Valuations which are implemented through a map to another valuation#
EXAMPLES:
Extensions of valuations over finite field extensions \(L=K[x]/(G)\) are realized through an infinite valuation on \(K[x]\) which maps \(G\) to infinity:
sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: v = K.valuation(0) # needs sage.rings.function_field
sage: w = v.extension(L); w # needs sage.rings.function_field
(x)-adic valuation
sage: w._base_valuation # needs sage.rings.function_field
[ Gauss valuation induced by (x)-adic valuation, v(y) = 1/2 , … ]
AUTHORS:
Julian Rüth (2016-11-10): initial version
- class sage.rings.valuation.mapped_valuation.FiniteExtensionFromInfiniteValuation(parent, base_valuation)#
Bases:
MappedValuation_base
,DiscreteValuation
A valuation on a quotient of the form \(L=K[x]/(G)\) with an irreducible \(G\) which is internally backed by a pseudo-valuations on \(K[x]\) which sends \(G\) to infinity.
INPUT:
parent
– the containing valuation space (usually the space of discrete valuations on \(L\))base_valuation
– an infinite valuation on \(K[x]\) which takes \(G\) to infinity
EXAMPLES:
sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: v = K.valuation(0) sage: w = v.extension(L); w (x)-adic valuation
- lower_bound(x)#
Return a lower bound of this valuation at
x
.Use this method to get an approximation of the valuation of
x
when speed is more important than accuracy.EXAMPLES:
sage: # needs sage.rings.number_field sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) sage: v = valuations.pAdicValuation(QQ, 5) sage: u,uu = v.extensions(L) sage: u.lower_bound(t + 2) 0 sage: u(t + 2) 1
- restriction(ring)#
Return the restriction of this valuation to
ring
.EXAMPLES:
sage: # needs sage.rings.number_field sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) sage: v = valuations.pAdicValuation(QQ, 2) sage: w = v.extension(L) sage: w.restriction(K) is v True
- simplify(x, error=None, force=False)#
Return a simplified version of
x
.Produce an element which differs from
x
by an element of valuation strictly greater than the valuation ofx
(or strictly greater thanerror
if set.)EXAMPLES:
sage: # needs sage.rings.number_field sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) sage: v = valuations.pAdicValuation(QQ, 5) sage: u,uu = v.extensions(L) sage: f = 125*t + 1 sage: u.simplify(f, error=u(f), force=True) 1
- upper_bound(x)#
Return an upper bound of this valuation at
x
.Use this method to get an approximation of the valuation of
x
when speed is more important than accuracy.EXAMPLES:
sage: # needs sage.rings.number_field sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) sage: v = valuations.pAdicValuation(QQ, 5) sage: u,uu = v.extensions(L) sage: u.upper_bound(t + 2) >= 1 True sage: u(t + 2) 1
- class sage.rings.valuation.mapped_valuation.FiniteExtensionFromLimitValuation(parent, approximant, G, approximants)#
Bases:
FiniteExtensionFromInfiniteValuation
An extension of a valuation on a finite field extensions \(L=K[x]/(G)\) which is induced by an infinite limit valuation on \(K[x]\).
EXAMPLES:
sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: v = K.valuation(1) 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]
- class sage.rings.valuation.mapped_valuation.MappedValuation_base(parent, base_valuation)#
Bases:
DiscretePseudoValuation
A valuation which is implemented through another proxy “base” valuation.
EXAMPLES:
sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: v = K.valuation(0) sage: w = v.extension(L); w (x)-adic valuation
- element_with_valuation(s)#
Return an element with valuation
s
.EXAMPLES:
sage: # needs sage.rings.number_field sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) sage: v = valuations.pAdicValuation(QQ, 5) sage: u,uu = v.extensions(L) sage: u.element_with_valuation(1) 5
- lift(F)#
Lift
F
from theresidue_field()
of this valuation into its domain.EXAMPLES:
sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: v = K.valuation(2) sage: w = v.extension(L) sage: w.lift(w.residue_field().gen()) y
- reduce(f)#
Return the reduction of
f
in theresidue_field()
of this valuation.EXAMPLES:
sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - (x - 2)) sage: v = K.valuation(0) sage: w = v.extension(L) sage: w.reduce(y) u1
- residue_ring()#
Return the residue ring of this valuation.
EXAMPLES:
sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) # needs sage.rings.number_field sage: v = valuations.pAdicValuation(QQ, 2) sage: v.extension(L).residue_ring() # needs sage.rings.number_field Finite Field of size 2
- simplify(x, error=None, force=False)#
Return a simplified version of
x
.Produce an element which differs from
x
by an element of valuation strictly greater than the valuation ofx
(or strictly greater thanerror
if set.)If
force
is not set, then expensive simplifications may be avoided.EXAMPLES:
sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: v = K.valuation(0) sage: w = v.extensions(L)[0]
As
_relative_size()
misses the bloated termx^32
, the following term does not get simplified:sage: w.simplify(y + x^32) # needs sage.rings.function_field y + x^32
In this case the simplification can be forced but this should not happen as a default as the recursive simplification can be quite costly:
sage: w.simplify(y + x^32, force=True) # needs sage.rings.function_field y
- uniformizer()#
Return a uniformizing element of this valuation.
EXAMPLES:
sage: K = QQ sage: R.<t> = K[] sage: L.<t> = K.extension(t^2 + 1) # needs sage.rings.number_field sage: v = valuations.pAdicValuation(QQ, 2) sage: v.extension(L).uniformizer() # needs sage.rings.number_field t + 1