Multivariate Power Series Rings¶

Construct a multivariate power series ring (in finitely many variables) over a given (commutative) base ring.

EXAMPLES:

Construct rings and elements:

Sage

sage: R.<t,u,v> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in t, u, v over Rational Field
sage: TestSuite(R).run()
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

sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2; g
1 + v + 3*t^2*u - 2*t^2*v^2
sage: g in R
True

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t', 'u', 'v',)); (t, u, v,) = R._first_ngens(3); R
Multivariate Power Series Ring in t, u, v over Rational Field
>>> TestSuite(R).run()
>>> p = -t + Integer(1)/Integer(2)*t**Integer(3)*u - Integer(1)/Integer(4)*t**Integer(4)*u + Integer(2)/Integer(3)*v**Integer(5) + R.O(Integer(6)); p
-t + 1/2*t^3*u - 1/4*t^4*u + 2/3*v^5 + O(t, u, v)^6
>>> p in R
True

>>> g = Integer(1) + v + Integer(3)*u*t**Integer(2) - Integer(2)*v**Integer(2)*t**Integer(2); g
1 + v + 3*t^2*u - 2*t^2*v^2
>>> g in R
True

Sage Live

R.<t,u,v> = PowerSeriesRing(QQ); R
TestSuite(R).run()
p = -t + 1/2*t^3*u - 1/4*t^4*u + 2/3*v^5 + R.O(6); p
p in R
g = 1 + v + 3*u*t^2 - 2*v^2*t^2; g
g in R

Add big O as with single variable power series:

Sage

sage: g.add_bigoh(3)
1 + v + O(t, u, v)^3
sage: g = g.O(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5

Python

>>> from sage.all import *
>>> g.add_bigoh(Integer(3))
1 + v + O(t, u, v)^3
>>> g = g.O(Integer(5)); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5

Sage Live

g.add_bigoh(3)
g = g.O(5); g

Sage keeps track of total-degree precision:

Sage

sage: f = (g-1)^2 - g + 1; f
-v + v^2 - 3*t^2*u + 6*t^2*u*v + 2*t^2*v^2 + O(t, u, v)^5
sage: f in R
True
sage: f.prec()
5
sage: ((g-1-v)^2).prec()
8

Python

>>> from sage.all import *
>>> f = (g-Integer(1))**Integer(2) - g + Integer(1); f
-v + v^2 - 3*t^2*u + 6*t^2*u*v + 2*t^2*v^2 + O(t, u, v)^5
>>> f in R
True
>>> f.prec()
5
>>> ((g-Integer(1)-v)**Integer(2)).prec()
8

Sage Live

f = (g-1)^2 - g + 1; f
f in R
f.prec()
((g-1-v)^2).prec()

Construct multivariate power series rings over various base rings.

Sage

sage: M = PowerSeriesRing(QQ, 4, 'k'); M
Multivariate Power Series Ring in k0, k1, k2, k3 over Rational Field
sage: loads(dumps(M)) is M
True
sage: TestSuite(M).run()

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
sage: TestSuite(H).run()
sage: loads(dumps(H)) is H
True

sage: z = H.base_ring().gens()
sage: f = H.gens()
sage: h = 4*z[1]^2 + 2*z[0]*z[2] + z[1]*z[2] + z[2]^2 \
+ (-z[2]^2 - 2*z[0] + z[2])*f[0]*f[2] \
+ (-22*z[0]^2 + 2*z[1]^2 - z[0]*z[2] + z[2]^2 - 1955*z[2])*f[1]*f[2] \
+ (-z[0]*z[1] - 2*z[1]^2)*f[2]*f[3] \
+ (2*z[0]*z[1] + z[1]*z[2] - z[2]^2 - z[1] + 3*z[2])*f[3]^2 \
+ H.O(3)
sage: h in H
True
sage: h
4*z1^2 + 2*z0*z2 + z1*z2 + z2^2 + (-z2^2 - 2*z0 + z2)*f0*f2
+ (-22*z0^2 + 2*z1^2 - z0*z2 + z2^2 - 1955*z2)*f1*f2
+ (-z0*z1 - 2*z1^2)*f2*f3 + (2*z0*z1 + z1*z2 - z2^2 - z1 + 3*z2)*f3^2
+ O(f0, f1, f2, f3)^3

