Power series implemented using PARI¶

EXAMPLES:

This implementation can be selected for any base ring supported by PARI by passing the keyword implementation='pari' to the PowerSeriesRing() constructor:

Sage

sage: R.<q> = PowerSeriesRing(ZZ, implementation='pari'); R
Power Series Ring in q over Integer Ring
sage: S.<t> = PowerSeriesRing(CC, implementation='pari'); S                         # needs sage.rings.real_mpfr
Power Series Ring in t over Complex Field with 53 bits of precision

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, implementation='pari', names=('q',)); (q,) = R._first_ngens(1); R
Power Series Ring in q over Integer Ring
>>> S = PowerSeriesRing(CC, implementation='pari', names=('t',)); (t,) = S._first_ngens(1); S                         # needs sage.rings.real_mpfr
Power Series Ring in t over Complex Field with 53 bits of precision

Sage Live

R.<q> = PowerSeriesRing(ZZ, implementation='pari'); R
S.<t> = PowerSeriesRing(CC, implementation='pari'); S                         # needs sage.rings.real_mpfr

Note that only the type of the elements depends on the implementation, not the type of the parents:

Sage

sage: type(R)
<class 'sage.rings.power_series_ring.PowerSeriesRing_domain_with_category'>
sage: type(q)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
sage: type(S)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_ring.PowerSeriesRing_over_field_with_category'>
sage: type(t)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_pari.PowerSeries_pari'>

Python

>>> from sage.all import *
>>> type(R)
<class 'sage.rings.power_series_ring.PowerSeriesRing_domain_with_category'>
>>> type(q)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
>>> type(S)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_ring.PowerSeriesRing_over_field_with_category'>
>>> type(t)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_pari.PowerSeries_pari'>

Sage Live

type(R)
type(q)
type(S)                                                                       # needs sage.rings.real_mpfr
type(t)                                                                       # needs sage.rings.real_mpfr

If \(k\) is a finite field implemented using PARI, this is the default implementation for power series over \(k\):

Sage

sage: k.<c> = GF(5^12)
sage: type(c)
<class 'sage.rings.finite_rings.element_pari_ffelt.FiniteFieldElement_pari_ffelt'>
sage: A.<x> = k[[]]
sage: type(x)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>

Python

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(12), names=('c',)); (c,) = k._first_ngens(1)
>>> type(c)
<class 'sage.rings.finite_rings.element_pari_ffelt.FiniteFieldElement_pari_ffelt'>
>>> A = k[['x']]; (x,) = A._first_ngens(1)
>>> type(x)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>

Sage Live

k.<c> = GF(5^12)
type(c)
A.<x> = k[[]]
type(x)

Warning

Because this implementation uses the PARI interface, the PARI variable ordering must be respected in the sense that the variable name of the power series ring must have higher priority than any variable names occurring in the base ring:

Sage

sage: R.<y> = QQ[]
sage: S.<x> = PowerSeriesRing(R, implementation='pari'); S
Power Series Ring in x over Univariate Polynomial Ring in y over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['y']; (y,) = R._first_ngens(1)
>>> S = PowerSeriesRing(R, implementation='pari', names=('x',)); (x,) = S._first_ngens(1); S
Power Series Ring in x over Univariate Polynomial Ring in y over Rational Field

Sage Live

R.<y> = QQ[]
S.<x> = PowerSeriesRing(R, implementation='pari'); S

Reversing the variable ordering leads to errors:

Sage

sage: R.<x> = QQ[]
sage: S.<y> = PowerSeriesRing(R, implementation='pari')
Traceback (most recent call last):
...
PariError: incorrect priority in gtopoly: variable x <= y

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> S = PowerSeriesRing(R, implementation='pari', names=('y',)); (y,) = S._first_ngens(1)
Traceback (most recent call last):
...
PariError: incorrect priority in gtopoly: variable x <= y

Sage Live

R.<x> = QQ[]
S.<y> = PowerSeriesRing(R, implementation='pari')

AUTHORS:

Peter Bruin (December 2013): initial version

class sage.rings.power_series_pari.PowerSeries_pari[source]¶

Bases: PowerSeries

A power series implemented using PARI.

