Trivial valuations¶

AUTHORS:

Julian Rüth (2016-10-14): initial version

EXAMPLES:

Sage

sage: v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field
sage: v(1)
0

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field
>>> v(Integer(1))
0

Sage Live

v = valuations.TrivialValuation(QQ); v
v(1)

class sage.rings.valuation.trivial_valuation.TrivialDiscretePseudoValuation(parent)[source]¶

Bases: TrivialDiscretePseudoValuation_base, InfiniteDiscretePseudoValuation

The trivial pseudo-valuation that is \(\infty\) everywhere.

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(QQ); v
Trivial pseudo-valuation on Rational Field

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(QQ); v
Trivial pseudo-valuation on Rational Field

Sage Live

v = valuations.TrivialPseudoValuation(QQ); v

lift(X)[source]¶

Return a lift of X to the domain of this valuation.

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(QQ)
sage: v.lift(v.residue_ring().zero())
0

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(QQ)
>>> v.lift(v.residue_ring().zero())
0

Sage Live

v = valuations.TrivialPseudoValuation(QQ)
v.lift(v.residue_ring().zero())

reduce(x)[source]¶

Reduce x modulo the positive elements of this valuation.

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(QQ)
sage: v.reduce(1)
0

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(QQ)
>>> v.reduce(Integer(1))
0

Sage Live

v = valuations.TrivialPseudoValuation(QQ)
v.reduce(1)

residue_ring()[source]¶

Return the residue ring of this valuation.

EXAMPLES:

Sage

sage: valuations.TrivialPseudoValuation(QQ).residue_ring()
Quotient of Rational Field by the ideal (1)

Python

>>> from sage.all import *
>>> valuations.TrivialPseudoValuation(QQ).residue_ring()
Quotient of Rational Field by the ideal (1)

Sage Live

valuations.TrivialPseudoValuation(QQ).residue_ring()

value_group()[source]¶

Return the value group of this valuation.

EXAMPLES:

A trivial discrete pseudo-valuation has no value group:

Sage

sage: v = valuations.TrivialPseudoValuation(QQ)
sage: v.value_group()
Traceback (most recent call last):
...
ValueError: The trivial pseudo-valuation that is infinity everywhere does not have a value group.

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(QQ)
>>> v.value_group()
Traceback (most recent call last):
...
ValueError: The trivial pseudo-valuation that is infinity everywhere does not have a value group.

Sage Live

v = valuations.TrivialPseudoValuation(QQ)
v.value_group()

class sage.rings.valuation.trivial_valuation.TrivialDiscretePseudoValuation_base(parent)[source]¶

Bases: DiscretePseudoValuation

Base class for code shared by trivial valuations.

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(ZZ); v
Trivial pseudo-valuation on Integer Ring

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(ZZ); v
Trivial pseudo-valuation on Integer Ring

Sage Live

v = valuations.TrivialPseudoValuation(ZZ); v

is_negative_pseudo_valuation()[source]¶

Return whether this valuation attains the value \(-\infty\).

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(QQ)
sage: v.is_negative_pseudo_valuation()
False

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(QQ)
>>> v.is_negative_pseudo_valuation()
False

Sage Live

v = valuations.TrivialPseudoValuation(QQ)
v.is_negative_pseudo_valuation()

is_trivial()[source]¶

Return whether this valuation is trivial.

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(QQ)
sage: v.is_trivial()
True

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(QQ)
>>> v.is_trivial()
True

Sage Live

v = valuations.TrivialPseudoValuation(QQ)
v.is_trivial()

uniformizer()[source]¶

Return a uniformizing element for this valuation.

EXAMPLES:

Sage

sage: v = valuations.TrivialPseudoValuation(ZZ)
sage: v.uniformizer()
Traceback (most recent call last):
...
ValueError: Trivial valuations do not define a uniformizing element

Python

>>> from sage.all import *
>>> v = valuations.TrivialPseudoValuation(ZZ)
>>> v.uniformizer()
Traceback (most recent call last):
...
ValueError: Trivial valuations do not define a uniformizing element

Sage Live

v = valuations.TrivialPseudoValuation(ZZ)
v.uniformizer()

class sage.rings.valuation.trivial_valuation.TrivialDiscreteValuation(parent)[source]¶

Bases: TrivialDiscretePseudoValuation_base, DiscreteValuation

The trivial valuation that is zero on nonzero elements.

EXAMPLES:

Sage

sage: v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field

Sage Live

v = valuations.TrivialValuation(QQ); v

extensions(ring)[source]¶

Return the unique extension of this valuation to ring.

EXAMPLES:

Sage

sage: v = valuations.TrivialValuation(ZZ)
sage: v.extensions(QQ)
[Trivial valuation on Rational Field]

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(ZZ)
>>> v.extensions(QQ)
[Trivial valuation on Rational Field]

Sage Live

v = valuations.TrivialValuation(ZZ)
v.extensions(QQ)

lift(X)[source]¶

Return a lift of X to the domain of this valuation.

EXAMPLES:

Sage

sage: v = valuations.TrivialValuation(QQ)
sage: v.lift(v.residue_ring().zero())
0

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ)
>>> v.lift(v.residue_ring().zero())
0

Sage Live

v = valuations.TrivialValuation(QQ)
v.lift(v.residue_ring().zero())

reduce(x)[source]¶

Reduce x modulo the positive elements of this valuation.

EXAMPLES:

Sage

sage: v = valuations.TrivialValuation(QQ)
sage: v.reduce(1)
1

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ)
>>> v.reduce(Integer(1))
1

Sage Live

v = valuations.TrivialValuation(QQ)
v.reduce(1)

residue_ring()[source]¶

Return the residue ring of this valuation.

EXAMPLES:

Sage

sage: valuations.TrivialValuation(QQ).residue_ring()
Rational Field

Python

>>> from sage.all import *
>>> valuations.TrivialValuation(QQ).residue_ring()
Rational Field

Sage Live

valuations.TrivialValuation(QQ).residue_ring()

value_group()[source]¶

Return the value group of this valuation.

EXAMPLES:

A trivial discrete valuation has a trivial value group:

Sage

sage: v = valuations.TrivialValuation(QQ)
sage: v.value_group()
Trivial Additive Abelian Group

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ)
>>> v.value_group()
Trivial Additive Abelian Group

Sage Live

v = valuations.TrivialValuation(QQ)
v.value_group()

class sage.rings.valuation.trivial_valuation.TrivialValuationFactory(clazz, parent, *args, **kwargs)[source]¶

Bases: UniqueFactory

Create a trivial valuation on domain.

EXAMPLES:

Sage

sage: v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field
sage: v(1)
0

Python

>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field
>>> v(Integer(1))
0

Sage Live

v = valuations.TrivialValuation(QQ); v
v(1)

create_key(domain)[source]¶

Create a key that identifies this valuation.

EXAMPLES:

Sage

sage: valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest
True

Python

>>> from sage.all import *
>>> valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest
True

Sage Live

valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest

create_object(version, key, **extra_args)[source]¶

Create a trivial valuation from key.

EXAMPLES:

Sage

sage: valuations.TrivialValuation(QQ) # indirect doctest
Trivial valuation on Rational Field

Python

>>> from sage.all import *
>>> valuations.TrivialValuation(QQ) # indirect doctest
Trivial valuation on Rational Field

Sage Live

valuations.TrivialValuation(QQ) # indirect doctest