Power Series Rings¶
Power series rings are constructed in the standard Sage fashion. See also Multivariate Power Series Rings.
EXAMPLES:
Construct rings and elements:
sage: R.<t> = PowerSeriesRing(QQ)
sage: R.random_element(6) # random
-4 - 1/2*t^2 - 1/95*t^3 + 1/2*t^4 - 12*t^5 + O(t^6)
sage: R.<t,u,v> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in t, u, v over Rational Field
sage: p = -t + 1/2*t^3*u - 1/4*t^4*u + 2/3*v^5 + R.O(6); p
-t + 1/2*t^3*u - 1/4*t^4*u + 2/3*v^5 + O(t, u, v)^6
sage: p in R
True
The default precision is specified at construction, but does not bound the precision of created elements.
sage: R.<t> = PowerSeriesRing(QQ, default_prec=5)
sage: R.random_element(6) # random
1/2 - 1/4*t + 2/3*t^2 - 5/2*t^3 + 2/3*t^5 + O(t^6)
Construct univariate power series from a list of coefficients:
sage: S = R([1, 3, 5, 7]); S
1 + 3*t + 5*t^2 + 7*t^3
An iterated example:
sage: R.<t> = PowerSeriesRing(ZZ)
sage: S.<t2> = PowerSeriesRing(R)
sage: S
Power Series Ring in t2 over Power Series Ring in t over Integer Ring
sage: S.base_ring()
Power Series Ring in t over Integer Ring
Sage can compute with power series over the symbolic ring.
sage: K.<t> = PowerSeriesRing(SR, default_prec=5)
sage: a, b, c = var('a,b,c')
sage: f = a + b*t + c*t^2 + O(t^3)
sage: f*f
a^2 + 2*a*b*t + (b^2 + 2*a*c)*t^2 + O(t^3)
sage: f = sqrt(2) + sqrt(3)*t + O(t^3)
sage: f^2
2 + 2*sqrt(3)*sqrt(2)*t + 3*t^2 + O(t^3)
Elements are first coerced to constants in base_ring, then coerced
into the PowerSeriesRing:
sage: R.<t> = PowerSeriesRing(ZZ)
sage: f = Mod(2, 3) * t; (f, f.parent())
(2*t, Power Series Ring in t over Ring of integers modulo 3)
We make a sparse power series.
sage: R.<x> = PowerSeriesRing(QQ, sparse=True); R
Sparse Power Series Ring in x over Rational Field
sage: f = 1 + x^1000000
sage: g = f*f
sage: g.degree()
2000000
We make a sparse Laurent series from a power series generator:
sage: R.<t> = PowerSeriesRing(QQ, sparse=True)
sage: latex(-2/3*(1/t^3) + 1/t + 3/5*t^2 + O(t^5))
\frac{-\frac{2}{3}}{t^{3}} + \frac{1}{t} + \frac{3}{5}t^{2} + O(t^{5})
sage: S = parent(1/t); S
Sparse Laurent Series Ring in t over Rational Field
AUTHORS:
- William Stein: the code
- Jeremy Cho (2006-05-17): some examples (above)
- Niles Johnson (2010-09): implement multivariate power series
- Simon King (2012-08): use category and coercion framework, trac ticket #13412
-
sage.rings.power_series_ring.PowerSeriesRing(base_ring, name=None, arg2=None, names=None, sparse=False, default_prec=None, order='negdeglex', num_gens=None, implementation=None)¶ Create a univariate or multivariate power series ring over a given (commutative) base ring.
INPUT:
base_ring- a commutative ringname,names- name(s) of the indeterminatedefault_prec- the default precision used if an exact object must- be changed to an approximate object in order to do an arithmetic
operation. If left as
None, it will be set to the global default (20) in the univariate case, and 12 in the multivariate case.
sparse- (default:False) whether power series are represented as sparse objects.order- (default:negdeglex) term ordering, for multivariate casenum_gens- number of generators, for multivariate case
There is a unique power series ring over each base ring with given variable name. Two power series over the same base ring with different variable names are not equal or isomorphic.
