Morphisms on affine schemes#
This module implements morphisms from affine schemes. A morphism from an affine scheme to an affine scheme is determined by rational functions that define what the morphism does on points in the ambient affine space. A morphism from an affine scheme to a projective scheme is determined by homogeneous polynomials.
EXAMPLES:
sage: A2.<x,y> = AffineSpace(QQ, 2)
sage: P2.<x0,x1,x2> = ProjectiveSpace(QQ, 2)
sage: A2.hom([x, x + y], A2)
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x, y) to (x, x + y)
sage: A2.hom([1, x, x + y], P2)
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x, y) to (1 : x : x + y)
AUTHORS:
David Kohel, William Stein: initial version
Volker Braun (2011-08-08): renamed classes, more documentation, misc cleanups
Ben Hutz (2013-03): iteration functionality and new directory structure for affine/projective
Kwankyu Lee (2020-02): added indeterminacy_locus() and image()
- class sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space(parent, polys, check=True)#
Bases:
SchemeMorphism_polynomial
A morphism of schemes determined by rational functions.
EXAMPLES:
sage: RA.<x,y> = QQ[] sage: A2 = AffineSpace(RA) sage: RP.<u,v,w> = QQ[] sage: P2 = ProjectiveSpace(RP) sage: H = A2.Hom(P2) sage: f = H([x, y, 1]) sage: f Scheme morphism: From: Affine Space of dimension 2 over Rational Field To: Projective Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y) to (x : y : 1)
- as_dynamical_system()#
Return this endomorphism as a
DynamicalSystem_affine
.OUTPUT:
EXAMPLES:
sage: A.<x,y,z> = AffineSpace(ZZ, 3) sage: H = End(A) sage: f = H([x^2, y^2, z^2]) sage: type(f.as_dynamical_system()) <class 'sage.dynamics.arithmetic_dynamics.affine_ds.DynamicalSystem_affine'>
sage: A.<x,y> = AffineSpace(ZZ, 2) sage: H = End(A) sage: f = H([x^2 - y^2, y^2]) sage: type(f.as_dynamical_system()) <class 'sage.dynamics.arithmetic_dynamics.affine_ds.DynamicalSystem_affine'>
sage: A.<x> = AffineSpace(GF(5), 1) # optional - sage.rings.finite_rings sage: H = End(A) # optional - sage.rings.finite_rings sage: f = H([x^2]) # optional - sage.rings.finite_rings sage: type(f.as_dynamical_system()) # optional - sage.rings.finite_rings <class 'sage.dynamics.arithmetic_dynamics.affine_ds.DynamicalSystem_affine_finite_field'>
sage: P.<x,y> = AffineSpace(RR, 2) sage: f = DynamicalSystem([x^2 + y^2, y^2], P) sage: g = f.as_dynamical_system() sage: g is f True
- degree()#
Return the degree of the affine morphism.
EXAMPLES:
sage: R.<x> = AffineSpace(QQ, 1) sage: H = Hom(R, R) sage: f = H([x^7]) sage: f.degree() 7
sage: R.<x,y,z> = AffineSpace(QQ, 3) sage: H = Hom(R, R) sage: f = H([x^3, y^2 + 5, z^4 + y]) sage: f.degree() 4
- global_height(prec=None)#
Take the height of the homogenization, and return the global height of the coefficients as a projective point.
INPUT:
prec
– desired floating point precision (default: default RealField precision).
OUTPUT: A real number.
EXAMPLES:
sage: A.<x> = AffineSpace(QQ, 1) sage: H = Hom(A, A) sage: f = H([1/1331*x^2 + 4000]) sage: f.global_height() 15.4877354584971
sage: R.<x> = PolynomialRing(QQ) sage: k.<w> = NumberField(x^2 + 5) # optional - sage.rings.number_field sage: A.<x,y> = AffineSpace(k, 2) # optional - sage.rings.number_field sage: H = Hom(A, A) # optional - sage.rings.number_field sage: f = H([13*w*x^2 + 4*y, 1/w*y^2]) # optional - sage.rings.number_field sage: f.global_height(prec=2) # optional - sage.rings.number_field 4.0
sage: A.<x> = AffineSpace(ZZ, 1) sage: H = Hom(A, A) sage: f = H([7*x^2 + 1513]) sage: f.global_height() 7.32184971378836
sage: A.<x> = AffineSpace(QQ, 1) sage: B.<y,z> = AffineSpace(QQ, 2) sage: H = Hom(A, B) sage: f = H([1/3*x^2 + 10, 7*x^3]) sage: f.global_height() 3.40119738166216
sage: P.<x,y> = AffineSpace(QQ, 2) sage: A.<z> = AffineSpace(QQ, 1) sage: H = Hom(P, A) sage: f = H([1/1331*x^2 + 4000*y]) sage: f.global_height() 15.4877354584971
- homogenize(n)#
Return the homogenization of this map.
