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)
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)
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)])
Traceback (most recent call last):
...
TypeError: polys (=[3/5*x^2, y^2/(2*x^2)]) must be rational functions in
Multivariate Polynomial Ring in x, y over Integer Ring
::
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() == source_ring.base_ring()
for p in polys):
raise TypeError(
"polys (=%s) must be rational functions in %s"
% (polys, source_ring))
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)
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)