def homogenize(self,n,newvar='h'):
r"""
Return the homogenization of ``self``. If ``self.domain()`` is a subscheme, the domain of
the homogenized map is the projective embedding of ``self.domain()``. The domain and codomain
can be homogenized at different coordinates: ``n[0]`` for the domain and ``n[1]`` for the codomain.
INPUT:
- ``newvar`` -- the name of the homogenization variable (only used when ``self.domain()`` is affine space)
- ``n`` -- a tuple of nonnegative integers. If ``n`` is an integer, then the two values of
the tuple are assumed to be the same.
OUTPUT:
- :class:`SchemMorphism_polynomial_projective_space`
EXAMPLES::
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/x^5,y^2])
sage: f.homogenize(2,'z')
Scheme endomorphism of Projective Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x : y : z) to
(x^2*z^5 - 2*z^7 : x^5*y^2 : x^5*z^2)
::
sage: A.<x,y>=AffineSpace(CC,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/(x*y),y^2-x])
sage: f.homogenize((2,0),'z')
Scheme endomorphism of Projective Space of dimension 2 over Complex
Field with 53 bits of precision
Defn: Defined on coordinates by sending (x : y : z) to
(x*y*z^2 : x^2*z^2 + (-2.00000000000000)*z^4 : x*y^3 - x^2*y*z)
::
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: X=A.subscheme([x-y^2])
sage: H=Hom(X,X)
sage: f=H([9*y^2,3*y])
sage: f.homogenize(2)
Scheme endomorphism of Closed subscheme of Projective Space of dimension 2 over Integer Ring defined by:
-x1^2 + x0*x2
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(9*x0*x2 : 3*x1*x2 : x2^2)
::
sage: R.<t>=PolynomialRing(ZZ)
sage: A.<x,y>=AffineSpace(R,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/y,y^2-x])
sage: f.homogenize((2,0),'z')
Scheme endomorphism of Projective Space of dimension 2 over Univariate
Polynomial Ring in t over Integer Ring
Defn: Defined on coordinates by sending (x : y : z) to
(y*z^2 : x^2*z + (-2)*z^3 : y^3 - x*y*z)
::
sage: A.<x>=AffineSpace(QQ,1)
sage: H=End(A)
sage: f=H([x^2-1])
sage: f.homogenize((1,0),'y')
Scheme endomorphism of Projective Space of dimension 1 over Rational
Field
Defn: Defined on coordinates by sending (x : y) to
(y^2 : x^2 - y^2)
"""
A=self.domain()
B=self.codomain()
N=A.ambient_space().dimension_relative()
NB=B.ambient_space().dimension_relative()
#it is possible to homogenize the domain and codomain at different coordinates
if isinstance(n,(tuple,list)):
ind=tuple(n)
else:
ind=(n,n)
#homogenize the domain
Vars=list(A.ambient_space().variable_names())
Vars.insert(ind[0],newvar)
S=PolynomialRing(A.base_ring(),Vars)
#find the denominators if a rational function
try:
l=lcm([self[i].denominator() for i in range(N)])
except Exception: #no lcm
l=prod([self[i].denominator() for i in range(N)])
from sage.rings.polynomial.polynomial_ring import PolynomialRing_general
from sage.rings.polynomial.multi_polynomial_ring_generic import MPolynomialRing_generic
if self.domain().base_ring()==RealField() or self.domain().base_ring()==ComplexField():
F=[S(((self[i]*l).numerator())._maxima_().divide(self[i].denominator())[0].sage()) for i in range(N)]
elif isinstance(self.domain().base_ring(),(PolynomialRing_general,MPolynomialRing_generic)):
F=[S(((self[i]*l).numerator())._maxima_().divide(self[i].denominator())[0].sage()) for i in range(N)]
else:
F=[S(self[i]*l) for i in range(N)]
#homogenize the codomain
F.insert(ind[1],S(l))
d=max([F[i].degree() for i in range(N+1)])