If it’s domain is a subscheme, the domain of the homogenized map is the projective embedding of the domain. The domain and codomain can be homogenized at different coordinates:
n[0]
for the domain andn[1]
for the codomain.INPUT:
n
– a tuple of nonnegative integers. Ifn
is an integer, then the two values of the tuple are assumed to be the same
OUTPUT: a morphism from the projective embedding of the domain of this map
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) Scheme endomorphism of Projective Space of dimension 2 over Integer Ring Defn: Defined on coordinates by sending (x0 : x1 : x2) to (x0^2*x2^5 - 2*x2^7 : x0^5*x1^2 : x0^5*x2^2)
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)) Scheme endomorphism of Projective Space of dimension 2 over Complex Field with 53 bits of precision Defn: Defined on coordinates by sending (x0 : x1 : x2) to (x0*x1*x2^2 : x0^2*x2^2 + (-2.00000000000000)*x2^4 : x0*x1^3 - x0^2*x1*x2)
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*x1^2 : 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)) Scheme endomorphism of Projective Space of dimension 2 over Univariate Polynomial Ring in t over Integer Ring Defn: Defined on coordinates by sending (x0 : x1 : x2) to (x1*x2^2 : x0^2*x2 + (-2)*x2^3 : x1^3 - x0*x1*x2)
sage: A.<x> = AffineSpace(QQ, 1) sage: H = End(A) sage: f = H([x^2 - 1]) sage: f.homogenize((1, 0)) Scheme endomorphism of Projective Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x0 : x1) to (x1^2 : x0^2 - x1^2)
sage: R.<a> = PolynomialRing(QQbar) # optional - sage.rings.number_field sage: A.<x,y> = AffineSpace(R, 2) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([QQbar(sqrt(2))*x*y, a*x^2]) # optional - sage.rings.number_field sage.symbolic sage: f.homogenize(2) # optional - sage.rings.number_field sage.symbolic Scheme endomorphism of Projective Space of dimension 2 over Univariate Polynomial Ring in a over Algebraic Field Defn: Defined on coordinates by sending (x0 : x1 : x2) to (1.414213562373095?*x0*x1 : a*x0^2 : x2^2)
sage: P.<x,y,z> = AffineSpace(QQ, 3) sage: H = End(P) sage: f = H([x^2 - 2*x*y + z*x, z^2 -y^2 , 5*z*y]) sage: f.homogenize(2).dehomogenize(2) == f True
sage: K.<c> = FunctionField(QQ) sage: A.<x> = AffineSpace(K, 1) sage: f = Hom(A, A)([x^2 + c]) sage: f.homogenize(1) Scheme endomorphism of Projective Space of dimension 1 over Rational function field in c over Rational Field Defn: Defined on coordinates by sending (x0 : x1) to (x0^2 + c*x1^2 : x1^2)
sage: A.<z> = AffineSpace(QQbar, 1) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([2*z / (z^2 + 2*z + 3)]) # optional - sage.rings.number_field sage: f.homogenize(1) # optional - sage.rings.number_field Scheme endomorphism of Projective Space of dimension 1 over Algebraic Field Defn: Defined on coordinates by sending (x0 : x1) to (x0*x1 : 1/2*x0^2 + x0*x1 + 3/2*x1^2)
sage: R.<c,d> = QQbar[] # optional - sage.rings.number_field sage: A.<x> = AffineSpace(R, 1) # optional - sage.rings.number_field sage: H = Hom(A, A) # optional - sage.rings.number_field sage: F = H([d*x^2 + c]) # optional - sage.rings.number_field sage: F.homogenize(1) # optional - sage.rings.number_field Scheme endomorphism of Projective Space of dimension 1 over Multivariate Polynomial Ring in c, d over Algebraic Field Defn: Defined on coordinates by sending (x0 : x1) to (d*x0^2 + c*x1^2 : x1^2)
- jacobian()#
Return the Jacobian matrix of partial derivative of this map.
