Power Series Methods¶

The class PowerSeries_poly provides additional methods for univariate power series.

class sage.rings.power_series_poly.BaseRingFloorDivAction[source]¶

Bases: Action

The floor division action of the base ring on a formal power series.

class sage.rings.power_series_poly.PowerSeries_poly[source]¶

Bases: PowerSeries

EXAMPLES:

Sage

sage: R.<q> = PowerSeriesRing(CC); R                                        # needs sage.rings.real_mpfr
Power Series Ring in q over Complex Field with 53 bits of precision
sage: loads(q.dumps()) == q                                                 # needs sage.rings.real_mpfr
True

sage: R.<t> = QQ[[]]
sage: f = 3 - t^3 + O(t^5)
sage: a = f^3; a
27 - 27*t^3 + O(t^5)
sage: b = f^-3; b
1/27 + 1/27*t^3 + O(t^5)
sage: a*b
1 + O(t^5)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(CC, names=('q',)); (q,) = R._first_ngens(1); R                                        # needs sage.rings.real_mpfr
Power Series Ring in q over Complex Field with 53 bits of precision
>>> loads(q.dumps()) == q                                                 # needs sage.rings.real_mpfr
True

>>> R = QQ[['t']]; (t,) = R._first_ngens(1)
>>> f = Integer(3) - t**Integer(3) + O(t**Integer(5))
>>> a = f**Integer(3); a
27 - 27*t^3 + O(t^5)
>>> b = f**-Integer(3); b
1/27 + 1/27*t^3 + O(t^5)
>>> a*b
1 + O(t^5)

Sage Live

R.<q> = PowerSeriesRing(CC); R                                        # needs sage.rings.real_mpfr
loads(q.dumps()) == q                                                 # needs sage.rings.real_mpfr
R.<t> = QQ[[]]
f = 3 - t^3 + O(t^5)
a = f^3; a
b = f^-3; b
a*b

Check that Issue #22216 is fixed:

Sage

sage: R.<T> = PowerSeriesRing(QQ)
sage: R(pari('1 + O(T)'))                                                   # needs sage.libs.pari
1 + O(T)
sage: R(pari('1/T + O(T)'))                                                 # needs sage.libs.pari
Traceback (most recent call last):
...
ValueError: series has negative valuation

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('T',)); (T,) = R._first_ngens(1)
>>> R(pari('1 + O(T)'))                                                   # needs sage.libs.pari
1 + O(T)
>>> R(pari('1/T + O(T)'))                                                 # needs sage.libs.pari
Traceback (most recent call last):
...
ValueError: series has negative valuation

Sage Live

R.<T> = PowerSeriesRing(QQ)
R(pari('1 + O(T)'))                                                   # needs sage.libs.pari
R(pari('1/T + O(T)'))                                                 # needs sage.libs.pari

degree()[source]¶

Return the degree of the underlying polynomial of self.

That is, if self is of the form \(f(x) + O(x^n)\), we return the degree of \(f(x)\). Note that if \(f(x)\) is \(0\), we return \(-1\), just as with polynomials.

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: (5 + t^3 + O(t^4)).degree()
3
sage: (5 + O(t^4)).degree()
0
sage: O(t^4).degree()
-1

Python

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> (Integer(5) + t**Integer(3) + O(t**Integer(4))).degree()
3
>>> (Integer(5) + O(t**Integer(4))).degree()
0
>>> O(t**Integer(4)).degree()
-1

Sage Live

R.<t> = ZZ[[]]
(5 + t^3 + O(t^4)).degree()
(5 + O(t^4)).degree()
O(t^4).degree()

dict()[source]¶

Return a dictionary of coefficients for self.

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

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: f = 1 + t^10 + O(t^12)
sage: f.monomial_coefficients()
{0: 1, 10: 1}

Python

>>> from sage.all import *
>>> R = ZZ[['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> = ZZ[[]]
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 integral of this power series.

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> = QQ[[]]
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 = QQ[['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> = QQ[[]]
(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, so in particular, need not have length equal to self.prec().

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: f = 1 - 5*t^3 + t^5 + O(t^7)
sage: f.list()
[1, 0, 0, -5, 0, 1]

Python

>>> from sage.all import *
>>> R = ZZ[['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]

Sage Live

R.<t> = ZZ[[]]
f = 1 - 5*t^3 + t^5 + O(t^7)
f.list()

monomial_coefficients()[source]¶

Return a dictionary of coefficients for self.