F=[F[i].homogenize(newvar)*S.gen(N)**(d-F[i].degree()) for i in range(N+1)]
from sage.schemes.affine.affine_space import is_AffineSpace
if is_AffineSpace(A)==True:
from sage.schemes.projective.projective_space import ProjectiveSpace
X=ProjectiveSpace(A.base_ring(),NB,Vars)
else:
X=A.projective_embedding(ind[1]).codomain()
phi=S.hom(X.ambient_space().gens(),X.ambient_space().coordinate_ring())
F=[phi(f) for f in F]
H=Hom(X,X)
return(H(F))
def homogenize(self,n,newvar='h'):
r"""
Return the homogenization of ``self``. If ``self.domain()`` is a subscheme, the domain of
the homogenized map is the projective embedding of ``self.domain()``
INPUT:
- ``newvar`` -- the name of the homogenization variable (only used when ``self.domain()`` is affine space)
- ``n`` -- the n-th projective embedding into projective space
OUTPUT:
- :class:`SchemMorphism_polynomial_projective_space`
EXAMPLES::
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/x^5,y^2])
sage: f.homogenize(2,'z')
Scheme endomorphism of Projective Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x : y : z) to
(x^2*z^5 - 2*z^7 : x^5*y^2 : x^5*z^2)
::
sage: A.<x,y>=AffineSpace(CC,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/(x*y),y^2-x])
sage: f.homogenize(0,'z')
Scheme endomorphism of Projective Space of dimension 2 over Complex
Field with 53 bits of precision
Defn: Defined on coordinates by sending (x : y : z) to
(x*y*z^2 : x^2*z^2 + (-2.00000000000000)*z^4 : x*y^3 - x^2*y*z)
::
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: X=A.subscheme([x-y^2])
sage: H=Hom(X,X)
sage: f=H([9*y^2,3*y])
sage: f.homogenize(2)
Scheme endomorphism of Closed subscheme of Projective Space of dimension 2 over Integer Ring defined by:
-x1^2 + x0*x2
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(9*x0*x2 : 3*x1*x2 : x2^2)
::
sage: R.<t>=PolynomialRing(ZZ)
sage: A.<x,y>=AffineSpace(R,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/y,y^2-x])
sage: f.homogenize(0,'z')
Scheme endomorphism of Projective Space of dimension 2 over Univariate
Polynomial Ring in t over Integer Ring
Defn: Defined on coordinates by sending (x : y : z) to
(y*z^2 : x^2*z + (-2)*z^3 : y^3 - x*y*z)
"""
A=self.domain()
B=self.codomain()
N=A.ambient_space().dimension_relative()
NB=B.ambient_space().dimension_relative()
Vars=list(A.ambient_space().variable_names())+[newvar]
S=PolynomialRing(A.base_ring(),Vars)
try:
l=lcm([self[i].denominator() for i in range(N)])
except Exception: #no lcm
l=prod([self[i].denominator() for i in range(N)])
from sage.rings.polynomial.polynomial_ring import PolynomialRing_general
from sage.rings.polynomial.multi_polynomial_ring_generic import MPolynomialRing_generic
if self.domain().base_ring()==RealField() or self.domain().base_ring()==ComplexField():
F=[S(((self[i]*l).numerator())._maxima_().divide(self[i].denominator())[0].sage()) for i in range(N)]
elif isinstance(self.domain().base_ring(),(PolynomialRing_general,MPolynomialRing_generic)):
F=[S(((self[i]*l).numerator())._maxima_().divide(self[i].denominator())[0].sage()) for i in range(N)]
else:
F=[S(self[i]*l) for i in range(N)]
F.insert(n,S(l))
d=max([F[i].degree() for i in range(N+1)])
F=[F[i].homogenize(newvar)*S.gen(N)**(d-F[i].degree()) for i in range(N+1)]
from sage.schemes.affine.affine_space import is_AffineSpace
if is_AffineSpace(A)==True:
from sage.schemes.projective.projective_space import ProjectiveSpace
