def reduce_base_field(self):
"""
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)[0][0])
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))
"""
g = self.homogenize(0).reduce_base_field().dehomogenize(0)
from sage.schemes.affine.affine_space import AffineSpace
new_domain = AffineSpace(g.domain().base_ring(),
self.domain().dimension_relative(),
self.domain().variable_names())
new_codomain = AffineSpace(g.codomain().base_ring(),
self.codomain().dimension_relative(),
self.codomain().variable_names())
R = new_domain.coordinate_ring()
H = Hom(new_domain, new_codomain)
if isinstance(g[0], FractionFieldElement):
return H([R(G.numerator()) / R(G.denominator()) for G in g])
return H([R(G) for G in g])