The \((i, j)\) entry of the Jacobian matrix is the partial derivative
diff(functions[i], variables[j])
.OUTPUT:
matrix with coordinates in the coordinate ring of the map.
EXAMPLES:
sage: A.<z> = AffineSpace(QQ, 1) sage: H = End(A) sage: f = H([z^2 - 3/4]) sage: f.jacobian() [2*z]
sage: A.<x,y> = AffineSpace(QQ, 2) sage: H = End(A) sage: f = H([x^3 - 25*x + 12*y, 5*y^2*x - 53*y + 24]) sage: f.jacobian() [ 3*x^2 - 25 12] [ 5*y^2 10*x*y - 53]
sage: A.<x,y> = AffineSpace(ZZ, 2) sage: H = End(A) sage: f = H([(x^2 - x*y)/(1+y), (5+y)/(2+x)]) sage: f.jacobian() [ (2*x - y)/(y + 1) (-x^2 - x)/(y^2 + 2*y + 1)] [ (-y - 5)/(x^2 + 4*x + 4) 1/(x + 2)]
- local_height(v, prec=None)#
Return the maximum of the local heights of the coefficients in any of the coordinate functions of this map.
INPUT:
v
– a prime or prime ideal of the base ring.prec
– desired floating point precision (default: default RealField precision).
OUTPUT:
a real number.
EXAMPLES:
sage: P.<x,y> = AffineSpace(QQ, 2) sage: H = Hom(P, P) sage: f = H([1/1331*x^2 + 1/4000*y^2, 210*x*y]) sage: f.local_height(1331) 7.19368581839511
sage: P.<x,y,z> = AffineSpace(QQ, 3) sage: H = Hom(P, P) sage: f = H([4*x^2 + 3/100*y^2, 8/210*x*y, 1/10000*z^2]) sage: f.local_height(2) 2.77258872223978
sage: P.<x,y,z> = AffineSpace(QQ, 3) sage: H = Hom(P, P) sage: f = H([4*x^2 + 3/100*y^2, 8/210*x*y, 1/10000*z^2]) sage: f.local_height(2, prec=2) 3.0
sage: R.<z> = PolynomialRing(QQ) sage: K.<w> = NumberField(z^2 - 2) # optional - sage.rings.number_field sage: P.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field sage: H = Hom(P, P) # optional - sage.rings.number_field sage: f = H([2*x^2 + w/3*y^2, 1/w*y^2]) # optional - sage.rings.number_field sage: f.local_height(K.ideal(3)) # optional - sage.rings.number_field 1.09861228866811
- local_height_arch(i, prec=None)#
Return the maximum of the local height at the
i
-th infinite place of the coefficients in any of the coordinate functions of this map.INPUT:
i
– an integer.prec
– desired floating point precision (default: default RealField precision).
OUTPUT:
a real number.
EXAMPLES:
sage: P.<x,y> = AffineSpace(QQ, 2) sage: H = Hom(P, P) sage: f = H([1/1331*x^2 + 1/4000*y^2, 210*x*y]); sage: f.local_height_arch(0) 5.34710753071747
sage: P.<x,y> = AffineSpace(QQ, 2) sage: H = Hom(P, P) sage: f = H([1/1331*x^2 + 1/4000*y^2, 210*x*y]); sage: f.local_height_arch(0, prec=5) 5.2
sage: R.<z> = PolynomialRing(QQ) sage: K.<w> = NumberField(z^2 - 2) # optional - sage.rings.number_field sage: P.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field sage: H = Hom(P, P) # optional - sage.rings.number_field sage: f = H([2*x^2 + w/3*y^2, 1/w*y^2]) # optional - sage.rings.number_field sage: f.local_height_arch(1) # optional - sage.rings.number_field 0.6931471805599453094172321214582
- class sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_field(parent, polys, check=True)#
Bases:
SchemeMorphism_polynomial_affine_space
- image()#
Return the scheme-theoretic image of the morphism.
OUTPUT: a subscheme of the ambient space of the codomain
EXAMPLES:
sage: A1.<w> = AffineSpace(QQ, 1) sage: A2.<x,y> = AffineSpace(QQ, 2) sage: f = A2.hom([x + y], A1) sage: f.image() Closed subscheme of Affine Space of dimension 1 over Rational Field defined by: (no polynomials) sage: f = A2.hom([x, x], A2) sage: f.image() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x - y sage: f = A2.hom([x^2, x^3], A2) sage: f.image() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^3 - y^2 sage: P2.<x0,x1,x2> = ProjectiveSpace(QQ, 2) sage: f = A2.hom([x, x^2, x^3], P2) sage: f.image() Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: x1^2 - x0*x2
- indeterminacy_locus()#
Return the indeterminacy locus of this map as a rational map on the domain.
