Laurent Series Rings¶

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.base_ring()
Rational Field
sage: S = LaurentSeriesRing(GF(17)['x'], 'y')
sage: S
Laurent Series Ring in y over
 Univariate Polynomial Ring in x over Finite Field of size 17
sage: S.base_ring()
Univariate Polynomial Ring in x over Finite Field of size 17

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.base_ring()
Rational Field
>>> S = LaurentSeriesRing(GF(Integer(17))['x'], 'y')
>>> S
Laurent Series Ring in y over
 Univariate Polynomial Ring in x over Finite Field of size 17
>>> S.base_ring()
Univariate Polynomial Ring in x over Finite Field of size 17

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.base_ring()
S = LaurentSeriesRing(GF(17)['x'], 'y')
S
S.base_ring()

See also

sage.misc.defaults.set_series_precision()

class sage.rings.laurent_series_ring.LaurentSeriesRing(power_series)[source]¶

Bases: UniqueRepresentation, CommutativeRing

Univariate Laurent Series Ring.

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, 'x'); R
Laurent Series Ring in x over Rational Field
sage: x = R.0
sage: g = 1 - x + x^2 - x^4 + O(x^8); g
1 - x + x^2 - x^4 + O(x^8)
sage: g = 10*x^(-3) + 2006 - 19*x + x^2 - x^4 +O(x^8); g
10*x^-3 + 2006 - 19*x + x^2 - x^4 + O(x^8)

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, 'x'); R
Laurent Series Ring in x over Rational Field
>>> x = R.gen(0)
>>> g = Integer(1) - x + x**Integer(2) - x**Integer(4) + O(x**Integer(8)); g
1 - x + x^2 - x^4 + O(x^8)
>>> g = Integer(10)*x**(-Integer(3)) + Integer(2006) - Integer(19)*x + x**Integer(2) - x**Integer(4) +O(x**Integer(8)); g
10*x^-3 + 2006 - 19*x + x^2 - x^4 + O(x^8)

Sage Live

R = LaurentSeriesRing(QQ, 'x'); R
x = R.0
g = 1 - x + x^2 - x^4 + O(x^8); g
g = 10*x^(-3) + 2006 - 19*x + x^2 - x^4 +O(x^8); g

You can also use more mathematical notation when the base is a field:

Sage

sage: Frac(QQ[['x']])
Laurent Series Ring in x over Rational Field
sage: Frac(GF(5)['y'])
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size 5

Python

>>> from sage.all import *
>>> Frac(QQ[['x']])
Laurent Series Ring in x over Rational Field
>>> Frac(GF(Integer(5))['y'])
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size 5

Sage Live

Frac(QQ[['x']])
Frac(GF(5)['y'])

When the base ring is a domain, the fraction field is the Laurent series ring over the fraction field of the base ring:

Sage

sage: Frac(ZZ[['t']])
Laurent Series Ring in t over Rational Field

Python

>>> from sage.all import *
>>> Frac(ZZ[['t']])
Laurent Series Ring in t over Rational Field

Sage Live

Frac(ZZ[['t']])

Laurent series rings are determined by their variable and the base ring, and are globally unique:

Sage

sage: # needs sage.rings.padics
sage: K = Qp(5, prec=5)
sage: L = Qp(5, prec=200)
sage: R.<x> = LaurentSeriesRing(K)
sage: S.<y> = LaurentSeriesRing(L)
sage: R is S
False
sage: T.<y> = LaurentSeriesRing(Qp(5, prec=200))
sage: S is T
True
sage: W.<y> = LaurentSeriesRing(Qp(5, prec=199))
sage: W is T
False

sage: K = LaurentSeriesRing(CC, 'q'); K                                         # needs sage.rings.real_mpfr
Laurent Series Ring in q over Complex Field with 53 bits of precision
sage: loads(K.dumps()) == K                                                     # needs sage.rings.real_mpfr
True
sage: P = QQ[['x']]
sage: F = Frac(P)
sage: TestSuite(F).run()

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = Qp(Integer(5), prec=Integer(5))
>>> L = Qp(Integer(5), prec=Integer(200))
>>> R = LaurentSeriesRing(K, names=('x',)); (x,) = R._first_ngens(1)
>>> S = LaurentSeriesRing(L, names=('y',)); (y,) = S._first_ngens(1)
>>> R is S
False
>>> T = LaurentSeriesRing(Qp(Integer(5), prec=Integer(200)), names=('y',)); (y,) = T._first_ngens(1)
>>> S is T
True
>>> W = LaurentSeriesRing(Qp(Integer(5), prec=Integer(199)), names=('y',)); (y,) = W._first_ngens(1)
>>> W is T
False