EXAMPLES (Univariate):
sage: R = PowerSeriesRing(QQ, 'x'); R Power Series Ring in x over Rational Field
sage: S = PowerSeriesRing(QQ, 'y'); S Power Series Ring in y over Rational Field
sage: R = PowerSeriesRing(QQ, 10) Traceback (most recent call last): ... ValueError: variable name '10' does not start with a letter
sage: S = PowerSeriesRing(QQ, 'x', default_prec = 15); S Power Series Ring in x over Rational Field sage: S.default_prec() 15
EXAMPLES (Multivariate) See also Multivariate Power Series Rings:
sage: R = PowerSeriesRing(QQ, 't,u,v'); R Multivariate Power Series Ring in t, u, v over Rational Field
sage: N = PowerSeriesRing(QQ,'w',num_gens=5); N Multivariate Power Series Ring in w0, w1, w2, w3, w4 over Rational Field
Number of generators can be specified before variable name without using keyword:
sage: M = PowerSeriesRing(QQ,4,'k'); M Multivariate Power Series Ring in k0, k1, k2, k3 over Rational Field
Multivariate power series can be constructed using angle bracket or double square bracket notation:
sage: R.<t,u,v> = PowerSeriesRing(QQ, 't,u,v'); R Multivariate Power Series Ring in t, u, v over Rational Field sage: ZZ[['s,t,u']] Multivariate Power Series Ring in s, t, u over Integer Ring
Sparse multivariate power series ring:
sage: M = PowerSeriesRing(QQ,4,'k',sparse=True); M Sparse Multivariate Power Series Ring in k0, k1, k2, k3 over Rational Field
Power series ring over polynomial ring:
sage: H = PowerSeriesRing(PolynomialRing(ZZ,3,'z'),4,'f'); H Multivariate Power Series Ring in f0, f1, f2, f3 over Multivariate Polynomial Ring in z0, z1, z2 over Integer Ring
Power series ring over finite field:
sage: S = PowerSeriesRing(GF(65537),'x,y'); S Multivariate Power Series Ring in x, y over Finite Field of size 65537
Power series ring with many variables:
sage: R = PowerSeriesRing(ZZ, ['x%s'%p for p in primes(100)]); R Multivariate Power Series Ring in x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97 over Integer Ring
Use
inject_variables()to make the variables available for interactive use.sage: R.inject_variables() Defining x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97 sage: f = x47 + 3*x11*x29 - x19 + R.O(3) sage: f in R True
Variable ordering determines how series are displayed:
sage: T.<a,b> = PowerSeriesRing(ZZ,order='deglex'); T Multivariate Power Series Ring in a, b over Integer Ring sage: T.term_order() Degree lexicographic term order sage: p = - 2*b^6 + a^5*b^2 + a^7 - b^2 - a*b^3 + T.O(9); p a^7 + a^5*b^2 - 2*b^6 - a*b^3 - b^2 + O(a, b)^9 sage: U = PowerSeriesRing(ZZ,'a,b',order='negdeglex'); U Multivariate Power Series Ring in a, b over Integer Ring sage: U.term_order() Negative degree lexicographic term order sage: U(p) -b^2 - a*b^3 - 2*b^6 + a^7 + a^5*b^2 + O(a, b)^9
-
class
sage.rings.power_series_ring.PowerSeriesRing_domain(base_ring, name=None, default_prec=None, sparse=False, implementation=None, category=None)¶ Bases:
sage.rings.power_series_ring.PowerSeriesRing_generic,sage.rings.ring.IntegralDomain
-
class
sage.rings.power_series_ring.PowerSeriesRing_generic(base_ring, name=None, default_prec=None, sparse=False, implementation=None, category=None)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.rings.ring.CommutativeRing,sage.structure.nonexact.NonexactA power series ring.
