Lazy Series Rings#

We provide lazy implementations for various \(\NN\)-graded rings.

`LazyLaurentSeriesRing`	The ring of lazy Laurent series.
`LazyPowerSeriesRing`	The ring of (possibly multivariate) lazy Taylor series.
`LazyCompletionGradedAlgebra`	The completion of a graded algebra consisting of formal series.
`LazySymmetricFunctions`	The ring of (possibly multivariate) lazy symmetric functions.
`LazyDirichletSeriesRing`	The ring of lazy Dirichlet series.

AUTHORS:

Kwankyu Lee (2019-02-24): initial version
Tejasvi Chebrolu, Martin Rubey, Travis Scrimshaw (2021-08): refactored and expanded functionality

class sage.rings.lazy_series_ring.LazyCompletionGradedAlgebra(basis, sparse=True, category=None)#

Bases: LazySeriesRing

The completion of a graded algebra consisting of formal series.

For a graded algebra \(A\), we can form a completion of \(A\) consisting of all formal series of \(A\) such that each homogeneous component is a finite linear combination of basis elements of \(A\).

INPUT:

basis – a graded algebra
names – name(s) of the alphabets
sparse – (default: True) whether we use a sparse or a dense representation

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: S = NCSF.Complete()
sage: L = S.formal_series_ring()
sage: L
Lazy completion of Non-Commutative Symmetric Functions over the Rational Field in the Complete basis

sage: f = 1 / (1 - L(S[1]))
sage: f
S[] + S[1] + (S[1,1]) + (S[1,1,1]) + (S[1,1,1,1]) + (S[1,1,1,1,1]) + (S[1,1,1,1,1,1]) + O^7
sage: g = 1 / (1 - L(S[2]))
sage: g
S[] + S[2] + (S[2,2]) + (S[2,2,2]) + O^7
sage: f * g
S[] + S[1] + (S[1,1]+S[2]) + (S[1,1,1]+S[1,2])
 + (S[1,1,1,1]+S[1,1,2]+S[2,2]) + (S[1,1,1,1,1]+S[1,1,1,2]+S[1,2,2])
 + (S[1,1,1,1,1,1]+S[1,1,1,1,2]+S[1,1,2,2]+S[2,2,2]) + O^7
sage: g * f
S[] + S[1] + (S[1,1]+S[2]) + (S[1,1,1]+S[2,1])
 + (S[1,1,1,1]+S[2,1,1]+S[2,2]) + (S[1,1,1,1,1]+S[2,1,1,1]+S[2,2,1])
 + (S[1,1,1,1,1,1]+S[2,1,1,1,1]+S[2,2,1,1]+S[2,2,2]) + O^7
sage: f * g - g * f
(S[1,2]-S[2,1]) + (S[1,1,2]-S[2,1,1])
 + (S[1,1,1,2]+S[1,2,2]-S[2,1,1,1]-S[2,2,1])
 + (S[1,1,1,1,2]+S[1,1,2,2]-S[2,1,1,1,1]-S[2,2,1,1]) + O^7

Element#: alias of LazyCompletionGradedAlgebraElement

some_elements()#

Return a list of elements of self.

EXAMPLES:

sage: m = SymmetricFunctions(GF(5)).m()
sage: L = LazySymmetricFunctions(m)
sage: L.some_elements()[:5]
[0, m[], 2*m[] + 2*m[1] + 3*m[2], 2*m[1] + 3*m[2],
 3*m[] + 2*m[1] + (m[1,1]+m[2])
       + (2*m[1,1,1]+m[3])
       + (2*m[1,1,1,1]+4*m[2,1,1]+2*m[2,2])
       + (3*m[2,1,1,1]+3*m[3,1,1]+4*m[3,2]+m[5])
       + (2*m[2,2,1,1]+m[2,2,2]+2*m[3,2,1]+2*m[3,3]+m[4,1,1]+3*m[4,2]+4*m[5,1]+4*m[6])
       + O^7]

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: S = NCSF.Complete()
sage: L = S.formal_series_ring()
sage: L.some_elements()[:4]
[0, S[], 2*S[] + 2*S[1] + (3*S[1,1]), 2*S[1] + (3*S[1,1])]

class sage.rings.lazy_series_ring.LazyDirichletSeriesRing(base_ring, names, sparse=True, category=None)#

Bases: LazySeriesRing

The ring of lazy Dirichlet series.