>>> K = LaurentSeriesRing(CC, 'q'); K                                         # needs sage.rings.real_mpfr
Laurent Series Ring in q over Complex Field with 53 bits of precision
>>> loads(K.dumps()) == K                                                     # needs sage.rings.real_mpfr
True
>>> P = QQ[['x']]
>>> F = Frac(P)
>>> TestSuite(F).run()

Sage Live

# needs sage.rings.padics
K = Qp(5, prec=5)
L = Qp(5, prec=200)
R.<x> = LaurentSeriesRing(K)
S.<y> = LaurentSeriesRing(L)
R is S
T.<y> = LaurentSeriesRing(Qp(5, prec=200))
S is T
W.<y> = LaurentSeriesRing(Qp(5, prec=199))
W is T
K = LaurentSeriesRing(CC, 'q'); K                                         # needs sage.rings.real_mpfr
loads(K.dumps()) == K                                                     # needs sage.rings.real_mpfr
P = QQ[['x']]
F = Frac(P)
TestSuite(F).run()

When the base ring \(k\) is a field, the ring \(k((x))\) is a CDVF, that is a field equipped with a discrete valuation for which it is complete. The appropriate (sub)category is automatically set in this case:

Sage

sage: k = GF(11)
sage: R.<x> = k[[]]
sage: F = Frac(R)
sage: F.category()
Join of
 Category of complete discrete valuation fields and
 Category of commutative algebras over (finite enumerated fields and
 subquotients of monoids and quotients of semigroups) and
 Category of infinite sets
sage: TestSuite(F).run()

Python

>>> from sage.all import *
>>> k = GF(Integer(11))
>>> R = k[['x']]; (x,) = R._first_ngens(1)
>>> F = Frac(R)
>>> F.category()
Join of
 Category of complete discrete valuation fields and
 Category of commutative algebras over (finite enumerated fields and
 subquotients of monoids and quotients of semigroups) and
 Category of infinite sets
>>> TestSuite(F).run()

Sage Live

k = GF(11)
R.<x> = k[[]]
F = Frac(R)
F.category()
TestSuite(F).run()

Element[source]¶: alias of LaurentSeries

base_extend(R)[source]¶

Return the Laurent series ring over R in the same variable as self, assuming there is a canonical coerce map from the base ring of self to R.

EXAMPLES:

Sage

sage: K.<x> = LaurentSeriesRing(QQ, default_prec=4)
sage: K.base_extend(QQ['t'])
Laurent Series Ring in x over Univariate Polynomial Ring in t over Rational Field

Python

>>> from sage.all import *
>>> K = LaurentSeriesRing(QQ, default_prec=Integer(4), names=('x',)); (x,) = K._first_ngens(1)
>>> K.base_extend(QQ['t'])
Laurent Series Ring in x over Univariate Polynomial Ring in t over Rational Field

Sage Live

K.<x> = LaurentSeriesRing(QQ, default_prec=4)
K.base_extend(QQ['t'])

change_ring(R)[source]¶

EXAMPLES:

Sage

sage: K.<x> = LaurentSeriesRing(QQ, default_prec=4)
sage: R = K.change_ring(ZZ); R
Laurent Series Ring in x over Integer Ring
sage: R.default_prec()
4

Python

>>> from sage.all import *
>>> K = LaurentSeriesRing(QQ, default_prec=Integer(4), names=('x',)); (x,) = K._first_ngens(1)
>>> R = K.change_ring(ZZ); R
Laurent Series Ring in x over Integer Ring
>>> R.default_prec()
4

Sage Live

K.<x> = LaurentSeriesRing(QQ, default_prec=4)
R = K.change_ring(ZZ); R
R.default_prec()

characteristic()[source]¶

EXAMPLES:

Sage

sage: R.<x> = LaurentSeriesRing(GF(17))
sage: R.characteristic()
17

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(GF(Integer(17)), names=('x',)); (x,) = R._first_ngens(1)
>>> R.characteristic()
17

Sage Live

R.<x> = LaurentSeriesRing(GF(17))
R.characteristic()

construction()[source]¶

Return the functorial construction of this Laurent power series ring.

The construction is given as the completion of the Laurent polynomials.