INPUT:

parent – the power series ring to use as the parent
f – object from which to construct a power series
prec – (default: infinity) precision of the element to be constructed
check – ignored, but accepted for compatibility with PowerSeries_poly

dict()[source]¶

Return a dictionary of coefficients for self.

This is simply a dict for the underlying polynomial; it need not have keys corresponding to every number smaller than self.prec().

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(ZZ, implementation='pari')
sage: f = 1 + t^10 + O(t^12)
sage: f.monomial_coefficients()
{0: 1, 10: 1}

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(1) + t**Integer(10) + O(t**Integer(12))
>>> f.monomial_coefficients()
{0: 1, 10: 1}

Sage Live

R.<t> = PowerSeriesRing(ZZ, implementation='pari')
f = 1 + t^10 + O(t^12)
f.monomial_coefficients()

dict is an alias:

Sage

sage: f.dict()
{0: 1, 10: 1}

Python

>>> from sage.all import *
>>> f.dict()
{0: 1, 10: 1}

Sage Live

f.dict()

integral(var=None)[source]¶

Return the formal integral of self.

By default, the integration variable is the variable of the power series. Otherwise, the integration variable is the optional parameter var.

Note

The integral is always chosen so the constant term is 0.

EXAMPLES:

Sage

sage: k.<w> = PowerSeriesRing(QQ, implementation='pari')
sage: (1+17*w+15*w^3+O(w^5)).integral()
w + 17/2*w^2 + 15/4*w^4 + O(w^6)
sage: (w^3 + 4*w^4 + O(w^7)).integral()
1/4*w^4 + 4/5*w^5 + O(w^8)
sage: (3*w^2).integral()
w^3

Python

>>> from sage.all import *
>>> k = PowerSeriesRing(QQ, implementation='pari', names=('w',)); (w,) = k._first_ngens(1)
>>> (Integer(1)+Integer(17)*w+Integer(15)*w**Integer(3)+O(w**Integer(5))).integral()
w + 17/2*w^2 + 15/4*w^4 + O(w^6)
>>> (w**Integer(3) + Integer(4)*w**Integer(4) + O(w**Integer(7))).integral()
1/4*w^4 + 4/5*w^5 + O(w^8)
>>> (Integer(3)*w**Integer(2)).integral()
w^3

Sage Live

k.<w> = PowerSeriesRing(QQ, implementation='pari')
(1+17*w+15*w^3+O(w^5)).integral()
(w^3 + 4*w^4 + O(w^7)).integral()
(3*w^2).integral()

list()[source]¶

Return the list of known coefficients for self.

This is just the list of coefficients of the underlying polynomial; it need not have length equal to self.prec().

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(ZZ, implementation='pari')
sage: f = 1 - 5*t^3 + t^5 + O(t^7)
sage: f.list()
[1, 0, 0, -5, 0, 1]

sage: # needs sage.rings.padics
sage: S.<u> = PowerSeriesRing(pAdicRing(5), implementation='pari')
sage: (2 + u).list()
[2 + O(5^20), 1 + O(5^20)]

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(1) - Integer(5)*t**Integer(3) + t**Integer(5) + O(t**Integer(7))
>>> f.list()
[1, 0, 0, -5, 0, 1]

>>> # needs sage.rings.padics
>>> S = PowerSeriesRing(pAdicRing(Integer(5)), implementation='pari', names=('u',)); (u,) = S._first_ngens(1)
>>> (Integer(2) + u).list()
[2 + O(5^20), 1 + O(5^20)]

Sage Live

R.<t> = PowerSeriesRing(ZZ, implementation='pari')
f = 1 - 5*t^3 + t^5 + O(t^7)
f.list()
# needs sage.rings.padics
S.<u> = PowerSeriesRing(pAdicRing(5), implementation='pari')
(2 + u).list()

monomial_coefficients()[source]¶

Return a dictionary of coefficients for self.

