Title Power Series Rings English 518.8ย KB 142
```                            Power Series Rings
Power Series
Power Series Methods
Power series implemented using PARI
Multivariate Power Series Rings
Multivariate Power Series
Laurent Series Rings
Laurent Series
Lazy Laurent Series
Lazy Laurent Series Rings
Lazy Laurent Series Operators
Puiseux Series Ring
Puiseux Series Ring Element
Tate algebras
Indices and Tables
Python Module Index
Index
```
##### Document Text Contents
Page 1

Sage 9.1 Reference Manual: Power
Series Rings and Laurent Series Rings

Release 9.1

The Sage Development Team

May 21, 2020

Page 71

Sage 9.1 Reference Manual: Power Series Rings and Laurent Series Rings, Release 9.1

sage: R.<a,b,c> = PowerSeriesRing(ZZ)
sage: f = 1 + a + b - a*b + R.O(3)
sage: f.trailing_monomial()
1
sage: f = a^2*b^3*f; f
a^2*b^3 + a^3*b^3 + a^2*b^4 - a^3*b^4 + O(a, b, c)^8
sage: f.trailing_monomial()
a^2*b^3

truncate(prec=+Infinity)
Return infinite precision multivariate power series formed by truncating self at precision prec.

EXAMPLES:

sage: M = PowerSeriesRing(QQ,4,'t'); M
Multivariate Power Series Ring in t0, t1, t2, t3 over Rational Field
sage: t = M.gens()
sage: f = 1/2*t[0]^3*t[1]^3*t[2]^2 + 2/3*t[0]*t[2]^6*t[3] - t[0]^
โห3*t[1]^3*t[3]^3 - 1/4*t[0]*t[1]*t[2]^7 + M.O(10)
sage: f
1/2*t0^3*t1^3*t2^2 + 2/3*t0*t2^6*t3 - t0^3*t1^3*t3^3
- 1/4*t0*t1*t2^7 + O(t0, t1, t2, t3)^10

sage: f.truncate()
1/2*t0^3*t1^3*t2^2 + 2/3*t0*t2^6*t3 - t0^3*t1^3*t3^3
- 1/4*t0*t1*t2^7
sage: f.truncate().parent()
Multivariate Power Series Ring in t0, t1, t2, t3 over Rational Field

Contrast with polynomial:

sage: f.polynomial()
1/2*t0^3*t1^3*t2^2 + 2/3*t0*t2^6*t3 - t0^3*t1^3*t3^3 - 1/4*t0*t1*t2^7
sage: f.polynomial().parent()
Multivariate Polynomial Ring in t0, t1, t2, t3 over Rational Field

valuation()
Return the valuation of self.

The valuation of a power series ๐ is the highest nonnegative integer ๐ less or equal to the precision of ๐
and such that the coefficient of ๐ before each term of degree < ๐ is zero. (If such an integer does not exist,
then the valuation is the precision of ๐ itself.)

EXAMPLES:

sage: R.<a,b> = PowerSeriesRing(GF(4949717)); R
Multivariate Power Series Ring in a, b over Finite Field of
size 4949717
sage: f = a^2 + a*b + a^3 + R.O(9)
sage: f.valuation()
2
sage: g = 1 + a + a^3
sage: g.valuation()
0
sage: R.zero().valuation()
+Infinity

valuation_zero_part()
Doesnโt make sense for multivariate power series; valuation zero with respect to which variable?

67

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Sage 9.1 Reference Manual: Power Series Rings and Laurent Series Rings, Release 9.1

variable()
Doesnโt make sense for multivariate power series.

variables()
Return tuple of variables occurring in self.

EXAMPLES:

sage: T = PowerSeriesRing(GF(3),5,'t'); T
Multivariate Power Series Ring in t0, t1, t2, t3, t4 over
Finite Field of size 3
sage: t = T.gens()
sage: w = t[0] - 2*t[0]*t[2] + 5*t[4]^3 - t[0]^3*t[2]^2 + T.O(6)
sage: w
t0 + t0*t2 - t4^3 - t0^3*t2^2 + O(t0, t1, t2, t3, t4)^6
sage: w.variables()
(t0, t2, t4)

sage.rings.multi_power_series_ring_element.is_MPowerSeries(f)
Return True if f is a multivariate power series.