EXAMPLES:

Sage

sage: L.<t> = LaurentSeriesRing(ZZ, default_prec=42)
sage: phi, arg = L.construction()
sage: phi
Completion[t, prec=42]
sage: arg
Univariate Laurent Polynomial Ring in t over Integer Ring
sage: phi(arg) is L
True

Python

>>> from sage.all import *
>>> L = LaurentSeriesRing(ZZ, default_prec=Integer(42), names=('t',)); (t,) = L._first_ngens(1)
>>> phi, arg = L.construction()
>>> phi
Completion[t, prec=42]
>>> arg
Univariate Laurent Polynomial Ring in t over Integer Ring
>>> phi(arg) is L
True

Sage Live

L.<t> = LaurentSeriesRing(ZZ, default_prec=42)
phi, arg = L.construction()
phi
arg
phi(arg) is L

Because of this construction, pushout is automatically available:

Sage

sage: 1/2 * t
1/2*t
sage: parent(1/2 * t)
Laurent Series Ring in t over Rational Field

sage: QQbar.gen() * t                                                       # needs sage.rings.number_field
I*t
sage: parent(QQbar.gen() * t)                                               # needs sage.rings.number_field
Laurent Series Ring in t over Algebraic Field

Python

>>> from sage.all import *
>>> Integer(1)/Integer(2) * t
1/2*t
>>> parent(Integer(1)/Integer(2) * t)
Laurent Series Ring in t over Rational Field

>>> QQbar.gen() * t                                                       # needs sage.rings.number_field
I*t
>>> parent(QQbar.gen() * t)                                               # needs sage.rings.number_field
Laurent Series Ring in t over Algebraic Field

Sage Live

1/2 * t
parent(1/2 * t)
QQbar.gen() * t                                                       # needs sage.rings.number_field
parent(QQbar.gen() * t)                                               # needs sage.rings.number_field

default_prec()[source]¶

Get the precision to which exact elements are truncated when necessary (most frequently when inverting).

EXAMPLES:

Sage

sage: R.<x> = LaurentSeriesRing(QQ, default_prec=5)
sage: R.default_prec()
5

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, default_prec=Integer(5), names=('x',)); (x,) = R._first_ngens(1)
>>> R.default_prec()
5

Sage Live

R.<x> = LaurentSeriesRing(QQ, default_prec=5)
R.default_prec()

fraction_field()[source]¶

Return the fraction field of this ring of Laurent series.

If the base ring is a field, then Laurent series are already a field. If the base ring is a domain, then the Laurent series over its fraction field is returned. Otherwise, raise a ValueError.

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(ZZ, 't', 30).fraction_field()
sage: R
Laurent Series Ring in t over Rational Field
sage: R.default_prec()
30

sage: LaurentSeriesRing(Zmod(4), 't').fraction_field()
Traceback (most recent call last):
...
ValueError: must be an integral domain

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(ZZ, 't', Integer(30)).fraction_field()
>>> R
Laurent Series Ring in t over Rational Field
>>> R.default_prec()
30

>>> LaurentSeriesRing(Zmod(Integer(4)), 't').fraction_field()
Traceback (most recent call last):
...
ValueError: must be an integral domain

Sage Live

R = LaurentSeriesRing(ZZ, 't', 30).fraction_field()
R
R.default_prec()
LaurentSeriesRing(Zmod(4), 't').fraction_field()

gen(n=0)[source]¶

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.gen()
x

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.gen()
x

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.gen()

is_dense()[source]¶

EXAMPLES:

Sage

sage: K.<x> = LaurentSeriesRing(QQ, sparse=True)
sage: K.is_dense()
False

Python

>>> from sage.all import *
>>> K = LaurentSeriesRing(QQ, sparse=True, names=('x',)); (x,) = K._first_ngens(1)
>>> K.is_dense()
False

Sage Live

K.<x> = LaurentSeriesRing(QQ, sparse=True)
K.is_dense()

is_exact()[source]¶

Laurent series rings are inexact.

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.is_exact()
False

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.is_exact()
False

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.is_exact()

is_field(proof=True)[source]¶: A Laurent series ring is a field if and only if the base ring is a field.

is_sparse()[source]¶

Return if self is a sparse implementation.

EXAMPLES:

Sage

sage: K.<x> = LaurentSeriesRing(QQ, sparse=True)
sage: K.is_sparse()
True

Python

>>> from sage.all import *
>>> K = LaurentSeriesRing(QQ, sparse=True, names=('x',)); (x,) = K._first_ngens(1)
>>> K.is_sparse()
True

Sage Live

K.<x> = LaurentSeriesRing(QQ, sparse=True)
K.is_sparse()

laurent_polynomial_ring()[source]¶

If this is the Laurent series ring \(R((t))\), return the Laurent polynomial ring \(R[t,1/t]\).