INPUT:

base_ring – base ring of this Dirichlet series ring
names – name of the generator of this Dirichlet series ring
sparse – (default: True) whether this series is sparse or not

Unlike formal univariate Laurent/power series (over a field), the ring of formal Dirichlet series is not a Wikipedia article discrete_valuation_ring. On the other hand, it is a Wikipedia article local_ring. The unique maximal ideal consists of all non-invertible series, i.e., series with vanishing constant term.

Todo

According to the answers in https://mathoverflow.net/questions/5522/dirichlet-series-with-integer-coefficients-as-a-ufd, (which, in particular, references arXiv math/0105219) the ring of formal Dirichlet series is actually a Wikipedia article Unique_factorization_domain over \(\ZZ\).

Note

An interesting valuation is described in Emil Daniel Schwab; Gheorghe Silberberg A note on some discrete valuation rings of arithmetical functions, Archivum Mathematicum, Vol. 36 (2000), No. 2, 103-109, http://dml.cz/dmlcz/107723. Let \(J_k\) be the ideal of Dirichlet series whose coefficient \(f[n]\) of \(n^s\) vanishes if \(n\) has less than \(k\) prime factors, counting multiplicities. For any Dirichlet series \(f\), let \(D(f)\) be the largest integer \(k\) such that \(f\) is in \(J_k\). Then \(D\) is surjective, \(D(f g) = D(f) + D(g)\) for nonzero \(f\) and \(g\), and \(D(f + g) \geq \min(D(f), D(g))\) provided that \(f + g\) is nonzero.

For example, \(J_1\) are series with no constant term, and \(J_2\) are series such that \(f[1]\) and \(f[p]\) for prime \(p\) vanish.

Since this is a chain of increasing ideals, the ring of formal Dirichlet series is not a Wikipedia article Noetherian_ring.

Evidently, this valuation cannot be computed for a given series.

EXAMPLES:

sage: LazyDirichletSeriesRing(ZZ, 't')
Lazy Dirichlet Series Ring in t over Integer Ring

The ideal generated by \(2^-s\) and \(3^-s\) is not principal:

sage: L = LazyDirichletSeriesRing(QQ, 's')
sage: L in PrincipalIdealDomains
False

Element#: alias of LazyDirichletSeries

one()#

Return the constant series \(1\).

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: L.one()
1
sage: ~L.one()
1 + O(1/(8^z))

some_elements()#

Return a list of elements of self.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: L.some_elements()
[0, 1,
 1/(4^z) + 1/(5^z) + 1/(6^z) + O(1/(7^z)),
 1/(2^z) - 1/(3^z) + 2/4^z - 2/5^z + 3/6^z - 3/7^z + 4/8^z - 4/9^z,
 1/(2^z) - 1/(3^z) + 2/4^z - 2/5^z + 3/6^z - 3/7^z + 4/8^z - 4/9^z + 1/(10^z) + 1/(11^z) + 1/(12^z) + O(1/(13^z)),
 1 + 4/2^z + 9/3^z + 16/4^z + 25/5^z + 36/6^z + 49/7^z + O(1/(8^z))]

sage: L = LazyDirichletSeriesRing(QQ, 'z')
sage: L.some_elements()
[0, 1,
 1/2/4^z + 1/2/5^z + 1/2/6^z + O(1/(7^z)),
 1/2 - 1/2/2^z + 2/3^z - 2/4^z + 1/(6^z) - 1/(7^z) + 42/8^z + 2/3/9^z,
 1/2 - 1/2/2^z + 2/3^z - 2/4^z + 1/(6^z) - 1/(7^z) + 42/8^z + 2/3/9^z + 1/2/10^z + 1/2/11^z + 1/2/12^z + O(1/(13^z)),
 1 + 4/2^z + 9/3^z + 16/4^z + 25/5^z + 36/6^z + 49/7^z + O(1/(8^z))]

class sage.rings.lazy_series_ring.LazyLaurentSeriesRing(base_ring, names, sparse=True, category=None)#

Bases: LazySeriesRing

The ring of lazy Laurent series.

The ring of Laurent series over a ring with the usual arithmetic where the coefficients are computed lazily.