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

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: f = 1 + t^10 + O(t^12)
sage: f.monomial_coefficients()
{0: 1, 10: 1}

Python

>>> from sage.all import *
>>> R = ZZ[['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> = ZZ[[]]
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()

pade(m, n)[source]¶

Return the Padé approximant of self of index \((m, n)\).

The Padé approximant of index \((m, n)\) of a formal power series \(f\) is the quotient \(Q/P\) of two polynomials \(Q\) and \(P\) such that \(\deg(Q)\leq m\), \(\deg(P)\leq n\) and

\[f(z) - Q(z)/P(z) = O(z^{m+n+1}).\]

The formal power series \(f\) must be known up to order \(n + m\).

See Wikipedia article Padé_approximant

INPUT:

m, n – integers, describing the degrees of the polynomials

OUTPUT: a ratio of two polynomials

ALGORITHM:

This method uses the formula as a quotient of two determinants.

See also

EXAMPLES:

Sage

sage: z = PowerSeriesRing(QQ, 'z').gen()
sage: exp(z).pade(4, 0)
1/24*z^4 + 1/6*z^3 + 1/2*z^2 + z + 1
sage: exp(z).pade(1, 1)
(-z - 2)/(z - 2)
sage: exp(z).pade(3, 3)
(-z^3 - 12*z^2 - 60*z - 120)/(z^3 - 12*z^2 + 60*z - 120)
sage: log(1-z).pade(4, 4)
(25/6*z^4 - 130/3*z^3 + 105*z^2 - 70*z)/(z^4 - 20*z^3 + 90*z^2
- 140*z + 70)
sage: sqrt(1+z).pade(3, 2)
(1/6*z^3 + 3*z^2 + 8*z + 16/3)/(z^2 + 16/3*z + 16/3)
sage: exp(2*z).pade(3, 3)
(-z^3 - 6*z^2 - 15*z - 15)/(z^3 - 6*z^2 + 15*z - 15)

Python

>>> from sage.all import *
>>> z = PowerSeriesRing(QQ, 'z').gen()
>>> exp(z).pade(Integer(4), Integer(0))
1/24*z^4 + 1/6*z^3 + 1/2*z^2 + z + 1
>>> exp(z).pade(Integer(1), Integer(1))
(-z - 2)/(z - 2)
>>> exp(z).pade(Integer(3), Integer(3))
(-z^3 - 12*z^2 - 60*z - 120)/(z^3 - 12*z^2 + 60*z - 120)
>>> log(Integer(1)-z).pade(Integer(4), Integer(4))
(25/6*z^4 - 130/3*z^3 + 105*z^2 - 70*z)/(z^4 - 20*z^3 + 90*z^2
- 140*z + 70)
>>> sqrt(Integer(1)+z).pade(Integer(3), Integer(2))
(1/6*z^3 + 3*z^2 + 8*z + 16/3)/(z^2 + 16/3*z + 16/3)
>>> exp(Integer(2)*z).pade(Integer(3), Integer(3))
(-z^3 - 6*z^2 - 15*z - 15)/(z^3 - 6*z^2 + 15*z - 15)

Sage Live

z = PowerSeriesRing(QQ, 'z').gen()
exp(z).pade(4, 0)
exp(z).pade(1, 1)
exp(z).pade(3, 3)
log(1-z).pade(4, 4)
sqrt(1+z).pade(3, 2)
exp(2*z).pade(3, 3)

polynomial()[source]¶

Return the underlying polynomial of self.

EXAMPLES:

Sage

sage: R.<t> = GF(7)[[]]
sage: f = 3 - t^3 + O(t^5)
sage: f.polynomial()
6*t^3 + 3

Python

>>> from sage.all import *
>>> R = GF(Integer(7))[['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> = GF(7)[[]]
f = 3 - t^3 + O(t^5)
f.polynomial()

reverse(precision=None)[source]¶

Return the reverse of \(f\), i.e., the series \(g\) such that \(g(f(x)) = x\).

Given an optional argument precision, return the reverse with given precision (note that the reverse can have precision at most f.prec()). If \(f\) has infinite precision, and the argument precision is not given, then the precision of the reverse defaults to the default precision of f.parent().

Note that this is only possible if the valuation of self is exactly 1.

ALGORITHM:

We first attempt to pass the computation to pari; if this fails, we use Lagrange inversion. Using sage: set_verbose(1) will print a message if passing to pari fails.

