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 , … ]

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - x)
v = K.valuation(0)                                                            # needs sage.rings.function_field
w = v.extension(L); w                                                         # needs sage.rings.function_field
w._base_valuation                                                             # needs sage.rings.function_field

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

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - x)
v = K.valuation(0)
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: # 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

# needs sage.rings.number_field
K = QQ
R.<t> = K[]
L.<t> = K.extension(t^2 + 1)
v = valuations.pAdicValuation(QQ, 5)
u,uu = v.extensions(L)
u.lower_bound(t + 2)
u(t + 2)

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

# needs sage.rings.number_field
K = QQ
R.<t> = K[]
L.<t> = K.extension(t^2 + 1)
v = valuations.pAdicValuation(QQ, 2)
w = v.extension(L)
w.restriction(K) is v

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: # 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

# needs sage.rings.number_field
K = QQ
R.<t> = K[]
L.<t> = K.extension(t^2 + 1)
v = valuations.pAdicValuation(QQ, 5)
u,uu = v.extensions(L)
f = 125*t + 1
u.simplify(f, error=u(f), force=True)

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

# needs sage.rings.number_field
K = QQ
R.<t> = K[]
L.<t> = K.extension(t^2 + 1)
v = valuations.pAdicValuation(QQ, 5)
u,uu = v.extensions(L)
u.upper_bound(t + 2) >= 1
u(t + 2)

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]

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - x)
v = K.valuation(1)
w = v.extensions(L); w

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

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - x)
v = K.valuation(0)
w = v.extension(L); w

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

# needs sage.rings.number_field
K = QQ
R.<t> = K[]
L.<t> = K.extension(t^2 + 1)
v = valuations.pAdicValuation(QQ, 5)
u,uu = v.extensions(L)
u.element_with_valuation(1)

lift(F)#

Lift F from the residue_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

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - x)
v = K.valuation(2)
w = v.extension(L)
w.lift(w.residue_field().gen())

reduce(f)#

Return the reduction of f in the residue_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

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - (x - 2))
v = K.valuation(0)
w = v.extension(L)
w.reduce(y)

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

K = QQ
R.<t> = K[]
L.<t> = K.extension(t^2 + 1)                                          # needs sage.rings.number_field
v = valuations.pAdicValuation(QQ, 2)
v.extension(L).residue_ring()                                         # needs sage.rings.number_field

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: # 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]

# needs sage.rings.function_field
K.<x> = FunctionField(QQ)
R.<y> = K[]
L.<y> = K.extension(y^2 - x)
v = K.valuation(0)
w = v.extensions(L)[0]

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

sage: w.simplify(y + x^32)                                                  # needs sage.rings.function_field
y + x^32

w.simplify(y + x^32)                                                  # needs sage.rings.function_field

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

w.simplify(y + x^32, force=True)                                      # needs sage.rings.function_field

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

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