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

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)

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
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)

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)

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
element_with_valuation(s)

Return an element with valuation s.

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.element_with_valuation(1)
5
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
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.)

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

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.extensions(L)

As _relative_size() misses the bloated term x^32, the following term does not get simplified:

sage: w.simplify(y + x^32)
y + x^32

In this case the simplification can be forced but this should not happen as a default as the recersive simplification can be quite costly:

sage: w.simplify(y + x^32, force=True)
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)
sage: v = valuations.pAdicValuation(QQ, 2)
sage: v.extension(L).uniformizer()
t + 1