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: 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

sage: w._base_valuation
[ 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: sage.rings.valuation.mapped_valuation.MappedValuation_base, sage.rings.valuation.valuation.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: 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 an 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: 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: 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 of x (or strictly greater than error if set.)

EXAMPLES:

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: 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: sage.rings.valuation.mapped_valuation.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: 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: sage.rings.valuation.valuation.DiscretePseudoValuation

A valuation which is implemented through another proxy “base” valuation.

EXAMPLES:

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

lift(F)

Lift F from the residue_field() of this valuation into its domain.

EXAMPLES:

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 the residue_field() of this valuation.

EXAMPLES:

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)
sage: v = valuations.pAdicValuation(QQ, 2)
sage: v.extension(L).residue_ring()
Finite Field of size 2

uniformizer()

Return a uniformizing element of this valuation.

EXAMPLES:

sage: K = QQ
sage: R.<t> = K[]
sage: L.<t> = K.extension(t^2 + 1)
sage: v = valuations.pAdicValuation(QQ, 2)
sage: v.extension(L).uniformizer()
t + 1