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19552: fix issues from review
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@bhutz
bhutz committed Nov 25, 2015
1 parent 45697b7 commit df4701b
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165 changes: 131 additions & 34 deletions src/sage/schemes/generic/algebraic_scheme.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -2211,11 +2211,10 @@ def is_smooth(self, point=None):

def orbit(self, f, N):
r"""
Returns the orbit of `self` by ``f``.
Returns the orbit of this scheme by ``f``.
If `N` is an integer it returns `[self,f(self),\ldots,f^N(self)]`.
If `N` is a list or tuple `N=[m,k]` it returns `[f^m(self),\ldots,f^k(self)`].
Perform the checks on subscheme initialization if ``check=True``
INPUT:
Expand All @@ -2225,7 +2224,7 @@ def orbit(self, f, N):
OUTPUT:
- a proejctive subscheme
- a list of projective subschemes
EXAMPLES::
Expand All @@ -2251,22 +2250,42 @@ def orbit(self, f, N):
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
x - w]
::
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: P1.<u,v> = ProjectiveSpace(QQ,1)
sage: H = Hom(PS,P1)
sage: f = H([x^2, y^2])
sage: X = PS.subscheme([x-y])
sage: X.orbit(f,2)
Traceback (most recent call last):
...
TypeError: map must be an endomorphism for iteration
::
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: H = End(PS)
sage: f = H([x^2, y^2, z^2])
sage: X = PS.subscheme([x-y])
sage: X.orbit(f,[-1,2])
Traceback (most recent call last):
...
TypeError: orbit bounds must be non-negative
"""
if not f.is_endomorphism():
raise TypeError("Map must be an endomorphism for iteration.")
if (isinstance(N,(list,tuple)) == False):
raise TypeError("map must be an endomorphism for iteration")
if not isinstance(N,(list,tuple)):
N = [0,N]
try:
N[0] = ZZ(N[0])
N[1] = ZZ(N[1])
except TypeError:
raise TypeError("Orbit bounds must be integers")
N[0] = ZZ(N[0])
N[1] = ZZ(N[1])
if N[0] < 0 or N[1] < 0:
raise TypeError("Orbit bounds must be non-negative")
raise TypeError("orbit bounds must be non-negative")
if N[0] > N[1]:
return([])

Q = copy(self)
Q = self
for i in range(1, N[0]+1):
Q = f(Q)
Orb = [Q]
Expand All @@ -2278,8 +2297,7 @@ def orbit(self, f, N):

def nth_iterate(self, f, n , **kwds):
r"""
For a subscheme ``self`` and a map `f` this function returns
the nth iterate of `self` by ``f``.
The nth forward image of this scheme by the map ``f``.
INPUT:
Expand All @@ -2301,28 +2319,57 @@ def nth_iterate(self, f, n , **kwds):
defined by:
y - z,
x - w
::
sage: PS.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: H = End(PS)
sage: f = H([x^2, y^2, z^2])
sage: X = PS.subscheme([x-y])
sage: X.nth_iterate(f,-2)
Traceback (most recent call last):
...
TypeError: must be a forward orbit
::
sage: PS.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: P2.<u,v,w>=ProjectiveSpace(QQ,2)
sage: H = Hom(PS,P2)
sage: f = H([x^2, y^2, z^2])
sage: X = PS.subscheme([x-y])
sage: X.nth_iterate(f,2)
Traceback (most recent call last):
...
TypeError: map must be an endomorphism for iteration
::
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: H = End(PS)
sage: f = H([x^2, y^2, z^2])
sage: X = PS.subscheme([x-y])
sage: X.nth_iterate(f,2.5)
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer
"""
if not f.is_endomorphism():
raise TypeError("Map must be an endomorphism for iteration.")
try:
n = ZZ(n)
except TypeError:
raise TypeError("Iterate number must be an integer")
raise TypeError("map must be an endomorphism for iteration")
n = ZZ(n)
if n < 0:
raise TypeError("Must be a forward orbit")
if n == 0:
return(self)
else:
Q = f(self)
for i in range(2,n+1):
Q = f(Q)
return(Q)
raise TypeError("must be a forward orbit")
Q = self
for i in range(n):
Q = f(Q)
return Q