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(QQ, Integer(4), 'k'); M
Multivariate Power Series Ring in k0, k1, k2, k3 over Rational Field
>>> loads(dumps(M)) is M
True
>>> TestSuite(M).run()

>>> H = PowerSeriesRing(PolynomialRing(ZZ, Integer(3), 'z'), Integer(4), 'f'); H
Multivariate Power Series Ring in f0, f1, f2, f3
 over Multivariate Polynomial Ring in z0, z1, z2 over Integer Ring
>>> TestSuite(H).run()
>>> loads(dumps(H)) is H
True

>>> z = H.base_ring().gens()
>>> f = H.gens()
>>> h = Integer(4)*z[Integer(1)]**Integer(2) + Integer(2)*z[Integer(0)]*z[Integer(2)] + z[Integer(1)]*z[Integer(2)] + z[Integer(2)]**Integer(2) \
+ (-z[2]^2 - 2*z[0] + z[2])*f[0]*f[2] \
+ (-22*z[0]^2 + 2*z[1]^2 - z[0]*z[2] + z[2]^2 - 1955*z[2])*f[1]*f[2] \
+ (-z[0]*z[1] - 2*z[1]^2)*f[2]*f[3] \
+ (2*z[0]*z[1] + z[1]*z[2] - z[2]^2 - z[1] + 3*z[2])*f[3]^2 \
+ H.O(3)
>>> h in H
True
>>> h
4*z1^2 + 2*z0*z2 + z1*z2 + z2^2 + (-z2^2 - 2*z0 + z2)*f0*f2
+ (-22*z0^2 + 2*z1^2 - z0*z2 + z2^2 - 1955*z2)*f1*f2
+ (-z0*z1 - 2*z1^2)*f2*f3 + (2*z0*z1 + z1*z2 - z2^2 - z1 + 3*z2)*f3^2
+ O(f0, f1, f2, f3)^3

Sage Live

M = PowerSeriesRing(QQ, 4, 'k'); M
loads(dumps(M)) is M
TestSuite(M).run()
H = PowerSeriesRing(PolynomialRing(ZZ, 3, 'z'), 4, 'f'); H
TestSuite(H).run()
loads(dumps(H)) is H
z = H.base_ring().gens()
f = H.gens()
h = 4*z[1]^2 + 2*z[0]*z[2] + z[1]*z[2] + z[2]^2 \
h in H
h

Use angle-bracket notation:

Sage

sage: # needs sage.rings.finite_rings
sage: S.<x,y> = PowerSeriesRing(GF(65537)); S
Multivariate Power Series Ring in x, y over Finite Field of size 65537
sage: s = -30077*x + 9485*x*y - 6260*y^3 + 12870*x^2*y^2 - 20289*y^4 + S.O(5); s
-30077*x + 9485*x*y - 6260*y^3 + 12870*x^2*y^2 - 20289*y^4 + O(x, y)^5
sage: s in S
True
sage: TestSuite(S).run()
sage: loads(dumps(S)) is S
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> S = PowerSeriesRing(GF(Integer(65537)), names=('x', 'y',)); (x, y,) = S._first_ngens(2); S
Multivariate Power Series Ring in x, y over Finite Field of size 65537
>>> s = -Integer(30077)*x + Integer(9485)*x*y - Integer(6260)*y**Integer(3) + Integer(12870)*x**Integer(2)*y**Integer(2) - Integer(20289)*y**Integer(4) + S.O(Integer(5)); s
-30077*x + 9485*x*y - 6260*y^3 + 12870*x^2*y^2 - 20289*y^4 + O(x, y)^5
>>> s in S
True
>>> TestSuite(S).run()
>>> loads(dumps(S)) is S
True

