# Discrete valuations on function fields¶

AUTHORS:

• Julian Rüth (2016-10-16): initial version

EXAMPLES:

We can create classical valuations that correspond to finite and infinite places on a rational function field:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(1); v
sage: v = K.valuation(x^2 + 1); v
sage: v = K.valuation(1/x); v
Valuation at the infinite place


Note that we can also specify valuations which do not correspond to a place of the function field:

sage: R.<x> = QQ[]
sage: w = valuations.GaussValuation(R, QQ.valuation(2))
sage: v = K.valuation(w); v


Valuations on a rational function field can then be extended to finite extensions:

sage: v = K.valuation(x - 1); v
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: w = v.extensions(L); w
[[ (x - 1)-adic valuation, v(y + 1) = 1 ]-adic valuation,
[ (x - 1)-adic valuation, v(y - 1) = 1 ]-adic valuation]


REFERENCES:

An overview of some computational tools relating to valuations on function fields can be found in Section 4.6 of [Rüt2014]. Most of this was originally developed for number fields in [Mac1936I] and [Mac1936II].

class sage.rings.function_field.function_field_valuation.ClassicalFunctionFieldValuation_base(parent)

Base class for discrete valuations on rational function fields that come from points on the projective line.

class sage.rings.function_field.function_field_valuation.DiscreteFunctionFieldValuation_base(parent)

Base class for discrete valuations on function fields.

extensions(L)

Return the extensions of this valuation to L.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(x)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: v.extensions(L)

class sage.rings.function_field.function_field_valuation.FiniteRationalFunctionFieldValuation(parent, base_valuation)

Valuation of a finite place of a function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(x + 1); v # indirect doctest


A finite place with residual degree:

sage: w = K.valuation(x^2 + 1); w


A finite place with ramification:

sage: K.<t> = FunctionField(GF(3))
sage: L.<x> = FunctionField(K)
sage: u = L.valuation(x^3 - t); u


A finite place with residual degree and ramification:

sage: q = L.valuation(x^6 - t); q

class sage.rings.function_field.function_field_valuation.FunctionFieldExtensionMappedValuation(parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain)

A valuation on a finite extensions of function fields $$L=K[y]/(G)$$ where $$K$$ is another function field which redirects to another base_valuation on an isomorphism function field $$M=K[y]/(H)$$.

The isomorphisms must be trivial on K.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 + y + x^3)
sage: v = K.valuation(1/x)
sage: w = v.extension(L)

sage: w(x)
-1
sage: w(y)
-3/2
sage: w.uniformizer()
1/x^2*y

restriction(ring)

Return the restriction of this valuation to ring.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 + y + x^3)
sage: v = K.valuation(1/x)
sage: w = v.extension(L)
sage: w.restriction(K) is v
True

class sage.rings.function_field.function_field_valuation.FunctionFieldFromLimitValuation(parent, approximant, G, approximants)

A valuation on a finite extensions of function fields $$L=K[y]/(G)$$ where $$K$$ is another function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - (x^2 + x + 1))
sage: v = K.valuation(x - 1) # indirect doctest
sage: w = v.extension(L); w

scale(scalar)

Return this valuation scaled by scalar.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - (x^2 + x + 1))
sage: v = K.valuation(x - 1) # indirect doctest
sage: w = v.extension(L)
sage: 3*w
3 * (x - 1)-adic valuation

class sage.rings.function_field.function_field_valuation.FunctionFieldMappedValuationRelative_base(parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain)

A valuation on a function field which relies on a base_valuation on an isomorphic function field and which is such that the map from and to the other function field is the identity on the constant field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: v = K.valuation(1/x); v
Valuation at the infinite place

restriction(ring)

Return the restriction of this valuation to ring.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: K.valuation(1/x).restriction(GF(2))
Trivial valuation on Finite Field of size 2

class sage.rings.function_field.function_field_valuation.FunctionFieldMappedValuation_base(parent, base_valuation, to_base_valuation_domain, from_base_valuation_domain)

A valuation on a function field which relies on a base_valuation on an isomorphic function field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: v = K.valuation(1/x); v
Valuation at the infinite place

is_discrete_valuation()

Return whether this is a discrete valuation.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^4 - 1)
sage: v = K.valuation(1/x)
sage: w0,w1 = v.extensions(L)
sage: w0.is_discrete_valuation()
True

scale(scalar)

Return this valuation scaled by scalar.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 + y + x^3)
sage: v = K.valuation(1/x)
sage: w = v.extension(L)
sage: 3*w
3 * (x)-adic valuation (in Rational function field in x over Finite Field of size 2 after x |--> 1/x)

class sage.rings.function_field.function_field_valuation.FunctionFieldValuationFactory

Create a valuation on domain corresponding to prime.