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.laurent_polynomial_ring()
Univariate Laurent Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.laurent_polynomial_ring()
Univariate Laurent Polynomial Ring in x over Rational Field

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.laurent_polynomial_ring()

ngens()[source]¶

Laurent series rings are univariate.

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.ngens()
1

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.ngens()
1

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.ngens()

polynomial_ring()[source]¶

If this is the Laurent series ring \(R((t))\), return the polynomial ring \(R[t]\).

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.polynomial_ring()
Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.polynomial_ring()
Univariate Polynomial Ring in x over Rational Field

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.polynomial_ring()

power_series_ring()[source]¶

If this is the Laurent series ring \(R((t))\), return the power series ring \(R[[t]]\).

EXAMPLES:

Sage

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.power_series_ring()
Power Series Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, "x")
>>> R.power_series_ring()
Power Series Ring in x over Rational Field

Sage Live

R = LaurentSeriesRing(QQ, "x")
R.power_series_ring()

random_element(algorithm='default')[source]¶

Return a random element of this Laurent series ring.

The optional algorithm parameter decides how elements are generated. Algorithms currently implemented:

'default': Choose an integer shift using the standard distribution on the integers. Then choose a list of coefficients using the random_element function of the base ring, and construct a new element based on those coefficients, so that the i-th coefficient corresponds to the (i+shift)-th power of the uniformizer. The amount of coefficients is determined by the default_prec of the ring. Note that this method only creates non-exact elements.

EXAMPLES:

Sage

sage: S.<s> = LaurentSeriesRing(GF(3))
sage: S.random_element()  # random
s^-8 + s^-7 + s^-6 + s^-5 + s^-1 + s + s^3 + s^4
+ s^5 + 2*s^6 + s^7 + s^11 + O(s^12)

Python

>>> from sage.all import *
>>> S = LaurentSeriesRing(GF(Integer(3)), names=('s',)); (s,) = S._first_ngens(1)
>>> S.random_element()  # random
s^-8 + s^-7 + s^-6 + s^-5 + s^-1 + s + s^3 + s^4
+ s^5 + 2*s^6 + s^7 + s^11 + O(s^12)

Sage Live

S.<s> = LaurentSeriesRing(GF(3))
S.random_element()  # random

residue_field()[source]¶

Return the residue field of this Laurent series field if it is a complete discrete valuation field (i.e. if the base ring is a field, in which base it is also the residue field).

EXAMPLES:

Sage

sage: R.<x> = LaurentSeriesRing(GF(17))
sage: R.residue_field()
Finite Field of size 17

sage: R.<x> = LaurentSeriesRing(ZZ)
sage: R.residue_field()
Traceback (most recent call last):
...
TypeError: the base ring is not a field

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(GF(Integer(17)), names=('x',)); (x,) = R._first_ngens(1)
>>> R.residue_field()
Finite Field of size 17

>>> R = LaurentSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> R.residue_field()
Traceback (most recent call last):
...
TypeError: the base ring is not a field

Sage Live

R.<x> = LaurentSeriesRing(GF(17))
R.residue_field()
R.<x> = LaurentSeriesRing(ZZ)
R.residue_field()

uniformizer()[source]¶

Return a uniformizer of this Laurent series field if it is a discrete valuation field (i.e. if the base ring is actually a field). Otherwise, an error is raised.

EXAMPLES:

Sage

sage: R.<t> = LaurentSeriesRing(QQ)
sage: R.uniformizer()
t

sage: R.<t> = LaurentSeriesRing(ZZ)
sage: R.uniformizer()
Traceback (most recent call last):
...
TypeError: the base ring is not a field

Python

>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> R.uniformizer()
t

>>> R = LaurentSeriesRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> R.uniformizer()
Traceback (most recent call last):
...
TypeError: the base ring is not a field

Sage Live

R.<t> = LaurentSeriesRing(QQ)
R.uniformizer()
R.<t> = LaurentSeriesRing(ZZ)
R.uniformizer()

sage.rings.laurent_series_ring.is_LaurentSeriesRing(x)[source]¶

Return True if this is a univariate Laurent series ring.

This is in keeping with the behavior of is_PolynomialRing versus is_MPolynomialRing.