Sage Live

# needs sage.rings.finite_rings
S.<x,y> = PowerSeriesRing(GF(65537)); S
s = -30077*x + 9485*x*y - 6260*y^3 + 12870*x^2*y^2 - 20289*y^4 + S.O(5); s
s in S
TestSuite(S).run()
loads(dumps(S)) is S

Use double square bracket notation:

Sage

sage: ZZ[['s,t,u']]
Multivariate Power Series Ring in s, t, u over Integer Ring
sage: GF(127931)[['x,y']]                                                           # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 127931

Python

>>> from sage.all import *
>>> ZZ[['s,t,u']]
Multivariate Power Series Ring in s, t, u over Integer Ring
>>> GF(Integer(127931))[['x,y']]                                                           # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 127931

Sage Live

ZZ[['s,t,u']]
GF(127931)[['x,y']]                                                           # needs sage.rings.finite_rings

Variable ordering determines how series are displayed.

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,order='deglex'); T
Multivariate Power Series Ring in a, b over Integer Ring
sage: TestSuite(T).run()
sage: loads(dumps(T)) is T
True
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

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,order='deglex', names=('a', 'b',)); (a, b,) = T._first_ngens(2); T
Multivariate Power Series Ring in a, b over Integer Ring
>>> TestSuite(T).run()
>>> loads(dumps(T)) is T
True
>>> T.term_order()
Degree lexicographic term order
>>> p = - Integer(2)*b**Integer(6) + a**Integer(5)*b**Integer(2) + a**Integer(7) - b**Integer(2) - a*b**Integer(3) + T.O(Integer(9)); p
a^7 + a^5*b^2 - 2*b^6 - a*b^3 - b^2 + O(a, b)^9

>>> U = PowerSeriesRing(ZZ,'a,b',order='negdeglex'); U
Multivariate Power Series Ring in a, b over Integer Ring
>>> U.term_order()
Negative degree lexicographic term order
>>> U(p)
-b^2 - a*b^3 - 2*b^6 + a^7 + a^5*b^2 + O(a, b)^9

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,order='deglex'); T
TestSuite(T).run()
loads(dumps(T)) is T
T.term_order()
p = - 2*b^6 + a^5*b^2 + a^7 - b^2 - a*b^3 + T.O(9); p
U = PowerSeriesRing(ZZ,'a,b',order='negdeglex'); U
U.term_order()
U(p)

Change from one base ring to another:

Sage

sage: R.<t,u,v> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in t, u, v over Rational Field
sage: R.base_extend(RR)                                                             # needs sage.rings.real_mpfr
Multivariate Power Series Ring in t, u, v
 over Real Field with 53 bits of precision
sage: R.change_ring(IntegerModRing(10))
Multivariate Power Series Ring in t, u, v
 over Ring of integers modulo 10

sage: S = PowerSeriesRing(GF(65537),2,'x,y'); S                                     # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 65537
sage: S.change_ring(GF(5))                                                          # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 5

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t', 'u', 'v',)); (t, u, v,) = R._first_ngens(3); R
Multivariate Power Series Ring in t, u, v over Rational Field
>>> R.base_extend(RR)                                                             # needs sage.rings.real_mpfr
Multivariate Power Series Ring in t, u, v
 over Real Field with 53 bits of precision
>>> R.change_ring(IntegerModRing(Integer(10)))
Multivariate Power Series Ring in t, u, v
 over Ring of integers modulo 10

>>> S = PowerSeriesRing(GF(Integer(65537)),Integer(2),'x,y'); S                                     # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 65537
>>> S.change_ring(GF(Integer(5)))                                                          # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 5