This is simply a dict for the underlying polynomial; it need not have keys corresponding to every number smaller than self.prec().

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(ZZ, implementation='pari')
sage: f = 1 + t^10 + O(t^12)
sage: f.monomial_coefficients()
{0: 1, 10: 1}

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(1) + t**Integer(10) + O(t**Integer(12))
>>> f.monomial_coefficients()
{0: 1, 10: 1}

Sage Live

R.<t> = PowerSeriesRing(ZZ, implementation='pari')
f = 1 + t^10 + O(t^12)
f.monomial_coefficients()

dict is an alias:

Sage

sage: f.dict()
{0: 1, 10: 1}

Python

>>> from sage.all import *
>>> f.dict()
{0: 1, 10: 1}

Sage Live

f.dict()

padded_list(n=None)[source]¶

Return a list of coefficients of self up to (but not including) \(q^n\).

The list is padded with zeroes on the right so that it has length \(n\).

INPUT:

n – nonnegative integer (optional); if \(n\) is not given, it will be taken to be the precision of self, unless this is +Infinity, in which case we just return self.list()

EXAMPLES:

Sage

sage: R.<q> = PowerSeriesRing(QQ, implementation='pari')
sage: f = 1 - 17*q + 13*q^2 + 10*q^4 + O(q^7)
sage: f.list()
[1, -17, 13, 0, 10]
sage: f.padded_list(7)
[1, -17, 13, 0, 10, 0, 0]
sage: f.padded_list(10)
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]
sage: f.padded_list(3)
[1, -17, 13]
sage: f.padded_list()
[1, -17, 13, 0, 10, 0, 0]
sage: g = 1 - 17*q + 13*q^2 + 10*q^4
sage: g.list()
[1, -17, 13, 0, 10]
sage: g.padded_list()
[1, -17, 13, 0, 10]
sage: g.padded_list(10)
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, implementation='pari', names=('q',)); (q,) = R._first_ngens(1)
>>> f = Integer(1) - Integer(17)*q + Integer(13)*q**Integer(2) + Integer(10)*q**Integer(4) + O(q**Integer(7))
>>> f.list()
[1, -17, 13, 0, 10]
>>> f.padded_list(Integer(7))
[1, -17, 13, 0, 10, 0, 0]
>>> f.padded_list(Integer(10))
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]
>>> f.padded_list(Integer(3))
[1, -17, 13]
>>> f.padded_list()
[1, -17, 13, 0, 10, 0, 0]
>>> g = Integer(1) - Integer(17)*q + Integer(13)*q**Integer(2) + Integer(10)*q**Integer(4)
>>> g.list()
[1, -17, 13, 0, 10]
>>> g.padded_list()
[1, -17, 13, 0, 10]
>>> g.padded_list(Integer(10))
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]

Sage Live

R.<q> = PowerSeriesRing(QQ, implementation='pari')
f = 1 - 17*q + 13*q^2 + 10*q^4 + O(q^7)
f.list()
f.padded_list(7)
f.padded_list(10)
f.padded_list(3)
f.padded_list()
g = 1 - 17*q + 13*q^2 + 10*q^4
g.list()
g.padded_list()
g.padded_list(10)

polynomial()[source]¶

Convert self to a polynomial.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(GF(7), implementation='pari')
sage: f = 3 - t^3 + O(t^5)
sage: f.polynomial()
6*t^3 + 3

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(GF(Integer(7)), implementation='pari', names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(3) - t**Integer(3) + O(t**Integer(5))
>>> f.polynomial()
6*t^3 + 3

Sage Live

R.<t> = PowerSeriesRing(GF(7), implementation='pari')
f = 3 - t^3 + O(t^5)
f.polynomial()

reverse(precision=None)[source]¶

Return the reverse of self.