The indeterminacy locus is the intersection of all the base indeterminacy locuses of maps that define the same rational map as by this map.
OUTPUT: a subscheme of the domain of the map
EXAMPLES:
sage: A.<x,y> = AffineSpace(QQ, 2) sage: H = End(A) sage: f = H([x - y, x^2 - y^2]) sage: f.indeterminacy_locus() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: 1
sage: A.<x,y> = AffineSpace(QQ, 2) sage: f = A.hom([x, x/y], A) sage: f.indeterminacy_locus() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: y
- indeterminacy_points(F=None)#
Return the points in the indeterminacy locus of this map.
If the dimension of the indeterminacy locus is not zero, an error is raised.
INPUT:
F
– a field; if not given, the base ring of the domain is assumed
OUTPUT: indeterminacy points of the map defined over
F
EXAMPLES:
sage: A.<x,y> = AffineSpace(QQ, 2) sage: H = End(A) sage: f = H([x - y, x^2 - y^2]) sage: f.indeterminacy_points() []
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: P2.<x0,x1,x2> = ProjectiveSpace(QQ, 2) sage: f = A2.hom([x*y, y, x], P2) sage: f.indeterminacy_points() [(0, 0)]
- reduce_base_field()#
Return this map defined over the field of definition of the coefficients.
The base field of the map could be strictly larger than the field where all of the coefficients are defined. This function reduces the base field to the minimal possible. This can be done when the base ring is a number field, QQbar, a finite field, or algebraic closure of a finite field.
OUTPUT: a scheme morphism
EXAMPLES:
sage: K.<t> = GF(5^4) # optional - sage.rings.finite_rings sage: A.<x> = AffineSpace(K, 1) # optional - sage.rings.finite_rings sage: A2.<a,b> = AffineSpace(K, 2) # optional - sage.rings.finite_rings sage: H = End(A) # optional - sage.rings.finite_rings sage: H2 = Hom(A, A2) # optional - sage.rings.finite_rings sage: H3 = Hom(A2, A) # optional - sage.rings.finite_rings sage: f = H([x^2 + 2*(t^3 + t^2 + t + 3)]) # optional - sage.rings.finite_rings sage: f.reduce_base_field() # optional - sage.rings.finite_rings Scheme endomorphism of Affine Space of dimension 1 over Finite Field in t2 of size 5^2 Defn: Defined on coordinates by sending (x) to (x^2 + (2*t2)) sage: f2 = H2([x^2 + 4, 2*x]) # optional - sage.rings.finite_rings sage: f2.reduce_base_field() # optional - sage.rings.finite_rings Scheme morphism: From: Affine Space of dimension 1 over Finite Field of size 5 To: Affine Space of dimension 2 over Finite Field of size 5 Defn: Defined on coordinates by sending (x) to (x^2 - 1, 2*x) sage: f3 = H3([a^2 + t*b]) # optional - sage.rings.finite_rings sage: f3.reduce_base_field() # optional - sage.rings.finite_rings Scheme morphism: From: Affine Space of dimension 2 over Finite Field in t of size 5^4 To: Affine Space of dimension 1 over Finite Field in t of size 5^4 Defn: Defined on coordinates by sending (a, b) to (a^2 + t*b)
sage: K.<v> = CyclotomicField(4) # optional - sage.rings.number_field sage: A.<x> = AffineSpace(K, 1) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([x^2 + v]) # optional - sage.rings.number_field sage: g = f.reduce_base_field(); g # optional - sage.rings.number_field Scheme endomorphism of Affine Space of dimension 1 over Cyclotomic Field of order 4 and degree 2 Defn: Defined on coordinates by sending (x) to (x^2 + v) sage: g.base_ring() is K # optional - sage.rings.number_field True
