Laurent Series Rings

EXAMPLES:

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

See also

class sage.rings.laurent_series_ring.LaurentSeriesRing(power_series)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.rings.ring.CommutativeRing

Univariate Laurent Series Ring.

EXAMPLES:

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)

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

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

Here the fraction field is not just the Laurent series ring, so you can’t use the Frac notation to make the Laurent series ring:

sage: Frac(ZZ[['t']])
Fraction Field of Power Series Ring in t over Integer Ring

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

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')
sage: K
Laurent Series Ring in q over Complex Field with 53 bits of precision
sage: loads(K.dumps()) == K
True
sage: P = QQ[['x']]
sage: F = Frac(P)
sage: 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: k = GF(11)
sage: R.<x> = k[[]]
sage: F = Frac(R)
sage: F.category()
Category of infinite complete discrete valuation fields
sage: TestSuite(F).run()
Element

alias of sage.rings.laurent_series_ring_element.LaurentSeries

base_extend(R)

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: 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
change_ring(R)

EXAMPLES:

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
characteristic()

EXAMPLES:

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

Return the functorial construction of this Laurent power series ring.

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

EXAMPLES:

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

Because of this construction, pushout is automatically available:

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

sage: QQbar.gen() * t
I*t
sage: parent(QQbar.gen() * t)
Laurent Series Ring in t over Algebraic Field
default_prec()

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

EXAMPLES:

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

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: 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
gen(n=0)

EXAMPLES:

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

EXAMPLES:

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

Laurent series rings are inexact.

EXAMPLES:

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

A Laurent series ring is a field if and only if the base ring is a field.

is_sparse()

Return if self is a sparse implementation.

EXAMPLES:

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

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

EXAMPLES:

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

Laurent series rings are univariate.

EXAMPLES:

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

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

EXAMPLES:

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

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

EXAMPLES:

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

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: 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
uniformizer()

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: 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
sage.rings.laurent_series_ring.is_LaurentSeriesRing(x)

Return True if this is a univariate Laurent series ring.

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