Morphisms of function fields

Maps and morphisms useful for computations with function fields.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: K.hom(1/x)
Function Field endomorphism of Rational function field in x over Rational Field
  Defn: x |--> 1/x
sage: L.<y> = K.extension(y^2 - x)
sage: K.hom(y)
Function Field morphism:
  From: Rational function field in x over Rational Field
  To:   Function field in y defined by y^2 - x
  Defn: x |--> y
sage: L.hom([y,x])
Function Field endomorphism of Function field in y defined by y^2 - x
  Defn: y |--> y
        x |--> x
sage: L.hom([x,y])
Traceback (most recent call last):
...
ValueError: invalid morphism

For global function fields, which have positive characteristics, the higher derivation is available:

sage: K.<x> = FunctionField(GF(2)); _.<Y>=K[]
sage: L.<y> = K.extension(Y^3+x+x^3*Y)
sage: h = L.higher_derivation()
sage: h(y^2, 2)
((x^7 + 1)/x^2)*y^2 + x^3*y

AUTHORS:

• William Stein (2010): initial version

• Julian Rüth (2011-09-14, 2014-06-23, 2017-08-21): refactored class hierarchy; added derivation classes; morphisms to/from fraction fields

• Kwankyu Lee (2017-04-30): added higher derivations and completions

class sage.rings.function_field.maps.FractionFieldToFunctionField

Bases: FunctionFieldVectorSpaceIsomorphism

Isomorphism from a fraction field of a polynomial ring to the isomorphic function field.

EXAMPLES:

sage: K = QQ['x'].fraction_field()
sage: L = K.function_field()
sage: f = L.coerce_map_from(K); f
Isomorphism:
    From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
    To:   Rational function field in x over Rational Field

See also

FunctionFieldToFractionField

section()

Return the inverse map of this isomorphism.

EXAMPLES:

sage: K = QQ['x'].fraction_field()
sage: L = K.function_field()
sage: f = L.coerce_map_from(K)
sage: f.section()
Isomorphism:
    From: Rational function field in x over Rational Field
    To:   Fraction Field of Univariate Polynomial Ring in x over Rational Field

class sage.rings.function_field.maps.FunctionFieldCompletion(field, place, name=None, prec=None, gen_name=None)

Bases: Map

Completions on function fields.

INPUT:

• field – function field

• place – place of the function field

• name – string for the name of the series variable

• prec – positive integer; default precision

• gen_name – string; name of the generator of the residue field; used only when place is non-rational

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: p = L.places_finite()[0]
sage: m = L.completion(p)
sage: m
Completion map:
  From: Function field in y defined by y^2 + y + (x^2 + 1)/x
  To:   Laurent Series Ring in s over Finite Field of size 2
sage: m(x)
s^2 + s^3 + s^4 + s^5 + s^7 + s^8 + s^9 + s^10 + s^12 + s^13
+ s^15 + s^16 + s^17 + s^19 + O(s^22)
sage: m(y)
s^-1 + 1 + s^3 + s^5 + s^7 + s^9 + s^13 + s^15 + s^17 + O(s^19)
sage: m(x*y) == m(x) * m(y)
True
sage: m(x+y) == m(x) + m(y)
True

The variable name of the series can be supplied. If the place is not rational such that the residue field is a proper extension of the constant field, you can also specify the generator name of the extension:

sage: p2 = L.places_finite(2)[0]
sage: p2
Place (x^2 + x + 1, x*y + 1)
sage: m2 = L.completion(p2, 't', gen_name='b')
sage: m2(x)
(b + 1) + t + t^2 + t^4 + t^8 + t^16 + O(t^20)
sage: m2(y)
b + b*t + b*t^3 + b*t^4 + (b + 1)*t^5 + (b + 1)*t^7 + b*t^9 + b*t^11
+ b*t^12 + b*t^13 + b*t^15 + b*t^16 + (b + 1)*t^17 + (b + 1)*t^19 + O(t^20)

default_precision()

Return the default precision.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: p = L.places_finite()[0]
sage: m = L.completion(p)
sage: m.default_precision()
20

class sage.rings.function_field.maps.FunctionFieldConversionToConstantBaseField(parent)

Bases: Map

Conversion map from the function field to its constant base field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: QQ.convert_map_from(K)
Conversion map:
  From: Rational function field in x over Rational Field
  To:   Rational Field

class sage.rings.function_field.maps.FunctionFieldDerivation(parent)

Bases: RingDerivationWithoutTwist

Base class for derivations on function fields.

