Points on affine varieties

This module implements scheme morphism for points on affine varieties.

AUTHORS:

  • David Kohel, William Stein (2006): initial version

  • Volker Braun (2011-08-08): renamed classes, more documentation, misc cleanups

  • Ben Hutz (2013): many improvements

class sage.schemes.affine.affine_point.SchemeMorphism_point_affine(X, v, check=True)[source]

Bases: SchemeMorphism_point

A rational point on an affine scheme.

INPUT:

  • X – a subscheme of an ambient affine space over a ring \(R\)

  • v – list/tuple/iterable of coordinates in \(R\)

  • check – boolean (default: True); whether to check the input for consistency

EXAMPLES:

sage: A = AffineSpace(2, QQ)
sage: A(1, 2)
(1, 2)

>>> from sage.all import *
>>> A = AffineSpace(Integer(2), QQ)
>>> A(Integer(1), Integer(2))
(1, 2)
global_height(prec=None)[source]

Return the logarithmic height of the point.

INPUT:

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: P.<x,y> = AffineSpace(QQ, 2)
sage: Q = P(41, 1/12)
sage: Q.global_height()                                                     # needs sage.rings.real_mpfr
3.71357206670431

>>> from sage.all import *
>>> P = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> Q = P(Integer(41), Integer(1)/Integer(12))
>>> Q.global_height()                                                     # needs sage.rings.real_mpfr
3.71357206670431


sage: P = AffineSpace(ZZ, 4, 'x')
sage: Q = P(3, 17, -51, 5)
sage: Q.global_height()                                                     # needs sage.rings.real_mpfr
3.93182563272433

>>> from sage.all import *
>>> P = AffineSpace(ZZ, Integer(4), 'x')
>>> Q = P(Integer(3), Integer(17), -Integer(51), Integer(5))
>>> Q.global_height()                                                     # needs sage.rings.real_mpfr
3.93182563272433


sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2 + 5)                                          # needs sage.rings.number_field
sage: A = AffineSpace(k, 2, 'z')                                            # needs sage.rings.number_field
sage: A([3, 5*w + 1]).global_height(prec=100)                               # needs sage.rings.number_field sage.rings.real_mpfr
2.4181409534757389986565376694

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> k = NumberField(x**Integer(2) + Integer(5), names=('w',)); (w,) = k._first_ngens(1)# needs sage.rings.number_field
>>> A = AffineSpace(k, Integer(2), 'z')                                            # needs sage.rings.number_field
>>> A([Integer(3), Integer(5)*w + Integer(1)]).global_height(prec=Integer(100))                               # needs sage.rings.number_field sage.rings.real_mpfr
2.4181409534757389986565376694

Todo

P-adic heights.

homogenize(n)[source]

Return the homogenization of the point at the nth coordinate.

INPUT:

  • n – integer between 0 and dimension of the map, inclusive

OUTPUT: a point in the projectivization of the codomain of the map

EXAMPLES:

sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: Q = A(2, 3)
sage: Q.homogenize(2).dehomogenize(2) == Q
True

::

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: Q = A(2, 3)
sage: P = A(0, 1)
sage: Q.homogenize(2).codomain() == P.homogenize(2).codomain()
True

>>> from sage.all import *
>>> A = AffineSpace(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> Q = A(Integer(2), Integer(3))
>>> Q.homogenize(Integer(2)).dehomogenize(Integer(2)) == Q
True

::

>>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> Q = A(Integer(2), Integer(3))
>>> P = A(Integer(0), Integer(1))
>>> Q.homogenize(Integer(2)).codomain() == P.homogenize(Integer(2)).codomain()
True
class sage.schemes.affine.affine_point.SchemeMorphism_point_affine_field(X, v, check=True)[source]

Bases: SchemeMorphism_point_affine

as_subscheme()[source]

Return the subscheme associated with this rational point.

EXAMPLES:

sage: A2.<x,y> = AffineSpace(QQ, 2)
sage: p1 = A2.point([0,0]).as_subscheme(); p1
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x, y
sage: p2 = A2.point([1,1]).as_subscheme(); p2
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x - 1, y - 1
sage: p1 + p2
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x - y, y^2 - y

>>> from sage.all import *
>>> A2 = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A2._first_ngens(2)
>>> p1 = A2.point([Integer(0),Integer(0)]).as_subscheme(); p1
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x, y
>>> p2 = A2.point([Integer(1),Integer(1)]).as_subscheme(); p2
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x - 1, y - 1
>>> p1 + p2
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x - y, y^2 - y
intersection_multiplicity(X)[source]

Return the intersection multiplicity of the codomain of this point and X at this point.

This uses the intersection_multiplicity implementations for projective/affine subschemes. This point must be a point on an affine subscheme.