Sage Live

R.<t,u,v> = PowerSeriesRing(QQ); R
R.base_extend(RR)                                                             # needs sage.rings.real_mpfr
R.change_ring(IntegerModRing(10))
S = PowerSeriesRing(GF(65537),2,'x,y'); S                                     # needs sage.rings.finite_rings
S.change_ring(GF(5))                                                          # needs sage.rings.finite_rings

Coercion from polynomial ring:

Sage

sage: R.<t,u,v> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in t, u, v over Rational Field
sage: A = PolynomialRing(ZZ,3,'t,u,v')
sage: g = A.gens()
sage: a = 2*g[0]*g[2] - 2*g[0] - 2; a
2*t*v - 2*t - 2
sage: R(a)
-2 - 2*t + 2*t*v
sage: R(a).O(4)
-2 - 2*t + 2*t*v + O(t, u, v)^4
sage: a.parent()
Multivariate Polynomial Ring in t, u, v over Integer Ring
sage: a in R
True

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t', 'u', 'v',)); (t, u, v,) = R._first_ngens(3); R
Multivariate Power Series Ring in t, u, v over Rational Field
>>> A = PolynomialRing(ZZ,Integer(3),'t,u,v')
>>> g = A.gens()
>>> a = Integer(2)*g[Integer(0)]*g[Integer(2)] - Integer(2)*g[Integer(0)] - Integer(2); a
2*t*v - 2*t - 2
>>> R(a)
-2 - 2*t + 2*t*v
>>> R(a).O(Integer(4))
-2 - 2*t + 2*t*v + O(t, u, v)^4
>>> a.parent()
Multivariate Polynomial Ring in t, u, v over Integer Ring
>>> a in R
True

Sage Live

R.<t,u,v> = PowerSeriesRing(QQ); R
A = PolynomialRing(ZZ,3,'t,u,v')
g = A.gens()
a = 2*g[0]*g[2] - 2*g[0] - 2; a
R(a)
R(a).O(4)
a.parent()
a in R

Coercion from polynomial ring in subset of variables:

Sage

sage: R.<t,u,v> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in t, u, v over Rational Field
sage: A = PolynomialRing(QQ,2,'t,v')
sage: g = A.gens()
sage: a = -2*g[0]*g[1] - 1/27*g[1]^2 + g[0] - 1/2*g[1]; a
-2*t*v - 1/27*v^2 + t - 1/2*v
sage: a in R
True

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t', 'u', 'v',)); (t, u, v,) = R._first_ngens(3); R
Multivariate Power Series Ring in t, u, v over Rational Field
>>> A = PolynomialRing(QQ,Integer(2),'t,v')
>>> g = A.gens()
>>> a = -Integer(2)*g[Integer(0)]*g[Integer(1)] - Integer(1)/Integer(27)*g[Integer(1)]**Integer(2) + g[Integer(0)] - Integer(1)/Integer(2)*g[Integer(1)]; a
-2*t*v - 1/27*v^2 + t - 1/2*v
>>> a in R
True

Sage Live

R.<t,u,v> = PowerSeriesRing(QQ); R
A = PolynomialRing(QQ,2,'t,v')
g = A.gens()
a = -2*g[0]*g[1] - 1/27*g[1]^2 + g[0] - 1/2*g[1]; a
a in R

Coercion from symbolic ring:

Sage

sage: # needs sage.symbolic
sage: x,y = var('x,y')
sage: S = PowerSeriesRing(GF(11),2,'x,y'); S
Multivariate Power Series Ring in x, y over Finite Field of size 11
sage: type(x)
<class 'sage.symbolic.expression.Expression'>
sage: type(S(x))
<class 'sage.rings.multi_power_series_ring.MPowerSeriesRing_generic_with_category.element_class'>
sage: f = S(2/7 -100*x^2 + 1/3*x*y + y^2).O(3); f
5 - x^2 + 4*x*y + y^2 + O(x, y)^3
sage: f.parent()
Multivariate Power Series Ring in x, y over Finite Field of size 11
sage: f.parent() == S
True