INPUT:

base_ring – base ring
names – name of the generator
sparse – (default: True) whether the implementation of the series is sparse or not

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: 1 / (1 - z)
1 + z + z^2 + O(z^3)
sage: 1 / (1 - z) == 1 / (1 - z)
True
sage: L in Fields
True

Lazy Laurent series ring over a finite field:

sage: L.<z> = LazyLaurentSeriesRing(GF(3)); L
Lazy Laurent Series Ring in z over Finite Field of size 3
sage: e = 1 / (1 + z)
sage: e.coefficient(100)
1
sage: e.coefficient(100).parent()
Finite Field of size 3

Series can be defined by specifying a coefficient function and a valuation:

sage: R.<x,y> = QQ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: def coeff(n):
....:     if n < 0:
....:         return -2 + n
....:     if n == 0:
....:         return 6
....:     return x + y^n
sage: f = L(coeff, valuation=-5)
sage: f
-7*z^-5 - 6*z^-4 - 5*z^-3 - 4*z^-2 - 3*z^-1 + 6 + (x + y)*z + O(z^2)
sage: 1 / (1 - f)
1/7*z^5 - 6/49*z^6 + 1/343*z^7 + 8/2401*z^8 + 64/16807*z^9
 + 17319/117649*z^10 + (1/49*x + 1/49*y - 180781/823543)*z^11 + O(z^12)
sage: L(coeff, valuation=-3, degree=3, constant=x)
-5*z^-3 - 4*z^-2 - 3*z^-1 + 6 + (x + y)*z + (y^2 + x)*z^2
 + x*z^3 + x*z^4 + x*z^5 + O(z^6)

We can also specify a polynomial or the initial coefficients. Additionally, we may specify that all coefficients are equal to a given constant, beginning at a given degree:

sage: L([1, x, y, 0, x+y])
1 + x*z + y*z^2 + (x + y)*z^4
sage: L([1, x, y, 0, x+y], constant=2)
1 + x*z + y*z^2 + (x + y)*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)
sage: L([1, x, y, 0, x+y], degree=7, constant=2)
1 + x*z + y*z^2 + (x + y)*z^4 + 2*z^7 + 2*z^8 + 2*z^9 + O(z^10)
sage: L([1, x, y, 0, x+y], valuation=-2)
z^-2 + x*z^-1 + y + (x + y)*z^2
sage: L([1, x, y, 0, x+y], valuation=-2, constant=3)
z^-2 + x*z^-1 + y + (x + y)*z^2 + 3*z^3 + 3*z^4 + 3*z^5 + O(z^6)
sage: L([1, x, y, 0, x+y], valuation=-2, degree=4, constant=3)
z^-2 + x*z^-1 + y + (x + y)*z^2 + 3*z^4 + 3*z^5 + 3*z^6 + O(z^7)

Some additional examples over the integer ring:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L in Fields
False
sage: 1 / (1 - 2*z)^3
1 + 6*z + 24*z^2 + 80*z^3 + 240*z^4 + 672*z^5 + 1792*z^6 + O(z^7)

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: L(x^-2 + 3 + x)
z^-2 + 3 + z
sage: L(x^-2 + 3 + x, valuation=-5, constant=2)
z^-5 + 3*z^-3 + z^-2 + 2*z^-1 + 2 + 2*z + O(z^2)
sage: L(x^-2 + 3 + x, valuation=-5, degree=0, constant=2)
z^-5 + 3*z^-3 + z^-2 + 2 + 2*z + 2*z^2 + O(z^3)

We can truncate a series, shift its coefficients, or replace all coefficients beginning with a given degree by a constant:

sage: f = 1 / (z + z^2)
sage: f
z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
sage: L(f, valuation=2)
z^2 - z^3 + z^4 - z^5 + z^6 - z^7 + z^8 + O(z^9)
sage: L(f, degree=3)
z^-1 - 1 + z - z^2
sage: L(f, degree=3, constant=2)
z^-1 - 1 + z - z^2 + 2*z^3 + 2*z^4 + 2*z^5 + O(z^6)
sage: L(f, valuation=1, degree=4)
z - z^2 + z^3
sage: L(f, valuation=1, degree=4, constant=5)
z - z^2 + z^3 + 5*z^4 + 5*z^5 + 5*z^6 + O(z^7)

Power series can be defined recursively (see sage.rings.lazy_series.LazyModuleElement.define() for more examples):

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: s = L.undefined(valuation=0)
sage: s.define(1 + z*s^2)
sage: s
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)