INPUT:

  • X – a subscheme in the same ambient space as that of the codomain of this point

OUTPUT: integer

EXAMPLES:

sage: # needs sage.libs.singular
sage: A.<x,y> = AffineSpace(GF(17), 2)
sage: X = A.subscheme([y^2 - x^3 + 2*x^2 - x])
sage: Y = A.subscheme([y - 2*x + 2])
sage: Q1 = Y([1,0])
sage: Q1.intersection_multiplicity(X)
2
sage: Q2 = X([4,6])
sage: Q2.intersection_multiplicity(Y)
1

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> A = AffineSpace(GF(Integer(17)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> X = A.subscheme([y**Integer(2) - x**Integer(3) + Integer(2)*x**Integer(2) - x])
>>> Y = A.subscheme([y - Integer(2)*x + Integer(2)])
>>> Q1 = Y([Integer(1),Integer(0)])
>>> Q1.intersection_multiplicity(X)
2
>>> Q2 = X([Integer(4),Integer(6)])
>>> Q2.intersection_multiplicity(Y)
1


sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: X = A.subscheme([x^2 - y*z^2, z - 2*w^2])
sage: Q = A([2,1,2,-1])
sage: Q.intersection_multiplicity(X)
Traceback (most recent call last):
...
TypeError: this point must be a point on an affine subscheme

>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = A._first_ngens(4)
>>> X = A.subscheme([x**Integer(2) - y*z**Integer(2), z - Integer(2)*w**Integer(2)])
>>> Q = A([Integer(2),Integer(1),Integer(2),-Integer(1)])
>>> Q.intersection_multiplicity(X)
Traceback (most recent call last):
...
TypeError: this point must be a point on an affine subscheme
multiplicity()[source]

Return the multiplicity of this point on its codomain.

Uses the subscheme multiplicity implementation. This point must be a point on an affine subscheme.

OUTPUT: integer

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([y^2 - x^7*z])
sage: Q1 = X([1,1,1])
sage: Q1.multiplicity()                                                     # needs sage.libs.singular
1
sage: Q2 = X([0,0,2])
sage: Q2.multiplicity()                                                     # needs sage.libs.singular
2

>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> X = A.subscheme([y**Integer(2) - x**Integer(7)*z])
>>> Q1 = X([Integer(1),Integer(1),Integer(1)])
>>> Q1.multiplicity()                                                     # needs sage.libs.singular
1
>>> Q2 = X([Integer(0),Integer(0),Integer(2)])
>>> Q2.multiplicity()                                                     # needs sage.libs.singular
2
weil_restriction()[source]

Compute the Weil restriction of this point 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. This functor applied to a point gives the equivalent point on the Weil restriction of its codomain.

OUTPUT: scheme point on the Weil restriction of the codomain of this point

EXAMPLES:

sage: # needs sage.libs.singular sage.rings.finite_rings
sage: A.<x,y,z> = AffineSpace(GF(5^3, 't'), 3)
sage: X = A.subscheme([y^2 - x*z, z^2 + y])
sage: Y = X.weil_restriction()
sage: P = X([1, -1, 1])
sage: Q = P.weil_restriction();Q
(1, 0, 0, 4, 0, 0, 1, 0, 0)
sage: Q.codomain() == Y
True

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.finite_rings
>>> A = AffineSpace(GF(Integer(5)**Integer(3), 't'), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> X = A.subscheme([y**Integer(2) - x*z, z**Integer(2) + y])
>>> Y = X.weil_restriction()
>>> P = X([Integer(1), -Integer(1), Integer(1)])
>>> Q = P.weil_restriction();Q
(1, 0, 0, 4, 0, 0, 1, 0, 0)
>>> Q.codomain() == Y
True


sage: # needs sage.libs.singular sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<w> = NumberField(x^5 - 2)
sage: R.<x> = K[]
sage: L.<v> = K.extension(x^2 + w)
sage: A.<x,y> = AffineSpace(L, 2)
sage: P = A([w^3 - v, 1 + w + w*v])
sage: P.weil_restriction()
(w^3, -1, w + 1, w)

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(5) - Integer(2), names=('w',)); (w,) = K._first_ngens(1)
>>> R = K['x']; (x,) = R._first_ngens(1)
>>> L = K.extension(x**Integer(2) + w, names=('v',)); (v,) = L._first_ngens(1)
>>> A = AffineSpace(L, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> P = A([w**Integer(3) - v, Integer(1) + w + w*v])
>>> P.weil_restriction()
(w^3, -1, w + 1, w)
class sage.schemes.affine.affine_point.SchemeMorphism_point_affine_finite_field(X, v, check=True)[source]

Bases: SchemeMorphism_point_affine_field