ring elements
                try:
                   polys = [source_ring(poly.lift()) for poly in polys]
                except (TypeError, AttributeError):
                    #must be a rational function since we cannot have
                    #rational functions for quotient rings
                    try:
                        if not all(p.base_ring().fraction_field()==source_ring.base_ring().fraction_field() for p in polys):
                            raise TypeError("polys (=%s) must be rational functions in %s"%(polys, source_ring))
                        K = FractionField(source_ring)
                        polys = [K(p) for p in polys]
                        #polys = [source_ring(poly.numerator())/source_ring(poly.denominator()) for poly in polys]
                    except TypeError: #can't seem to coerce
                        raise TypeError("polys (=%s) must be rational functions in %s"%(polys, source_ring))
        self._is_prime_finite_field = is_PrimeFiniteField(polys[0].base_ring()) # Needed for _fast_eval and _fastpolys
        SchemeMorphism_polynomial.__init__(self, parent, polys, False)