-
base_extend(R)¶ Return the power series ring over R in the same variable as self, assuming there is a canonical coerce map from the base ring of self to R.
EXAMPLES:
sage: R.<T> = GF(7)[[]]; R Power Series Ring in T over Finite Field of size 7 sage: R.change_ring(ZZ) Power Series Ring in T over Integer Ring sage: R.base_extend(ZZ) Traceback (most recent call last): ... TypeError: no base extension defined
-
change_ring(R)¶ Return the power series ring over R in the same variable as self.
EXAMPLES:
sage: R.<T> = QQ[[]]; R Power Series Ring in T over Rational Field sage: R.change_ring(GF(7)) Power Series Ring in T over Finite Field of size 7 sage: R.base_extend(GF(7)) Traceback (most recent call last): ... TypeError: no base extension defined sage: R.base_extend(QuadraticField(3,'a')) Power Series Ring in T over Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
-
change_var(var)¶ Return the power series ring in variable
varover the same base ring.EXAMPLES:
sage: R.<T> = QQ[[]]; R Power Series Ring in T over Rational Field sage: R.change_var('D') Power Series Ring in D over Rational Field
-
characteristic()¶ Return the characteristic of this power series ring, which is the same as the characteristic of the base ring of the power series ring.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: R.characteristic() 0 sage: R.<w> = Integers(2^50)[[]]; R Power Series Ring in w over Ring of integers modulo 1125899906842624 sage: R.characteristic() 1125899906842624
-
construction()¶ Return the functorial construction of self, namely, completion of the univariate polynomial ring with respect to the indeterminate (to a given precision).
EXAMPLES:
sage: R = PowerSeriesRing(ZZ, 'x') sage: c, S = R.construction(); S Univariate Polynomial Ring in x over Integer Ring sage: R == c(S) True
-
gen(n=0)¶ Return the generator of this power series ring.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: R.gen() t sage: R.gen(3) Traceback (most recent call last): ... IndexError: generator n>0 not defined
-
is_dense()¶ EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: t.is_dense() True sage: R.<t> = PowerSeriesRing(ZZ, sparse=True) sage: t.is_dense() False
-
is_exact()¶ Return False since the ring of power series over any ring is not exact.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: R.is_exact() False
-
is_field(proof=True)¶ Return False since the ring of power series over any ring is never a field.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: R.is_field() False
-
is_finite()¶ Return False since the ring of power series over any ring is never finite.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: R.is_finite() False
-
is_sparse()¶ EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ) sage: t.is_sparse() False sage: R.<t> = PowerSeriesRing(ZZ, sparse=True) sage: t.is_sparse() True
-
laurent_series_ring()¶ If this is the power series ring \(R[[t]]\), return the Laurent series ring \(R((t))\).
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ,default_prec=5) sage: S = R.laurent_series_ring(); S Laurent Series Ring in t over Integer Ring sage: S.default_prec() 5 sage: f = 1+t; g=1/f; g 1 - t + t^2 - t^3 + t^4 + O(t^5)
-
ngens()¶ Return the number of generators of this power series ring.
This is always 1.
EXAMPLES:
sage: R.<t> = ZZ[[]] sage: R.ngens() 1
-
random_element(prec=None, *args, **kwds)¶ Return a random power series.