def _forward_image(self, f):
"""
Compute the forward image of the subscheme ``self`` by the map ``f``.
Compute the forward image of this subscheme by the morphism ``f``.
The forward image is computed through elimination.
The forward image is computed through elimination and ``f`` must be
a morphism for this to be well defined.
In particular, let $X = V(h_1,\ldots, h_t)$ and define the ideal
$I = (h_1,\ldots,h_t,y_0-f_0(\bar{x}), \ldots, y_n-f_n(\bar{x}))$.
Then the elimination ideal $I_{n+1} = I \cap K[y_0,\ldots,y_n]$ is a homogeneous
Expand Down Expand Up @@ -2408,11 +2455,36 @@ def _forward_image(self, f):
of Multivariate Polynomial Ring in y0, y1, y2, y3 over Rational Field
defined by:
x*z + (-y1)*z^2
::
sage: PS.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: H = End(PS)
sage: f = H([x^3, x*y^2, x*z^2])
sage: X = PS.subscheme([x-y])
sage: X._forward_image(f)
Traceback (most recent call last):
...
TypeError: map must be a morphism
::
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: P1.<u,v> = ProjectiveSpace(QQ,1)
sage: Y= P1.subscheme([u-v])
sage: H = End(PS)
sage: f = H([x^2, y^2, z^2])
sage: Y._forward_image(f)
Traceback (most recent call last):
...
TypeError: subscheme must be in ambient space of domain of map
"""
if not f.is_morphism():
raise TypeError("map must be a morphism")
dom = f.domain()
codom = f.codomain()
if self.ambient_space() != dom:
raise TypeError("Subscheme must be in ambient space of domain of map.")
raise TypeError("subscheme must be in ambient space of domain of map")
CR_dom = dom.coordinate_ring()
CR_codom = codom.coordinate_ring()
n = CR_dom.ngens()
Expand All @@ -2422,7 +2494,7 @@ def _forward_image(self, f):
R = PolynomialRing(f.base_ring(), n+m, 'tempvar', order = 'lex')
Rvars = R.gens()[0 : n]
phi = CR_dom.hom(Rvars,R)
zero = [0 for _ in range(n)]
zero = n*[0]
psi = R.hom(zero + list(CR_codom.gens()),CR_codom)
#set up ideal
L = R.ideal([phi(t) for t in self.defining_polynomials()] + [R.gen(n+i) - phi(f[i]) for i in range(m)])
Expand All @@ -2437,7 +2509,7 @@ def _forward_image(self, f):