If the base ring has positive characteristic, then we attempt to lift to a characteristic zero ring and perform the reverse there. If this fails, an error is raised.

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(QQ)
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)
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)
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)
sage: B.<s> = A[[]]
sage: f = (1 - 3*t + 4*t^3 + O(t^4))*s + (2 + t + t^2 + O(t^3))*s^2 + O(s^3)
sage: from sage.misc.verbose import set_verbose
sage: set_verbose(1)
sage: g = f.reverse(); g
verbose 1 (<module>) passing to pari failed; trying Lagrange inversion
(1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*s + (-2 - 19*t - 118*t^2 + O(t^3))*s^2 + O(s^3)
sage: set_verbose(0)
sage: f(g) == g(f) == s
True

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, 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, 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, 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, names=('t',)); (t,) = A._first_ngens(1)
>>> B = A[['s']]; (s,) = B._first_ngens(1)
>>> f = (Integer(1) - Integer(3)*t + Integer(4)*t**Integer(3) + O(t**Integer(4)))*s + (Integer(2) + t + t**Integer(2) + O(t**Integer(3)))*s**Integer(2) + O(s**Integer(3))
>>> from sage.misc.verbose import set_verbose
>>> set_verbose(Integer(1))
>>> g = f.reverse(); g
verbose 1 (<module>) passing to pari failed; trying Lagrange inversion
(1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*s + (-2 - 19*t - 118*t^2 + O(t^3))*s^2 + O(s^3)
>>> set_verbose(Integer(0))
>>> f(g) == g(f) == s
True

Sage Live

R.<x> = PowerSeriesRing(QQ)
f = 2*x + 3*x^2 - x^4 + O(x^5)
g = f.reverse()
g
f(g)
g(f)
A.<t> = PowerSeriesRing(ZZ)
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)
f = t + b*t^2 + c*t^3 + O(t^4)
g = f.reverse(); g
f(g)
g(f)
A.<t> = PowerSeriesRing(ZZ)
B.<s> = A[[]]
f = (1 - 3*t + 4*t^3 + O(t^4))*s + (2 + t + t^2 + O(t^3))*s^2 + O(s^3)
from sage.misc.verbose import set_verbose
set_verbose(1)
g = f.reverse(); g
set_verbose(0)
f(g) == g(f) == s

If the leading coefficient is not a unit, we pass to its fraction field if possible:

Sage

sage: A.<t> = PowerSeriesRing(ZZ)
sage: a = 2*t - 4*t^2 + t^4 - t^5 + O(t^6)
sage: a.reverse()
1/2*t + 1/2*t^2 + t^3 + 79/32*t^4 + 437/64*t^5 + O(t^6)

sage: B.<b> = PolynomialRing(ZZ)
sage: A.<t> = PowerSeriesRing(B)
sage: f = 2*b*t + b*t^2 + 3*b^2*t^3 + O(t^4)
sage: g = f.reverse(); g
1/(2*b)*t - 1/(8*b^2)*t^2 + ((-3*b + 1)/(16*b^3))*t^3 + O(t^4)
sage: f(g)
t + O(t^4)
sage: g(f)
t + O(t^4)

Python

>>> from sage.all import *
>>> A = PowerSeriesRing(ZZ, names=('t',)); (t,) = A._first_ngens(1)
>>> a = Integer(2)*t - Integer(4)*t**Integer(2) + t**Integer(4) - t**Integer(5) + O(t**Integer(6))
>>> a.reverse()
1/2*t + 1/2*t^2 + t^3 + 79/32*t^4 + 437/64*t^5 + O(t^6)

>>> B = PolynomialRing(ZZ, names=('b',)); (b,) = B._first_ngens(1)
>>> A = PowerSeriesRing(B, names=('t',)); (t,) = A._first_ngens(1)
>>> f = Integer(2)*b*t + b*t**Integer(2) + Integer(3)*b**Integer(2)*t**Integer(3) + O(t**Integer(4))
>>> g = f.reverse(); g
1/(2*b)*t - 1/(8*b^2)*t^2 + ((-3*b + 1)/(16*b^3))*t^3 + O(t^4)
>>> f(g)
t + O(t^4)
>>> g(f)
t + O(t^4)

Sage Live

A.<t> = PowerSeriesRing(ZZ)
a = 2*t - 4*t^2 + t^4 - t^5 + O(t^6)
a.reverse()
B.<b> = PolynomialRing(ZZ)
A.<t> = PowerSeriesRing(B)
f = 2*b*t + b*t^2 + 3*b^2*t^3 + O(t^4)
g = f.reverse(); g
f(g)
g(f)