68 Chapter 6. Multivariate Power Series

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Sage 9.1 Reference Manual: Power Series Rings and Laurent Series Rings, Release 9.1

sage.rings.power_series_ring (module), 1
sage.rings.power_series_ring_element (module), 11
sage.rings.puiseux_series_ring (module), 103
sage.rings.puiseux_series_ring_element (module), 107
sage.rings.tate_algebra (module), 115
series() (sage.rings.lazy_laurent_series_ring.LazyLaurentSeriesRing method), 96
shift() (sage.rings.laurent_series_ring_element.LaurentSeries method), 85
shift() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 66
shift() (sage.rings.power_series_ring_element.PowerSeries method), 24
shift() (sage.rings.puiseux_series_ring_element.PuiseuxSeries method), 113
sin() (sage.rings.power_series_ring_element.PowerSeries method), 25
solve_linear_de() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 66
solve_linear_de() (sage.rings.power_series_ring_element.PowerSeries method), 25
some_elements() (sage.rings.tate_algebra.TateAlgebra_generic method), 122
some_elements() (sage.rings.tate_algebra.TateTermMonoid method), 125
sqrt() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 66
sqrt() (sage.rings.power_series_ring_element.PowerSeries method), 26
square_root() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 66
square_root() (sage.rings.power_series_ring_element.PowerSeries method), 28

T
tan() (sage.rings.power_series_ring_element.PowerSeries method), 28
TateAlgebra_generic (class in sage.rings.tate_algebra), 118
TateAlgebraFactory (class in sage.rings.tate_algebra), 117
TateTermMonoid (class in sage.rings.tate_algebra), 123
term_order() (sage.rings.multi_power_series_ring.MPowerSeriesRing_generic method), 51
term_order() (sage.rings.tate_algebra.TateAlgebra_generic method), 122
term_order() (sage.rings.tate_algebra.TateTermMonoid method), 125
trailing_monomial() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 66
truncate() (sage.rings.laurent_series_ring_element.LaurentSeries method), 85
truncate() (sage.rings.lazy_laurent_series.LazyLaurentSeries method), 93
truncate() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 67
truncate() (sage.rings.power_series_poly.PowerSeries_poly method), 37
truncate() (sage.rings.power_series_ring_element.PowerSeries method), 29
truncate() (sage.rings.puiseux_series_ring_element.PuiseuxSeries method), 113
truncate_laurentseries() (sage.rings.laurent_series_ring_element.LaurentSeries method), 86
truncate_neg() (sage.rings.laurent_series_ring_element.LaurentSeries method), 86
truncate_powerseries() (sage.rings.power_series_poly.PowerSeries_poly method), 37

U
uniformizer() (sage.rings.laurent_series_ring.LaurentSeriesRing method), 73
uniformizer() (sage.rings.power_series_ring.PowerSeriesRing_generic method), 8
uniformizer() (sage.rings.puiseux_series_ring.PuiseuxSeriesRing method), 105
unpickle_multi_power_series_ring_v0() (in module sage.rings.multi_power_series_ring), 52
unpickle_power_series_ring_v0() (in module sage.rings.power_series_ring), 9

V
V() (sage.rings.laurent_series_ring_element.LaurentSeries method), 76
V() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 57
V() (sage.rings.power_series_ring_element.PowerSeries method), 12

Index 137

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Sage 9.1 Reference Manual: Power Series Rings and Laurent Series Rings, Release 9.1

valuation() (sage.rings.laurent_series_ring_element.LaurentSeries method), 86
valuation() (sage.rings.lazy_laurent_series.LazyLaurentSeries method), 93
valuation() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 67
valuation() (sage.rings.power_series_pari.PowerSeries_pari method), 43
valuation() (sage.rings.power_series_poly.PowerSeries_poly method), 38
valuation() (sage.rings.power_series_ring_element.PowerSeries method), 29
valuation() (sage.rings.puiseux_series_ring_element.PuiseuxSeries method), 113
valuation_zero_part() (sage.rings.laurent_series_ring_element.LaurentSeries method), 86
valuation_zero_part() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 67
valuation_zero_part() (sage.rings.power_series_ring_element.PowerSeries method), 30
variable() (sage.rings.laurent_series_ring_element.LaurentSeries method), 87
variable() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 68
variable() (sage.rings.power_series_ring_element.PowerSeries method), 30
variable() (sage.rings.puiseux_series_ring_element.PuiseuxSeries method), 114
variable_names() (sage.rings.tate_algebra.TateAlgebra_generic method), 123
variable_names() (sage.rings.tate_algebra.TateTermMonoid method), 125
variable_names_recursive() (sage.rings.power_series_ring.PowerSeriesRing_generic method), 9
variables() (sage.rings.multi_power_series_ring_element.MPowerSeries method), 68
verschiebung() (sage.rings.laurent_series_ring_element.LaurentSeries method), 87

Z
zero() (sage.rings.lazy_laurent_series_ring.LazyLaurentSeriesRing method), 97

138 Index