sage: A.<x> = AffineSpace(QQbar, 1) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([(QQbar(sqrt(2))*x^2 + 1/QQbar(sqrt(3))) / (5*x)]) # optional - sage.rings.number_field sage: f.reduce_base_field() # optional - sage.rings.number_field Scheme endomorphism of Affine Space of dimension 1 over Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a = ...? Defn: Defined on coordinates by sending (x) to (((a^3 - 3*a)*x^2 + (-1/3*a^2 + 2/3))/(5*x))
sage: R.<x> = PolynomialRing(QQ) sage: A.<x> = AffineSpace(QQbar, 1) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([QQbar(3^(1/3))*x^2 + QQbar(sqrt(-2))]) # optional - sage.rings.number_field sage: f.reduce_base_field() # optional - sage.rings.number_field Scheme endomorphism of Affine Space of dimension 1 over Number Field in a with defining polynomial y^6 + 6*y^4 - 6*y^3 + 12*y^2 + 36*y + 17 with a = 1.442249570307409? + 1.414213562373095?*I Defn: Defined on coordinates by sending (x) to ((-48/269*a^5 + 27/269*a^4 - 320/269*a^3 + 468/269*a^2 - 772/269*a - 1092/269)*x^2 + (48/269*a^5 - 27/269*a^4 + 320/269*a^3 - 468/269*a^2 + 1041/269*a + 1092/269))
sage: R.<x> = PolynomialRing(QQ) sage: K.<a> = NumberField(x^3 - x + 1, # optional - sage.rings.number_field ....: embedding=(x^3+x+1).roots(ring=CC)[0][0]) sage: A.<x> = AffineSpace(K, 1) # optional - sage.rings.number_field sage: A2.<u,v> = AffineSpace(K, 2) # optional - sage.rings.number_field sage: H = Hom(A, A2) # optional - sage.rings.number_field sage: f = H([x^2 + a*x + 3, 5*x]) # optional - sage.rings.number_field sage: f.reduce_base_field() # optional - sage.rings.number_field Scheme morphism: From: Affine Space of dimension 1 over Number Field in a with defining polynomial x^3 - x + 1 with a = -1.324717957244746? To: Affine Space of dimension 2 over Number Field in a with defining polynomial x^3 - x + 1 with a = -1.324717957244746? Defn: Defined on coordinates by sending (x) to (x^2 + a*x + 3, 5*x)
sage: K.<v> = QuadraticField(2) # optional - sage.rings.number_field sage: A.<x> = AffineSpace(K, 1) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([3*x^2 + x + 1]) # optional - sage.rings.number_field sage: f.reduce_base_field() # optional - sage.rings.number_field Scheme endomorphism of Affine Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x) to (3*x^2 + x + 1)
sage: K.<t> = GF(5^6) # optional - sage.rings.finite_rings sage: A.<x> = AffineSpace(K, 1) # optional - sage.rings.finite_rings sage: H = End(A) # optional - sage.rings.finite_rings sage: f = H([x^2 + x*(t^3 + 2*t^2 + 4*t) + (t^5 + 3*t^4 + t^2 + 4*t)]) # optional - sage.rings.finite_rings sage: f.reduce_base_field() # optional - sage.rings.finite_rings Scheme endomorphism of Affine Space of dimension 1 over Finite Field in t of size 5^6 Defn: Defined on coordinates by sending (x) to (x^2 + (t^3 + 2*t^2 - t)*x + (t^5 - 2*t^4 + t^2 - t))
- weil_restriction()#
Compute the Weil restriction of this morphism over some extension field.
If the field is a finite field, then this computes the Weil restriction to the prime subfield.
A Weil restriction of scalars - denoted \(Res_{L/k}\) - is a functor which, for any finite extension of fields \(L/k\) and any algebraic variety \(X\) over \(L\), produces another corresponding variety \(Res_{L/k}(X)\), defined over \(k\). It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields. Since it is a functor it also applied to morphisms. In particular, the functor applied to a morphism gives the equivalent morphism from the Weil restriction of the domain to the Weil restriction of the codomain.
- OUTPUT: Scheme morphism on the Weil restrictions of the domain
and codomain of the map.