We can handle some base rings of positive characteristic:

Sage

sage: A8.<t> = PowerSeriesRing(Zmod(8))
sage: a = t - 15*t^2 - 2*t^4 + t^5 + O(t^6)
sage: b = a.reverse(); b
t + 7*t^2 + 2*t^3 + 5*t^4 + t^5 + O(t^6)
sage: a(b)
t + O(t^6)
sage: b(a)
t + O(t^6)

Python

>>> from sage.all import *
>>> A8 = PowerSeriesRing(Zmod(Integer(8)), names=('t',)); (t,) = A8._first_ngens(1)
>>> a = t - Integer(15)*t**Integer(2) - Integer(2)*t**Integer(4) + t**Integer(5) + O(t**Integer(6))
>>> b = a.reverse(); b
t + 7*t^2 + 2*t^3 + 5*t^4 + t^5 + O(t^6)
>>> a(b)
t + O(t^6)
>>> b(a)
t + O(t^6)

Sage Live

A8.<t> = PowerSeriesRing(Zmod(8))
a = t - 15*t^2 - 2*t^4 + t^5 + O(t^6)
b = a.reverse(); b
a(b)
b(a)

The optional argument precision sets the precision of the output:

Sage

sage: R.<x> = PowerSeriesRing(QQ)
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, 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)
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)
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), 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)
(x - x^2).reverse() # get some Catalan numbers
(x - x^2).reverse(precision=3)

truncate(prec='infinity')[source]¶

The polynomial obtained from power series by truncation at precision prec.

EXAMPLES:

Sage

sage: R.<I> = GF(2)[[]]
sage: f = 1/(1+I+O(I^8)); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
sage: f.truncate(5)
I^4 + I^3 + I^2 + I + 1

Python

>>> from sage.all import *
>>> R = GF(Integer(2))[['I']]; (I,) = R._first_ngens(1)
>>> f = Integer(1)/(Integer(1)+I+O(I**Integer(8))); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
>>> f.truncate(Integer(5))
I^4 + I^3 + I^2 + I + 1

Sage Live

R.<I> = GF(2)[[]]
f = 1/(1+I+O(I^8)); f
f.truncate(5)

truncate_powerseries(prec)[source]¶

Given input prec = \(n\), returns the power series of degree \(< n\) which is equivalent to self modulo \(x^n\).

EXAMPLES:

Sage

sage: R.<I> = GF(2)[[]]
sage: f = 1/(1+I+O(I^8)); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
sage: f.truncate_powerseries(5)
1 + I + I^2 + I^3 + I^4 + O(I^5)

Python

>>> from sage.all import *
>>> R = GF(Integer(2))[['I']]; (I,) = R._first_ngens(1)
>>> f = Integer(1)/(Integer(1)+I+O(I**Integer(8))); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
>>> f.truncate_powerseries(Integer(5))
1 + I + I^2 + I^3 + I^4 + O(I^5)

Sage Live

R.<I> = GF(2)[[]]
f = 1/(1+I+O(I^8)); f
f.truncate_powerseries(5)

valuation()[source]¶

Return the valuation of self.

EXAMPLES:

Sage

sage: R.<t> = QQ[[]]
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 = QQ[['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> = QQ[[]]
(5 - t^8 + O(t^11)).valuation()
(-t^8 + O(t^11)).valuation()
O(t^7).valuation()
R(0).valuation()

sage.rings.power_series_poly.make_powerseries_poly_v0(parent, f, prec, is_gen)[source]¶

Return the power series specified by f, prec, and is_gen.

This function exists for the purposes of pickling. Do not delete this function – if you change the internal representation, instead make a new function and make sure that both kinds of objects correctly unpickle as the new type.

EXAMPLES:

Sage

sage: R.<t> = QQ[[]]
sage: sage.rings.power_series_poly.make_powerseries_poly_v0(R, t, infinity, True)
t

Python

>>> from sage.all import *
>>> R = QQ[['t']]; (t,) = R._first_ngens(1)
>>> sage.rings.power_series_poly.make_powerseries_poly_v0(R, t, infinity, True)
t

Sage Live

R.<t> = QQ[[]]
sage.rings.power_series_poly.make_powerseries_poly_v0(R, t, infinity, True)