def preimage(self, f):
r"""
Given a subscheme ``self``, return the subscheme that maps to ``self`` by ``f``.
The subscheme that maps to this scheme by the map ``f``.
In particular, `f^{-1}(V(h_1,\ldots,h_t)) = V(h_1 \circ f, \ldots, h_t \circ f)`.
Expand All @@ -2464,18 +2536,43 @@ def preimage(self, f):
sage: P.<x,y,z,w,t> = ProjectiveSpace(QQ, 4)
sage: H = End(P)
sage: f = H([x^2-y^2, z*y, z^2, w^2, t^2+w^2])
sage: f = H([x^2-y^2, y^2, z^2, w^2, t^2+w^2])
sage: f.rational_preimages(P.subscheme([x-z, t^2, w-t]))
Closed subscheme of Projective Space of dimension 4 over Rational Field
defined by:
x^2 - y^2 - z^2,
w^4 + 2*w^2*t^2 + t^4,
-t^2
::
sage: PS.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: H = End(PS)
sage: f = H([x^2, x^2, x^2])
sage: X = PS.subscheme([x-y])
sage: X.preimage(f)
Traceback (most recent call last):
...
TypeError: map must be a morphism
::
sage: PS.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: P1.<u,v> = ProjectiveSpace(ZZ,1)
sage: Y= P1.subscheme([u^2-v^2])
sage: H = End(PS)
sage: f = H([x^2, y^2, z^2])
sage: Y.preimage(f)
Traceback (most recent call last):
...
TypeError: subscheme must be in ambient space of codomain
"""
if not f.is_morphism():
raise TypeError("map must be a morphism")
dom = f.domain()
codom = f.codomain()
if self.ambient_space() != codom:
raise TypeError("Subscheme must be in ambient space of codomain.")
raise TypeError("subscheme must be in ambient space of codomain")
R = codom.coordinate_ring()
dict = {}
for i in range(codom.dimension_relative()+1):
Expand Down
70 changes: 58 additions & 12 deletions src/sage/schemes/projective/projective_morphism.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -228,22 +228,73 @@ def __call__(self, x, check=True):
EXAMPLES::
sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
sage: H=Hom(P,P)
sage: f=H([x^2+y^2,y^2,z^2 + y*z])
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: H = End(P)
sage: f = H([x^2+y^2, y^2, z^2 + y*z])
sage: f(P([1,1,1]))
(1 : 1/2 : 1)
::
sage: PS.<x,y,z,w>=ProjectiveSpace(QQ,3)
sage: H=End(PS)
sage: f=H([y^2,x^2,w^2,z^2])
sage: X=PS.subscheme([z^2+y*w])
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: P1.<u,v> = ProjectiveSpace(QQ,1)
sage: H = End(P1)
sage: f = H([u^2, v^2])
sage: f(PS([0,1,1]))
Traceback (most recent call last):
...
TypeError: (0 : 1 : 1) fails to convert into the map's domain Projective Space of
dimension 1 over Rational Field, but a `pushforward` method is not properly implemented
::
sage: PS.<x,y> = ProjectiveSpace(QQ, 1)
sage: P1.<u,v> = ProjectiveSpace(QQ, 1)
sage: H = End(P1)
sage: f = H([u^2, v^2])
sage: f([0,1])
(0 : 1)
sage: f(PS([0,1]))
(0 : 1)
::
sage: PS.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: H = End(PS)
sage: f = H([y^2, x^2, w^2, z^2])
sage: X = PS.subscheme([z^2+y*w])
sage: f(X)
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
x*z - w^2
::
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: P1.<u,v> = ProjectiveSpace(ZZ, 1)
sage: H = End(PS)
sage: f = H([x^2, y^2, z^2])
sage: X = P1.subscheme([u-v])
sage: f(X)
Traceback (most recent call last):
...
TypeError: subscheme must be in ambient space of domain of map
::
sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: P1.<u,v> = ProjectiveSpace(ZZ, 1)
sage: H = End(P1)
sage: f = H([u^2, v^2])
sage: f([u-v])
Closed subscheme of Projective Space of dimension 1 over Integer Ring defined by:
u - v
sage: X = PS.subscheme([x-z])
sage: f([x-z])
Traceback (most recent call last):
...
TypeError: [x - z] fails to convert into the map's domain Projective Space of
dimension 1 over Integer Ring, but a `pushforward` method is not properly implemented
"""
from sage.schemes.projective.projective_point import SchemeMorphism_point_projective_ring
if check:
Expand All @@ -256,11 +307,6 @@ def __call__(self, x, check=True):
raise TypeError("%s fails to convert into the map's domain %s, but a `pushforward` method is not properly implemented"%(x, self.domain()))
#else pass it onto the eval below
elif isinstance(x, AlgebraicScheme_subscheme_projective):
if self.domain() != x.ambient_space():
try:
x = self.domain().subscheme(x.defining_polynomials())
except (TypeError, NotImplementedError):
raise TypeError("%s fails to convert into the map's domain %s, but a `pushforward` method is not properly implemented"%(x, self.domain()))
return x._forward_image(self) #call subscheme eval
else: #not a projective point or subscheme
try:
Expand Down

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