EXAMPLES:
sage: K.<v> = QuadraticField(5) # optional - sage.rings.number_field sage: A.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field sage: H = End(A) # optional - sage.rings.number_field sage: f = H([x^2 - y^2, y^2]) # optional - sage.rings.number_field sage: f.weil_restriction() # optional - sage.rings.number_field Scheme endomorphism of Affine Space of dimension 4 over Rational Field Defn: Defined on coordinates by sending (z0, z1, z2, z3) to (z0^2 + 5*z1^2 - z2^2 - 5*z3^2, 2*z0*z1 - 2*z2*z3, z2^2 + 5*z3^2, 2*z2*z3)
sage: K.<v> = QuadraticField(5) # optional - sage.rings.number_field sage: PS.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field sage: H = Hom(PS, PS) # optional - sage.rings.number_field sage: f = H([x, y]) # optional - sage.rings.number_field sage: F = f.weil_restriction() # optional - sage.rings.number_field sage: P = PS(2, 1) # optional - sage.rings.number_field sage: Q = P.weil_restriction() # optional - sage.rings.number_field sage: f(P).weil_restriction() == F(Q) # optional - sage.rings.number_field True
- class sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_finite_field(parent, polys, check=True)#
- class sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_subscheme_field(parent, polys, check=True)#
Bases:
SchemeMorphism_polynomial_affine_space_field
Morphisms from subschemes of affine spaces defined over fields.
- image()#
Return the scheme-theoretic image of the morphism.
OUTPUT: a subscheme of the ambient space of the codomain
EXAMPLES:
sage: A1.<w> = AffineSpace(QQ, 1) sage: A2.<x,y> = AffineSpace(QQ, 2) sage: X = A2.subscheme(0) sage: f = X.hom([x + y], A1) sage: f.image() Closed subscheme of Affine Space of dimension 1 over Rational Field defined by: (no polynomials)
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: X = A2.subscheme([x*y^2 - y^3 - 1]) sage: f = X.hom([y, y/x], A2) sage: f.image() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: -x^3*y + x^3 - y
- indeterminacy_locus()#
Return the indeterminacy locus of this map.
The map defines a rational map on the domain. The output is the subscheme of the domain on which the rational map is not defined by any representative of the rational map. See
representatives()
.EXAMPLES:
sage: A2.<x1,x2> = AffineSpace(QQ, 2) sage: X = A2.subscheme(0) sage: A1.<x> = AffineSpace(QQ, 1) sage: f = X.hom([x1/x2], A1) sage: f.indeterminacy_locus() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x2
sage: A2.<x1,x2> = AffineSpace(QQ, 2) sage: X = A2.subscheme(0) sage: P1.<a,b> = ProjectiveSpace(QQ, 1) sage: f = X.hom([x1,x2], P1) sage: L = f.indeterminacy_locus() sage: L.rational_points() [(0, 0)]
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: X = A2.subscheme([x^2 - y^2 - y]) sage: A1.<a> = AffineSpace(QQ,1) sage: f = X.hom([x/y], A1) sage: f.indeterminacy_locus() Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: y, x
sage: A3.<x,y,z> = AffineSpace(QQ, 3) sage: X = A3.subscheme(x^2 - y*z - x) sage: A2.<a,b> = AffineSpace(QQ, 2) sage: f = X.hom([y, y/x], A2) sage: L = f.indeterminacy_locus() sage: L Closed subscheme of Affine Space of dimension 3 over Rational Field defined by: x, y*z sage: L.dimension() 1
- is_morphism()#
Return
True
if the map is defined everywhere on the domain.EXAMPLES:
sage: P2.<x,y,z> = ProjectiveSpace(QQ,2) sage: P1.<a,b> = ProjectiveSpace(QQ,1) sage: X = P2.subscheme([x^2 - y^2 - y*z]) sage: f = X.hom([x,y], P1) sage: f.is_morphism() True
- representatives()#
Return all maps representing the same rational map as by this map.
EXAMPLES:
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: X = A2.subscheme(0) sage: f = X.hom([x, x/y], A2) sage: f.representatives() [Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: 0 To: Affine Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y) to (x, x/y)]
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: A1.<a> = AffineSpace(QQ, 1) sage: X = A2.subscheme([x^2 - y^2 - y]) sage: f = X.hom([x/y], A1) sage: f.representatives() [Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^2 - y^2 - y To: Affine Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x, y) to (x/y), Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^2 - y^2 - y To: Affine Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x, y) to ((y + 1)/x)] sage: g = _[1] sage: g.representatives() [Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^2 - y^2 - y To: Affine Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x, y) to (x/y), Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^2 - y^2 - y To: Affine Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x, y) to ((y + 1)/x)]
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: P1.<a,b> = ProjectiveSpace(QQ, 1) sage: X = A2.subscheme([x^2 - y^2 - y]) sage: f = X.hom([x, y], P1) sage: f.representatives() [Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^2 - y^2 - y To: Projective Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x, y) to (x : y), Scheme morphism: From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: x^2 - y^2 - y To: Projective Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x, y) to (y + 1 : x)]