# 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)

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)
sage: H = End(A)
sage: f = H([x^2])
sage: type(f.as_dynamical_system())
<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

global_height(prec=None)

Returns the maximum of the heights of the coefficients in any of the coordinate functions of the affine morphism.

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()
8.29404964010203

sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2 + 5)
sage: A.<x,y> = AffineSpace(k, 2)
sage: H = Hom(A, A)
sage: f = H([13*w*x^2 + 4*y, 1/w*y^2]);
sage: f.global_height(prec=100)
3.3696683136785869233538671082

sage: A.<x> = AffineSpace(ZZ, 1)
sage: H = Hom(A, A)
sage: f = H([7*x^2 + 1513]);
sage: f.global_height()
7.32184971378836

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 for the domain and n for the codomain.

INPUT:

• 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: 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)
sage: A.<x,y> = AffineSpace(R, 2)
sage: H = End(A)
sage: f = H([QQbar(sqrt(2))*x*y, a*x^2])
sage: f.homogenize(2)
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)
sage: H = End(A)
sage: f = H([2*z / (z^2 + 2*z + 3)])
sage: f.homogenize(1)
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[]
sage: A.<x> = AffineSpace(R, 1)
sage: H = Hom(A, A)
sage: F = H([d*x^2 + c])
sage: F.homogenize(1)
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)]

class sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_field(parent, polys, check=True)
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)
sage: A.<x> = AffineSpace(K, 1)
sage: A2.<a,b> = AffineSpace(K, 2)
sage: H = End(A)
sage: H2 = Hom(A,A2)
sage: H3 = Hom(A2,A)
sage: f = H([x^2 + 2*(t^3 + t^2 + t + 3)])
sage: f.reduce_base_field()
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])
sage: f2.reduce_base_field()
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])
sage: f3.reduce_base_field()
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)
sage: A.<x> = AffineSpace(K, 1)
sage: H = End(A)
sage: f = H([x^2 + v])
sage: g = f.reduce_base_field();g
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
True

sage: A.<x> = AffineSpace(QQbar, 1)
sage: H = End(A)
sage: f = H([(QQbar(sqrt(2))*x^2 + 1/QQbar(sqrt(3))) / (5*x)])
sage: f.reduce_base_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 = 1.931851652578137?
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)
sage: H = End(A)
sage: f = H([QQbar(3^(1/3))*x^2 + QQbar(sqrt(-2))])
sage: f.reduce_base_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, embedding=(x^3+x+1).roots(ring=CC))
sage: A.<x> = AffineSpace(K,1)
sage: A2.<u,v> = AffineSpace(K,2)
sage: H = Hom(A, A2)
sage: f = H([x^2 + a*x + 3, 5*x])
sage: f.reduce_base_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)
sage: A.<x> =AffineSpace(K,1)
sage: H = End(A)
sage: f = H([3*x^2 + x + 1])
sage: f.reduce_base_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)
sage: A.<x> = AffineSpace(K, 1)
sage: H = End(A)
sage: f = H([x^2 + x*(t^3 + 2*t^2 + 4*t) + (t^5 + 3*t^4 + t^2 + 4*t)])
sage: f.reduce_base_field()
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)
sage: A.<x,y> = AffineSpace(K, 2)
sage: H = End(A)
sage: f = H([x^2-y^2, y^2])
sage: f.weil_restriction()
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)
sage: PS.<x,y> = AffineSpace(K, 2)
sage: H = Hom(PS, PS)
sage: f = H([x, y])
sage: F = f.weil_restriction()
sage: P = PS(2, 1)
sage: Q = P.weil_restriction()
sage: f(P).weil_restriction() == F(Q)
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)

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 = _
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)]