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> x,y = var('x,y')
>>> S = PowerSeriesRing(GF(Integer(11)),Integer(2),'x,y'); S
Multivariate Power Series Ring in x, y over Finite Field of size 11
>>> type(x)
<class 'sage.symbolic.expression.Expression'>
>>> type(S(x))
<class 'sage.rings.multi_power_series_ring.MPowerSeriesRing_generic_with_category.element_class'>
>>> f = S(Integer(2)/Integer(7) -Integer(100)*x**Integer(2) + Integer(1)/Integer(3)*x*y + y**Integer(2)).O(Integer(3)); f
5 - x^2 + 4*x*y + y^2 + O(x, y)^3
>>> f.parent()
Multivariate Power Series Ring in x, y over Finite Field of size 11
>>> f.parent() == S
True

Sage Live

# needs sage.symbolic
x,y = var('x,y')
S = PowerSeriesRing(GF(11),2,'x,y'); S
type(x)
type(S(x))
f = S(2/7 -100*x^2 + 1/3*x*y + y^2).O(3); f
f.parent()
f.parent() == S

The implementation of the multivariate power series ring uses a combination of multivariate polynomials and univariate power series. Namely, in order to construct the multivariate power series ring \(R[[x_1, x_2, \cdots, x_n]]\), we consider the univariate power series ring \(S[[T]]\) over the multivariate polynomial ring \(S := R[x_1, x_2, \cdots, x_n]\), and in it we take the subring formed by all power series whose \(i\)-th coefficient has degree \(i\) for all \(i \geq 0\). This subring is isomorphic to \(R[[x_1, x_2, \cdots, x_n]]\). This is how \(R[[x_1, x_2, \cdots, x_n]]\) is implemented in this class. The ring \(S\) is called the foreground polynomial ring, and the ring \(S[[T]]\) is called the background univariate power series ring.

AUTHORS:

Niles Johnson (2010-07): initial code
Simon King (2012-08, 2013-02): Use category and coercion framework, Issue #13412 and Issue #14084

class sage.rings.multi_power_series_ring.MPowerSeriesRing_generic(base_ring, num_gens, name_list, order='negdeglex', default_prec=10, sparse=False)[source]¶

Bases: PowerSeriesRing_generic, Nonexact

A multivariate power series ring. This class is implemented as a single variable power series ring in the variable T over a multivariable polynomial ring in the specified generators. Each generator g of the multivariable polynomial ring (called the “foreground ring”) is mapped to g*T in the single variable power series ring (called the “background ring”). The background power series ring is used to do arithmetic and track total-degree precision. The foreground polynomial ring is used to display elements.

For usage and examples, see above, and PowerSeriesRing().

Element[source]¶: alias of MPowerSeries

O(prec)[source]¶

Return big oh with precision prec. This function is an alias for bigoh.

EXAMPLES:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2); T
Multivariate Power Series Ring in a, b over Integer Ring
sage: T.O(10)
0 + O(a, b)^10
sage: T.bigoh(10)
0 + O(a, b)^10

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2); T
Multivariate Power Series Ring in a, b over Integer Ring
>>> T.O(Integer(10))
0 + O(a, b)^10
>>> T.bigoh(Integer(10))
0 + O(a, b)^10

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2); T
T.O(10)
T.bigoh(10)

bigoh(prec)[source]¶

Return big oh with precision prec. The function O does the same thing.

EXAMPLES:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2); T
Multivariate Power Series Ring in a, b over Integer Ring
sage: T.bigoh(10)
0 + O(a, b)^10
sage: T.O(10)
0 + O(a, b)^10

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2); T
Multivariate Power Series Ring in a, b over Integer Ring
>>> T.bigoh(Integer(10))
0 + O(a, b)^10
>>> T.O(Integer(10))
0 + O(a, b)^10

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2); T
T.bigoh(10)
T.O(10)

change_ring(R)[source]¶

Return the power series ring over \(R\) in the same variable as self. This function ignores the question of whether the base ring of self is or can extend to the base ring of \(R\); for the latter, use base_extend.