INPUT:
prec- Integer specifying precision of output (default: default precision of self)*args, **kwds- Passed on to therandom_elementmethod for the base ring
OUTPUT:
- Power series with precision
precwhose coefficients are random elements from the base ring, randomized subject to the arguments*argsand**kwds
ALGORITHM:
Call the
random_elementmethod on the underlying polynomial ring.EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ) sage: R.random_element(5) # random -4 - 1/2*t^2 - 1/95*t^3 + 1/2*t^4 + O(t^5) sage: R.random_element(10) # random -1/2 + 2*t - 2/7*t^2 - 25*t^3 - t^4 + 2*t^5 - 4*t^7 - 1/3*t^8 - t^9 + O(t^10)
If given no argument,
random_elementuses default precision of self:sage: T = PowerSeriesRing(ZZ,'t') sage: T.default_prec() 20 sage: T.random_element() # random 4 + 2*t - t^2 - t^3 + 2*t^4 + t^5 + t^6 - 2*t^7 - t^8 - t^9 + t^11 - 6*t^12 + 2*t^14 + 2*t^16 - t^17 - 3*t^18 + O(t^20) sage: S = PowerSeriesRing(ZZ,'t', default_prec=4) sage: S.random_element() # random 2 - t - 5*t^2 + t^3 + O(t^4)
Further arguments are passed to the underlying base ring (trac ticket #9481):
sage: SZ = PowerSeriesRing(ZZ,'v') sage: SQ = PowerSeriesRing(QQ,'v') sage: SR = PowerSeriesRing(RR,'v') sage: SZ.random_element(x=4, y=6) # random 4 + 5*v + 5*v^2 + 5*v^3 + 4*v^4 + 5*v^5 + 5*v^6 + 5*v^7 + 4*v^8 + 5*v^9 + 4*v^10 + 4*v^11 + 5*v^12 + 5*v^13 + 5*v^14 + 5*v^15 + 5*v^16 + 5*v^17 + 4*v^18 + 5*v^19 + O(v^20) sage: SZ.random_element(3, x=4, y=6) # random 5 + 4*v + 5*v^2 + O(v^3) sage: SQ.random_element(3, num_bound=3, den_bound=100) # random 1/87 - 3/70*v - 3/44*v^2 + O(v^3) sage: SR.random_element(3, max=10, min=-10) # random 2.85948321262904 - 9.73071330911226*v - 6.60414378519265*v^2 + O(v^3)
-
residue_field()¶ Return the residue field of this power series ring.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(GF(17)) sage: R.residue_field() Finite Field of size 17 sage: R.<x> = PowerSeriesRing(Zp(5)) sage: R.residue_field() Finite Field of size 5
-
uniformizer()¶ Return a uniformizer of this power series ring if it is a discrete valuation ring (i.e., if the base ring is actually a field). Otherwise, an error is raised.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ) sage: R.uniformizer() t sage: R.<t> = PowerSeriesRing(ZZ) sage: R.uniformizer() Traceback (most recent call last): ... TypeError: The base ring is not a field
-
variable_names_recursive(depth=None)¶ Return the list of variable names of this and its base rings.
EXAMPLES:
sage: R = QQ[['x']][['y']][['z']] sage: R.variable_names_recursive() ('x', 'y', 'z') sage: R.variable_names_recursive(2) ('y', 'z')
-
-
class
sage.rings.power_series_ring.PowerSeriesRing_over_field(base_ring, name=None, default_prec=None, sparse=False, implementation=None, category=None)¶ Bases:
sage.rings.power_series_ring.PowerSeriesRing_domain-
fraction_field()¶ Return the fraction field of this power series ring, which is defined since this is over a field.
This fraction field is just the Laurent series ring over the base field.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(GF(7)) sage: R.fraction_field() Laurent Series Ring in t over Finite Field of size 7 sage: Frac(R) Laurent Series Ring in t over Finite Field of size 7
-
-
sage.rings.power_series_ring.is_PowerSeriesRing(R)¶ Return True if this is a univariate power series ring. This is in keeping with the behavior of
is_PolynomialRingversusis_MPolynomialRing.EXAMPLES:
sage: from sage.rings.power_series_ring import is_PowerSeriesRing sage: is_PowerSeriesRing(10) False sage: is_PowerSeriesRing(QQ[['x']]) True
-
sage.rings.power_series_ring.unpickle_power_series_ring_v0(base_ring, name, default_prec, sparse)¶ Unpickle (deserialize) a univariate power series ring according to the given inputs.
EXAMPLES:
sage: P.<x> = PowerSeriesRing(QQ) sage: loads(dumps(P)) == P # indirect doctest True