If the series is not specified by a finite number of initial coefficients and a constant for the remaining coefficients, then equality checking will depend on the coefficients which have already been computed. If this information is not enough to check that two series are different we raise an error:

sage: f = 1 / (z + z^2); f
z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
sage: f2 = f * 2  # currently no coefficients computed
sage: f3 = f * 3  # currently no coefficients computed
sage: f2 == f3
Traceback (most recent call last):
...
ValueError: undecidable
sage: f2  # computes some of the coefficients of f2
2*z^-1 - 2 + 2*z - 2*z^2 + 2*z^3 - 2*z^4 + 2*z^5 + O(z^6)
sage: f3  # computes some of the coefficients of f3
3*z^-1 - 3 + 3*z - 3*z^2 + 3*z^3 - 3*z^4 + 3*z^5 + O(z^6)
sage: f2 == f3
False

The implementation of the ring can be either be a sparse or a dense one. The default is a sparse implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L.is_sparse()
True
sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: L.is_sparse()
False

Element#: alias of LazyLaurentSeries

euler()#

Return the Euler function as an element of self.

The Euler function is defined as

\[\phi(z) = (z; z)_{\infty} = \sum_{n=0}^{\infty} (-1)^n q^{(3n^2-n)/2}.\]

EXAMPLES:

sage: L.<q> = LazyLaurentSeriesRing(ZZ)
sage: phi = q.euler()
sage: phi
1 - q - q^2 + q^5 + O(q^7)

We verify that \(1 / phi\) gives the generating function for all partitions:

sage: P = 1 / phi; P
1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + O(q^7)
sage: P[:20] == [Partitions(n).cardinality() for n in range(20)]
True

REFERENCES:

Wikipedia article Euler_function

gen(n=0)#

Return the n-th generator of self.

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.gen()
z
sage: L.gen(3)
Traceback (most recent call last):
...
IndexError: there is only one generator

gens()#

Return the generators of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L.gens()
(z,)
sage: 1/(1 - z)
1 + z + z^2 + O(z^3)

ngens()#

Return the number of generators of self.

This is always 1.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L.ngens()
1

q_pochhammer(q=None)#

Return the infinite q-Pochhammer symbol \((a; q)_{\infty}\), where \(a\) is the variable of self.

This is also one version of the quantum dilogarithm or the \(q\)-Exponential function.

INPUT:

q – (default: \(q \in \QQ(q)\)) the parameter \(q\)

EXAMPLES:

sage: q = ZZ['q'].fraction_field().gen()
sage: L.<z> = LazyLaurentSeriesRing(q.parent())
sage: qpoch = L.q_pochhammer(q)
sage: qpoch
1
 + (-1/(-q + 1))*z
 + (q/(q^3 - q^2 - q + 1))*z^2
 + (-q^3/(-q^6 + q^5 + q^4 - q^2 - q + 1))*z^3
 + (q^6/(q^10 - q^9 - q^8 + 2*q^5 - q^2 - q + 1))*z^4
 + (-q^10/(-q^15 + q^14 + q^13 - q^10 - q^9 - q^8 + q^7 + q^6 + q^5 - q^2 - q + 1))*z^5
 + (q^15/(q^21 - q^20 - q^19 + q^16 + 2*q^14 - q^12 - q^11 - q^10 - q^9 + 2*q^7 + q^5 - q^2 - q + 1))*z^6
 + O(z^7)

We show that \((z; q)_n = \frac{(z; q)_{\infty}}{(q^n z; q)_{\infty}}\):

sage: qpoch / qpoch(q*z)
1 - z + O(z^7)
sage: qpoch / qpoch(q^2*z)
1 + (-q - 1)*z + q*z^2 + O(z^7)
sage: qpoch / qpoch(q^3*z)
1 + (-q^2 - q - 1)*z + (q^3 + q^2 + q)*z^2 - q^3*z^3 + O(z^7)
sage: qpoch / qpoch(q^4*z)
1 + (-q^3 - q^2 - q - 1)*z + (q^5 + q^4 + 2*q^3 + q^2 + q)*z^2
 + (-q^6 - q^5 - q^4 - q^3)*z^3 + q^6*z^4 + O(z^7)

We can also construct part of Euler’s function:

sage: M.<a> = LazyLaurentSeriesRing(QQ)
sage: phi = sum(qpoch[i](q=a)*a^i for i in range(10))
sage: phi[:20] == M.euler()[:20]
True

REFERENCES:

residue_field()#

Return the residue field of the ring of integers of self.