The reverse of a power series \(f\) is the power series \(g\) such that \(g(f(x)) = x\). This exists if and only if the valuation of self is exactly 1 and the coefficient of \(x\) is a unit.

If the optional argument precision is given, the reverse is returned with this precision. If f has infinite precision and the argument precision is not given, then the reverse is returned with the default precision of f.parent().

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(QQ, implementation='pari')
sage: f = 2*x + 3*x^2 - x^4 + O(x^5)
sage: g = f.reverse()
sage: g
1/2*x - 3/8*x^2 + 9/16*x^3 - 131/128*x^4 + O(x^5)
sage: f(g)
x + O(x^5)
sage: g(f)
x + O(x^5)

sage: A.<t> = PowerSeriesRing(ZZ, implementation='pari')
sage: a = t - t^2 - 2*t^4 + t^5 + O(t^6)
sage: b = a.reverse(); b
t + t^2 + 2*t^3 + 7*t^4 + 25*t^5 + O(t^6)
sage: a(b)
t + O(t^6)
sage: b(a)
t + O(t^6)

sage: B.<b,c> = PolynomialRing(ZZ)
sage: A.<t> = PowerSeriesRing(B, implementation='pari')
sage: f = t + b*t^2 + c*t^3 + O(t^4)
sage: g = f.reverse(); g
t - b*t^2 + (2*b^2 - c)*t^3 + O(t^4)
sage: f(g)
t + O(t^4)
sage: g(f)
t + O(t^4)

sage: A.<t> = PowerSeriesRing(ZZ, implementation='pari')
sage: B.<x> = PowerSeriesRing(A, implementation='pari')
sage: f = (1 - 3*t + 4*t^3 + O(t^4))*x + (2 + t + t^2 + O(t^3))*x^2 + O(x^3)
sage: g = f.reverse(); g
(1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*x + (-2 - 19*t - 118*t^2 + O(t^3))*x^2 + O(x^3)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, implementation='pari', names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(2)*x + Integer(3)*x**Integer(2) - x**Integer(4) + O(x**Integer(5))
>>> g = f.reverse()
>>> g
1/2*x - 3/8*x^2 + 9/16*x^3 - 131/128*x^4 + O(x^5)
>>> f(g)
x + O(x^5)
>>> g(f)
x + O(x^5)

>>> A = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = A._first_ngens(1)
>>> a = t - t**Integer(2) - Integer(2)*t**Integer(4) + t**Integer(5) + O(t**Integer(6))
>>> b = a.reverse(); b
t + t^2 + 2*t^3 + 7*t^4 + 25*t^5 + O(t^6)
>>> a(b)
t + O(t^6)
>>> b(a)
t + O(t^6)

>>> B = PolynomialRing(ZZ, names=('b', 'c',)); (b, c,) = B._first_ngens(2)
>>> A = PowerSeriesRing(B, implementation='pari', names=('t',)); (t,) = A._first_ngens(1)
>>> f = t + b*t**Integer(2) + c*t**Integer(3) + O(t**Integer(4))
>>> g = f.reverse(); g
t - b*t^2 + (2*b^2 - c)*t^3 + O(t^4)
>>> f(g)
t + O(t^4)
>>> g(f)
t + O(t^4)

>>> A = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = A._first_ngens(1)
>>> B = PowerSeriesRing(A, implementation='pari', names=('x',)); (x,) = B._first_ngens(1)
>>> f = (Integer(1) - Integer(3)*t + Integer(4)*t**Integer(3) + O(t**Integer(4)))*x + (Integer(2) + t + t**Integer(2) + O(t**Integer(3)))*x**Integer(2) + O(x**Integer(3))
>>> g = f.reverse(); g
(1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*x + (-2 - 19*t - 118*t^2 + O(t^3))*x^2 + O(x^3)