X=ProjectiveSpace(A.base_ring(),NB,Vars)
else:
X=A.projective_embedding(n).codomain()
phi=S.hom(X.ambient_space().gens(),X.ambient_space().coordinate_ring())
F=[phi(f) for f in F]
H=Hom(X,X)
return(H(F))
def homogenize(self, n, newvar='h'):
r"""
Return the homogenization of ``self``. If ``self.domain()`` is a subscheme, the domain of
the homogenized map is the projective embedding of ``self.domain()``
INPUT:
- ``newvar`` -- the name of the homogenization variable (only used when ``self.domain()`` is affine space)
- ``n`` -- the n-th projective embedding into projective space
OUTPUT:
- :class:`SchemMorphism_polynomial_projective_space`
EXAMPLES::
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/x^5,y^2])
sage: f.homogenize(2,'z')
Scheme endomorphism of Projective Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x : y : z) to
(x^2*z^5 - 2*z^7 : x^5*y^2 : x^5*z^2)
::
sage: A.<x,y>=AffineSpace(CC,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/(x*y),y^2-x])
sage: f.homogenize(0,'z')
Scheme endomorphism of Projective Space of dimension 2 over Complex
Field with 53 bits of precision
Defn: Defined on coordinates by sending (x : y : z) to
(x*y*z^2 : x^2*z^2 + (-2.00000000000000)*z^4 : x*y^3 - x^2*y*z)
::
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: X=A.subscheme([x-y^2])
sage: H=Hom(X,X)
sage: f=H([9*y^2,3*y])
sage: f.homogenize(2)
Scheme endomorphism of Closed subscheme of Projective Space of dimension 2 over Integer Ring defined by:
-x1^2 + x0*x2
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(9*x0*x2 : 3*x1*x2 : x2^2)
::
sage: R.<t>=PolynomialRing(ZZ)
sage: A.<x,y>=AffineSpace(R,2)
sage: H=Hom(A,A)
sage: f=H([(x^2-2)/y,y^2-x])
sage: f.homogenize(0,'z')
Scheme endomorphism of Projective Space of dimension 2 over Univariate
Polynomial Ring in t over Integer Ring
Defn: Defined on coordinates by sending (x : y : z) to
(y*z^2 : x^2*z + (-2)*z^3 : y^3 - x*y*z)
"""
A = self.domain()
B = self.codomain()
N = A.ambient_space().dimension_relative()
NB = B.ambient_space().dimension_relative()
Vars = list(A.ambient_space().variable_names()) + [newvar]
S = PolynomialRing(A.base_ring(), Vars)
try:
l = lcm([self[i].denominator() for i in range(N)])
except Exception: #no lcm
l = prod([self[i].denominator() for i in range(N)])
from sage.rings.polynomial.polynomial_ring import PolynomialRing_general
from sage.rings.polynomial.multi_polynomial_ring_generic import MPolynomialRing_generic
if self.domain().base_ring() == RealField() or self.domain().base_ring(
) == ComplexField():
F = [
S(((self[i] * l).numerator())._maxima_().divide(
self[i].denominator())[0].sage()) for i in range(N)
]
elif isinstance(self.domain().base_ring(),
(PolynomialRing_general, MPolynomialRing_generic)):
F = [
S(((self[i] * l).numerator())._maxima_().divide(
self[i].denominator())[0].sage()) for i in range(N)
]
else:
F = [S(self[i] * l) for i in range(N)]
F.insert(n, S(l))
d = max([F[i].degree() for i in range(N + 1)])
F = [
F[i].homogenize(newvar) * S.gen(N)**(d - F[i].degree())
for i in range(N + 1)
]
from sage.schemes.affine.affine_space import is_AffineSpace
if is_AffineSpace(A) == True:
from sage.schemes.projective.projective_space import ProjectiveSpace
X = ProjectiveSpace(A.base_ring(), NB, Vars)
else:
X = A.projective_embedding(n).codomain()
phi = S.hom(X.ambient_space().gens(),
X.ambient_space().coordinate_ring())
F = [phi(f) for f in F]
H = Hom(X, X)
return (H(F))