INPUT:

• domain – a function field
• prime – a place of the function field, a valuation on a subring, or a valuation on another function field together with information for isomorphisms to and from that function field

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(1); v # indirect doctest
sage: v(x)
0
sage: v(x - 1)
1


See sage.rings.function_field.function_field.FunctionField.valuation() for further examples.

create_key_and_extra_args(domain, prime)

Create a unique key which identifies the valuation given by prime on domain.

create_key_and_extra_args_from_place(domain, generator)

Create a unique key which identifies the valuation at the place specified by generator.

create_key_and_extra_args_from_valuation(domain, valuation)

Create a unique key which identifies the valuation which extends valuation.

create_key_and_extra_args_from_valuation_on_isomorphic_field(domain, valuation, to_valuation_domain, from_valuation_domain)

Create a unique key which identifies the valuation which is valuation after mapping through to_valuation_domain.

create_object(version, key, **extra_args)

Create the valuation specified by key.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: R.<x> = QQ[]
sage: w = valuations.GaussValuation(R, QQ.valuation(2))
sage: v = K.valuation(w); v # indirect doctest

class sage.rings.function_field.function_field_valuation.FunctionFieldValuation_base(parent)

Abstract base class for any discrete (pseudo-)valuation on a function field.

class sage.rings.function_field.function_field_valuation.InducedRationalFunctionFieldValuation_base(parent, base_valuation)

Base class for function field valuation induced by a valuation on the underlying polynomial ring.

extensions(L)

Return all extensions of this valuation to L which has a larger constant field than the domain of this valuation.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(x^2 + 1)
sage: L.<x> = FunctionField(GaussianIntegers().fraction_field())
sage: v.extensions(L) # indirect doctest

lift(F)

Return a lift of F to the domain of this valuation such that reduce() returns the original element.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(x)
sage: v.lift(0)
0
sage: v.lift(1)
1

reduce(f)

Return the reduction of f in residue_ring().

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(x^2 + 1)
sage: v.reduce(x)
u1

residue_ring()

Return the residue field of this valuation.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.valuation(x).residue_ring()
Rational Field

restriction(ring)

Return the restriction of this valuation to ring.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.valuation(x).restriction(QQ)
Trivial valuation on Rational Field

simplify(f, error=None, force=False)

Return a simplified version of f.

Produce an element which differs from f by an element of valuation strictly greater than the valuation of f (or strictly greater than error if set.)

If force is not set, then expensive simplifications may be avoided.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(2)
sage: f = (x + 1)/(x - 1)


As the coefficients of this fraction are small, we do not simplify as this could be very costly in some cases:

sage: v.simplify(f)
(x + 1)/(x - 1)


However, simplification can be forced:

sage: v.simplify(f, force=True)
3

uniformizer()

Return a uniformizing element for this valuation.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.valuation(x).uniformizer()
x

value_group()

Return the value group of this valuation.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.valuation(x).value_group()
Additive Abelian Group generated by 1

class sage.rings.function_field.function_field_valuation.InfiniteRationalFunctionFieldValuation(parent)

Valuation of the infinite place of a function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = K.valuation(1/x) # indirect doctest

class sage.rings.function_field.function_field_valuation.NonClassicalRationalFunctionFieldValuation(parent, base_valuation)

Valuation induced by a valuation on the underlying polynomial ring which is non-classical.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = GaussValuation(QQ['x'], QQ.valuation(2))
sage: w = K.valuation(v); w # indirect doctest

residue_ring()

Return the residue field of this valuation.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: v = valuations.GaussValuation(QQ['x'], QQ.valuation(2))
sage: w = K.valuation(v)
sage: w.residue_ring()
Rational function field in x over Finite Field of size 2

sage: R.<x> = QQ[]
sage: vv = v.augmentation(x, 1)
sage: w = K.valuation(vv)
sage: w.residue_ring()
Rational function field in x over Finite Field of size 2

sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 + 2*x)
sage: w.extension(L).residue_ring()
Function field in u2 defined by u2^2 + x

class sage.rings.function_field.function_field_valuation.RationalFunctionFieldMappedValuation(parent, base_valuation, to_base_valuation_doain, from_base_valuation_domain)

Valuation on a rational function field that is implemented after a map to an isomorphic rational function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: R.<x> = QQ[]
sage: w = GaussValuation(R, QQ.valuation(2)).augmentation(x, 1)
sage: w = K.valuation(w)
sage: v = K.valuation((w, K.hom([~K.gen()]), K.hom([~K.gen()]))); v
Valuation on rational function field induced by [ Gauss valuation induced by 2-adic valuation, v(x) = 1 ] (in Rational function field in x over Rational Field after x |--> 1/x)

class sage.rings.function_field.function_field_valuation.RationalFunctionFieldValuation_base(parent)

Base class for valuations on rational function fields.

element_with_valuation(s)

Return an element with valuation s.

EXAMPLES:

sage: K.<a> = NumberField(x^3+6)
sage: v = K.valuation(2)
sage: R.<x> = K[]
sage: w = GaussValuation(R, v).augmentation(x, 1/123)
sage: K.<x> = FunctionField(K)
sage: w = w.extension(K)
sage: w.element_with_valuation(122/123)
2/x
sage: w.element_with_valuation(1)
2