Points on affine varieties#
Scheme morphism for points on affine varieties.
AUTHORS:
David Kohel, William Stein
Volker Braun (2011-08-08): Renamed classes, more documentation, misc cleanups.
Ben Hutz (2013)
- class sage.schemes.affine.affine_point.SchemeMorphism_point_affine(X, v, check=True)#
Bases:
SchemeMorphism_point
A rational point on an affine scheme.
INPUT:
X
– a subscheme of an ambient affine space over a ring \(R\)v
– a list/tuple/iterable of coordinates in \(R\)check
– boolean (optional, default:True
); whether to check the input for consistency
EXAMPLES:
sage: A = AffineSpace(2, QQ) sage: A(1, 2) (1, 2)
- global_height(prec=None)#
Returns 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
sage: P = AffineSpace(ZZ, 4, 'x') sage: Q = P(3, 17, -51, 5) sage: 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
Todo
P-adic heights.
- homogenize(n)#
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
- class sage.schemes.affine.affine_point.SchemeMorphism_point_affine_field(X, v, check=True)#
Bases:
SchemeMorphism_point_affine
- intersection_multiplicity(X)#
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
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
- multiplicity()#
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: an 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
- weil_restriction()#
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
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)
- class sage.schemes.affine.affine_point.SchemeMorphism_point_affine_finite_field(X, v, check=True)#