EXAMPLES:

sage: L = LazyLaurentSeriesRing(QQ, 'z')
sage: L.residue_field()
Rational Field

series(coefficient, valuation, degree=None, constant=None)#

Return a lazy Laurent series.

INPUT:

coefficient – Python function that computes coefficients or a list
valuation – integer; approximate valuation of the series
degree – (optional) integer
constant – (optional) an element of the base ring

Let the coefficient of index \(i\) mean the coefficient of the term of the series with exponent \(i\).

Python function coefficient returns the value of the coefficient of index \(i\) from input \(s\) and \(i\) where \(s\) is the series itself.

Let valuation be \(n\). All coefficients of index below \(n\) are zero. If constant is not specified, then the coefficient function is responsible to compute the values of all coefficients of index \(\ge n\). If degree or constant is a pair \((c,m)\), then the coefficient function is responsible to compute the values of all coefficients of index \(\ge n\) and \(< m\) and all the coefficients of index \(\ge m\) is the constant \(c\).

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.series(lambda s, i: i, 5, (1,10))
5*z^5 + 6*z^6 + 7*z^7 + 8*z^8 + 9*z^9 + z^10 + z^11 + z^12 + O(z^13)

sage: def g(s, i):
....:     if i < 0:
....:         return 1
....:     else:
....:         return s.coefficient(i - 1) + i
sage: e = L.series(g, -5); e
z^-5 + z^-4 + z^-3 + z^-2 + z^-1 + 1 + 2*z + O(z^2)
sage: f = e^-1; f
z^5 - z^6 - z^11 + O(z^12)
sage: f.coefficient(10)
0
sage: f.coefficient(20)
9
sage: f.coefficient(30)
-219

Alternatively, the coefficient can be a list of elements of the base ring. Then these elements are read as coefficients of the terms of degrees starting from the valuation. In this case, constant may be just an element of the base ring instead of a tuple or can be simply omitted if it is zero.

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: f = L.series([1,2,3,4], -5); f
z^-5 + 2*z^-4 + 3*z^-3 + 4*z^-2
sage: g = L.series([1,3,5,7,9], 5, constant=-1); g
z^5 + 3*z^6 + 5*z^7 + 7*z^8 + 9*z^9 - z^10 - z^11 - z^12 + O(z^13)

some_elements()#

Return a list of elements of self.

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.some_elements()[:7]
[0, 1, z,
 -3*z^-4 + z^-3 - 12*z^-2 - 2*z^-1 - 10 - 8*z + z^2 + z^3,
 z^-2 + z^3 + z^4 + z^5 + O(z^6),
 -2*z^-3 - 2*z^-2 + 4*z^-1 + 11 - z - 34*z^2 - 31*z^3 + O(z^4),
 4*z^-2 + z^-1 + z + 4*z^2 + 9*z^3 + 16*z^4 + O(z^5)]

sage: L = LazyLaurentSeriesRing(GF(2), 'z')
sage: L.some_elements()[:7]
[0, 1, z,
 z^-4 + z^-3 + z^2 + z^3,
 z^-2,
 1 + z + z^3 + z^4 + z^6 + O(z^7),
 z^-1 + z + z^3 + O(z^5)]

sage: L = LazyLaurentSeriesRing(GF(3), 'z')
sage: L.some_elements()[:7]
[0, 1, z,
 z^-3 + z^-1 + 2 + z + z^2 + z^3,
 z^-2,
 z^-3 + z^-2 + z^-1 + 2 + 2*z + 2*z^2 + O(z^3),
 z^-2 + z^-1 + z + z^2 + z^4 + O(z^5)]

uniformizer()#

Return a uniformizer of self..

EXAMPLES:

sage: L = LazyLaurentSeriesRing(QQ, 'z')
sage: L.uniformizer()
z

class sage.rings.lazy_series_ring.LazyPowerSeriesRing(base_ring, names, sparse=True, category=None)#

Bases: LazySeriesRing

The ring of (possibly multivariate) lazy Taylor series.

