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

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

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)
<class 'sage.rings.power_series_ring.PowerSeriesRing_over_field_with_category'>
sage: type(t)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>

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

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'>

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: 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

Reversing the variable ordering leads to errors:

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

AUTHORS:

  • Peter Bruin (December 2013): initial version

class sage.rings.power_series_pari.PowerSeries_pari

Bases: sage.rings.power_series_ring_element.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()

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: R.<t> = PowerSeriesRing(ZZ, implementation='pari')
sage: f = 1 + t^10 + O(t^12)
sage: f.dict()
{0: 1, 10: 1}
integral(var=None)

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: 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
list()

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: 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: S.<u> = PowerSeriesRing(pAdicRing(5), implementation='pari')
sage: (2 + u).list()
[2 + O(5^20), 1 + O(5^20)]
padded_list(n=None)

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 – a non-negative 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: 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]
polynomial()

Convert self to a polynomial.

EXAMPLES:

sage: R.<t> = PowerSeriesRing(GF(7), implementation='pari')
sage: f = 3 - t^3 + O(t^5)
sage: f.polynomial()
6*t^3 + 3
reverse(precision=None)

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: 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)

The optional argument precision sets the precision of the output:

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)

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

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)
valuation()

Return the valuation of self.

EXAMPLES:

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