A derivation on \(R\) is a map \(R \to R\) with \(D(\alpha+\beta)=D(\alpha)+D(\beta)\) and \(D(\alpha\beta)=\beta D(\alpha)+\alpha D(\beta)\) for all \(\alpha,\beta\in R\).

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: d
d/dx

is_injective()

Return False since a derivation is never injective.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: d.is_injective()
False

class sage.rings.function_field.maps.FunctionFieldDerivation_inseparable(parent, u=None)

Bases: FunctionFieldDerivation

Initialize this derivation.

INPUT:

• parent – the parent of this derivation

• u – a parameter describing the derivation

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: d = L.derivation()

This also works for iterated non-monic extensions:

sage: K.<x> = FunctionField(GF(2))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - 1/x)
sage: R.<z> = L[]
sage: M.<z> = L.extension(z^2*y - x^3)
sage: M.derivation()
d/dz

We can also create a multiple of the canonical derivation:

sage: M.derivation([x])
x*d/dz

class sage.rings.function_field.maps.FunctionFieldDerivation_rational(parent, u=None)

Bases: FunctionFieldDerivation

Derivations on rational function fields.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.derivation()
d/dx

class sage.rings.function_field.maps.FunctionFieldDerivation_separable(parent, d)

Bases: FunctionFieldDerivation

Derivations of separable extensions.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: L.derivation()
d/dx

class sage.rings.function_field.maps.FunctionFieldHigherDerivation(field)

Bases: Map

Base class of higher derivations on function fields.

INPUT:

• field – function field on which the derivation operates

EXAMPLES:

sage: F.<x> = FunctionField(GF(2))
sage: F.higher_derivation()
Higher derivation map:
  From: Rational function field in x over Finite Field of size 2
  To:   Rational function field in x over Finite Field of size 2

class sage.rings.function_field.maps.FunctionFieldHigherDerivation_char_zero(field)

Bases: FunctionFieldHigherDerivation

Higher derivations of function fields of characteristic zero.

INPUT:

• field – function field on which the derivation operates

EXAMPLES:

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: h = L.higher_derivation()
sage: h
Higher derivation map:
  From: Function field in y defined by y^3 + x^3*y + x
  To:   Function field in y defined by y^3 + x^3*y + x
sage: h(y,1) == -(3*x^2*y+1)/(3*y^2+x^3)
True
sage: h(y^2,1) == -2*y*(3*x^2*y+1)/(3*y^2+x^3)
True
sage: e = L.random_element()
sage: h(h(e,1),1) == 2*h(e,2)
True
sage: h(h(h(e,1),1),1) == 3*2*h(e,3)
True

class sage.rings.function_field.maps.FunctionFieldHigherDerivation_global(field)

Bases: FunctionFieldHigherDerivation

Higher derivations of global function fields.

INPUT:

• field – function field on which the derivation operates

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: h = L.higher_derivation()
sage: h
Higher derivation map:
  From: Function field in y defined by y^3 + x^3*y + x
  To:   Function field in y defined by y^3 + x^3*y + x
sage: h(y^2, 2)
((x^7 + 1)/x^2)*y^2 + x^3*y

class sage.rings.function_field.maps.FunctionFieldLinearMap

Bases: SetMorphism

Linear map to function fields.

class sage.rings.function_field.maps.FunctionFieldLinearMapSection

Bases: SetMorphism

Section of linear map from function fields.