INPUT:

base_ring – base ring of this Taylor series ring
names – name(s) of the generator of this Taylor series ring
sparse – (default: True) whether this series is sparse or not

EXAMPLES:

sage: LazyPowerSeriesRing(ZZ, 't')
Lazy Taylor Series Ring in t over Integer Ring

sage: L.<x, y> = LazyPowerSeriesRing(QQ); L
Multivariate Lazy Taylor Series Ring in x, y over Rational Field

Element#: alias of LazyPowerSeries

fraction_field()#

Return the fraction field of self.

If this is with a single variable over a field, then the fraction field is the field of (lazy) formal Laurent series.

Todo

Implement other fraction fields.

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: L.fraction_field()
Lazy Laurent Series Ring in x over Rational Field

gen(n=0)#

Return the n-th generator of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.gen()
z
sage: L.gen(3)
Traceback (most recent call last):
...
IndexError: there is only one generator

gens()#

Return the generators of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(ZZ, 'x,y')
sage: L.gens()
(x, y)

ngens()#

Return the number of generators of self.

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: L.ngens()
1

residue_field()#

Return the residue field of the ring of integers of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ, 'x')
sage: L.residue_field()
Rational Field

some_elements()#

Return a list of elements of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.some_elements()[:6]
[0, 1, z + z^2 + z^3 + O(z^4),
 -12 - 8*z + z^2 + z^3,
 1 + z - 2*z^2 - 7*z^3 - z^4 + 20*z^5 + 23*z^6 + O(z^7),
 z + 4*z^2 + 9*z^3 + 16*z^4 + 25*z^5 + 36*z^6 + O(z^7)]

sage: L = LazyPowerSeriesRing(GF(3)["q"], 'z')
sage: L.some_elements()[:6]
[0, 1, z + q*z^2 + q*z^3 + q*z^4 + O(z^5),
 z + z^2 + z^3,
 1 + z + z^2 + 2*z^3 + 2*z^4 + 2*z^5 + O(z^6),
 z + z^2 + z^4 + z^5 + O(z^7)]

sage: L = LazyPowerSeriesRing(GF(3), 'q, t')
sage: L.some_elements()[:6]
[0, 1, q,
 q + q^2 + q^3,
 1 + q + q^2 + (-q^3) + (-q^4) + (-q^5) + (-q^6) + O(q,t)^7,
 1 + (q+t) + (q^2-q*t+t^2) + (q^3+t^3)
   + (q^4+q^3*t+q*t^3+t^4)
   + (q^5-q^4*t+q^3*t^2+q^2*t^3-q*t^4+t^5)
   + (q^6-q^3*t^3+t^6) + O(q,t)^7]

uniformizer()#

Return a uniformizer of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ, 'x')
sage: L.uniformizer()
x

class sage.rings.lazy_series_ring.LazySeriesRing#

Bases: UniqueRepresentation, Parent

Abstract base class for lazy series.

characteristic()#

Return the characteristic of this lazy power series ring, which is the same as the characteristic of its base ring.

EXAMPLES:

sage: L.<t> = LazyLaurentSeriesRing(ZZ)
sage: L.characteristic()
0

sage: R.<w> = LazyLaurentSeriesRing(GF(11)); R
Lazy Laurent Series Ring in w over Finite Field of size 11
sage: R.characteristic()
11

sage: R.<x, y> = LazyPowerSeriesRing(GF(7)); R
Multivariate Lazy Taylor Series Ring in x, y over Finite Field of size 7
sage: R.characteristic()
7

sage: L = LazyDirichletSeriesRing(ZZ, "s")
sage: L.characteristic()
0

is_exact()#

Return if self is exact or not.

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.is_exact()
True
sage: L = LazyLaurentSeriesRing(RR, 'z')
sage: L.is_exact()
False

is_sparse()#

Return whether self is sparse or not.

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z', sparse=False)
sage: L.is_sparse()
False

sage: L = LazyLaurentSeriesRing(ZZ, 'z', sparse=True)
sage: L.is_sparse()
True

one()#

Return the constant series \(1\).

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.one()
1

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.one()
1

sage: m = SymmetricFunctions(ZZ).m()
sage: L = LazySymmetricFunctions(m)
sage: L.one()
m[]

options = Current options for lazy series rings - constant_length: 3 - display_length: 7 - halting_precision: None#

undefined(valuation=None)#

Return an uninitialized series.

INPUT:

valuation – integer; a lower bound for the valuation of the series

Power series can be defined recursively (see sage.rings.lazy_series.LazyModuleElement.define() for more examples).