Sage Live

R.<x> = PowerSeriesRing(QQ, implementation='pari')
f = 2*x + 3*x^2 - x^4 + O(x^5)
g = f.reverse()
g
f(g)
g(f)
A.<t> = PowerSeriesRing(ZZ, implementation='pari')
a = t - t^2 - 2*t^4 + t^5 + O(t^6)
b = a.reverse(); b
a(b)
b(a)
B.<b,c> = PolynomialRing(ZZ)
A.<t> = PowerSeriesRing(B, implementation='pari')
f = t + b*t^2 + c*t^3 + O(t^4)
g = f.reverse(); g
f(g)
g(f)
A.<t> = PowerSeriesRing(ZZ, implementation='pari')
B.<x> = PowerSeriesRing(A, implementation='pari')
f = (1 - 3*t + 4*t^3 + O(t^4))*x + (2 + t + t^2 + O(t^3))*x^2 + O(x^3)
g = f.reverse(); g

The optional argument precision sets the precision of the output:

Sage

sage: R.<x> = PowerSeriesRing(QQ, implementation='pari')
sage: f = 2*x + 3*x^2 - 7*x^3 + x^4 + O(x^5)
sage: g = f.reverse(precision=3); g
1/2*x - 3/8*x^2 + O(x^3)
sage: f(g)
x + O(x^3)
sage: g(f)
x + O(x^3)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, implementation='pari', names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(2)*x + Integer(3)*x**Integer(2) - Integer(7)*x**Integer(3) + x**Integer(4) + O(x**Integer(5))
>>> g = f.reverse(precision=Integer(3)); g
1/2*x - 3/8*x^2 + O(x^3)
>>> f(g)
x + O(x^3)
>>> g(f)
x + O(x^3)

Sage Live

R.<x> = PowerSeriesRing(QQ, implementation='pari')
f = 2*x + 3*x^2 - 7*x^3 + x^4 + O(x^5)
g = f.reverse(precision=3); g
f(g)
g(f)

If the input series has infinite precision, the precision of the output is automatically set to the default precision of the parent ring:

Sage

sage: R.<x> = PowerSeriesRing(QQ, default_prec=20, implementation='pari')
sage: (x - x^2).reverse()  # get some Catalan numbers
x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8
 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 + 208012*x^13
 + 742900*x^14 + 2674440*x^15 + 9694845*x^16 + 35357670*x^17
 + 129644790*x^18 + 477638700*x^19 + O(x^20)
sage: (x - x^2).reverse(precision=3)
x + x^2 + O(x^3)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, default_prec=Integer(20), implementation='pari', names=('x',)); (x,) = R._first_ngens(1)
>>> (x - x**Integer(2)).reverse()  # get some Catalan numbers
x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8
 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 + 208012*x^13
 + 742900*x^14 + 2674440*x^15 + 9694845*x^16 + 35357670*x^17
 + 129644790*x^18 + 477638700*x^19 + O(x^20)
>>> (x - x**Integer(2)).reverse(precision=Integer(3))
x + x^2 + O(x^3)

Sage Live

R.<x> = PowerSeriesRing(QQ, default_prec=20, implementation='pari')
(x - x^2).reverse()  # get some Catalan numbers
(x - x^2).reverse(precision=3)

valuation()[source]¶

Return the valuation of self.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ, implementation='pari')
sage: (5 - t^8 + O(t^11)).valuation()
0
sage: (-t^8 + O(t^11)).valuation()
8
sage: O(t^7).valuation()
7
sage: R(0).valuation()
+Infinity

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1)
>>> (Integer(5) - t**Integer(8) + O(t**Integer(11))).valuation()
0
>>> (-t**Integer(8) + O(t**Integer(11))).valuation()
8
>>> O(t**Integer(7)).valuation()
7
>>> R(Integer(0)).valuation()
+Infinity

Sage Live

R.<t> = PowerSeriesRing(QQ, implementation='pari')
(5 - t^8 + O(t^11)).valuation()
(-t^8 + O(t^11)).valuation()
O(t^7).valuation()
R(0).valuation()