class sage.rings.function_field.maps.FunctionFieldMorphism(parent, im_gen, base_morphism)

Bases: RingHomomorphism

Base class for morphisms between function fields.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: f = K.hom(1/x); f
Function Field endomorphism of Rational function field in x over Rational Field
  Defn: x |--> 1/x

class sage.rings.function_field.maps.FunctionFieldMorphism_polymod(parent, im_gen, base_morphism)

Bases: FunctionFieldMorphism

Morphism from a finite extension of a function field to a function field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^3 + 6*x^3 + x)
sage: f = L.hom(y*2); f
Function Field endomorphism of Function field in y defined by y^3 + 6*x^3 + x
  Defn: y |--> 2*y
sage: factor(L.polynomial())
y^3 + 6*x^3 + x
sage: f(y).charpoly('y')
y^3 + 6*x^3 + x

class sage.rings.function_field.maps.FunctionFieldMorphism_rational(parent, im_gen, base_morphism)

Bases: FunctionFieldMorphism

Morphism from a rational function field to a function field.

class sage.rings.function_field.maps.FunctionFieldRingMorphism

Bases: SetMorphism

Ring homomorphism.

class sage.rings.function_field.maps.FunctionFieldToFractionField

Bases: FunctionFieldVectorSpaceIsomorphism

Isomorphism from rational function field to the isomorphic fraction field of a polynomial ring.

EXAMPLES:

sage: K = QQ['x'].fraction_field()
sage: L = K.function_field()
sage: f = K.coerce_map_from(L); f
Isomorphism:
  From: Rational function field in x over Rational Field
  To:   Fraction Field of Univariate Polynomial Ring in x over Rational Field

See also

FractionFieldToFunctionField

section()

Return the inverse map of this isomorphism.

EXAMPLES:

sage: K = QQ['x'].fraction_field()
sage: L = K.function_field()
sage: f = K.coerce_map_from(L)
sage: f.section()
Isomorphism:
    From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
    To:   Rational function field in x over Rational Field

class sage.rings.function_field.maps.FunctionFieldVectorSpaceIsomorphism

Bases: Morphism

Base class for isomorphisms between function fields and vector spaces.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: isinstance(f, sage.rings.function_field.maps.FunctionFieldVectorSpaceIsomorphism)
True

is_injective()

Return True, since the isomorphism is injective.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.is_injective()
True

is_surjective()

Return True, since the isomorphism is surjective.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.is_surjective()
True

class sage.rings.function_field.maps.MapFunctionFieldToVectorSpace(K, V)

Bases: FunctionFieldVectorSpaceIsomorphism

Isomorphism from a function field to a vector space.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space(); t
Isomorphism:
  From: Function field in y defined by y^2 - x*y + 4*x^3
  To:   Vector space of dimension 2 over Rational function field in x over Rational Field

class sage.rings.function_field.maps.MapVectorSpaceToFunctionField(V, K)

Bases: FunctionFieldVectorSpaceIsomorphism

Isomorphism from a vector space to a function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space(); f
Isomorphism:
  From: Vector space of dimension 2 over Rational function field in x over Rational Field
  To:   Function field in y defined by y^2 - x*y + 4*x^3

codomain()

Return the function field which is the codomain of the isomorphism.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.codomain()
Function field in y defined by y^2 - x*y + 4*x^3

domain()

Return the vector space which is the domain of the isomorphism.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.domain()
Vector space of dimension 2 over Rational function field in x over Rational Field

class sage.rings.function_field.maps.RationalFunctionFieldHigherDerivation_global(field)

Bases: FunctionFieldHigherDerivation

Higher derivations of rational function fields over finite fields.

INPUT:

• field – function field on which the derivation operates

EXAMPLES:

sage: F.<x> = FunctionField(GF(2))
sage: h = F.higher_derivation()
sage: h
Higher derivation map:
  From: Rational function field in x over Finite Field of size 2
  To:   Rational function field in x over Finite Field of size 2
sage: h(x^2,2)
1