EXAMPLES:

Sage

sage: R.<t,u,v> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in t, u, v over Rational Field
sage: R.base_extend(RR)                                                     # needs sage.rings.real_mpfr
Multivariate Power Series Ring in t, u, v over Real Field with
53 bits of precision
sage: R.change_ring(IntegerModRing(10))
Multivariate Power Series Ring in t, u, v over Ring of integers
modulo 10
sage: R.base_extend(IntegerModRing(10))
Traceback (most recent call last):
...
TypeError: no base extension defined


sage: S = PowerSeriesRing(GF(65537),2,'x,y'); S                             # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size
65537
sage: S.change_ring(GF(5))                                                  # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 5

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t', 'u', 'v',)); (t, u, v,) = R._first_ngens(3); R
Multivariate Power Series Ring in t, u, v over Rational Field
>>> R.base_extend(RR)                                                     # needs sage.rings.real_mpfr
Multivariate Power Series Ring in t, u, v over Real Field with
53 bits of precision
>>> R.change_ring(IntegerModRing(Integer(10)))
Multivariate Power Series Ring in t, u, v over Ring of integers
modulo 10
>>> R.base_extend(IntegerModRing(Integer(10)))
Traceback (most recent call last):
...
TypeError: no base extension defined


>>> S = PowerSeriesRing(GF(Integer(65537)),Integer(2),'x,y'); S                             # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size
65537
>>> S.change_ring(GF(Integer(5)))                                                  # needs sage.rings.finite_rings
Multivariate Power Series Ring in x, y over Finite Field of size 5

Sage Live

R.<t,u,v> = PowerSeriesRing(QQ); R
R.base_extend(RR)                                                     # needs sage.rings.real_mpfr
R.change_ring(IntegerModRing(10))
R.base_extend(IntegerModRing(10))
S = PowerSeriesRing(GF(65537),2,'x,y'); S                             # needs sage.rings.finite_rings
S.change_ring(GF(5))                                                  # needs sage.rings.finite_rings

characteristic()[source]¶

Return characteristic of base ring, which is characteristic of self.

EXAMPLES:

Sage

sage: H = PowerSeriesRing(GF(65537),4,'f'); H                               # needs sage.rings.finite_rings
Multivariate Power Series Ring in f0, f1, f2, f3 over
Finite Field of size 65537
sage: H.characteristic()                                                    # needs sage.rings.finite_rings
65537

Python

>>> from sage.all import *
>>> H = PowerSeriesRing(GF(Integer(65537)),Integer(4),'f'); H                               # needs sage.rings.finite_rings
Multivariate Power Series Ring in f0, f1, f2, f3 over
Finite Field of size 65537
>>> H.characteristic()                                                    # needs sage.rings.finite_rings
65537

Sage Live

H = PowerSeriesRing(GF(65537),4,'f'); H                               # needs sage.rings.finite_rings
H.characteristic()                                                    # needs sage.rings.finite_rings

construction()[source]¶

Return a functor \(F\) and base ring \(R\) such that F(R) == self.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(QQ, 4, 'f'); M
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field

sage: (c,R) = M.construction(); (c,R)
(Completion[('f0', 'f1', 'f2', 'f3'), prec=12],
 Multivariate Polynomial Ring in f0, f1, f2, f3 over Rational Field)
sage: c
Completion[('f0', 'f1', 'f2', 'f3'), prec=12]
sage: c(R)
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field
sage: c(R) == M
True

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(QQ, Integer(4), 'f'); M
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field

>>> (c,R) = M.construction(); (c,R)
(Completion[('f0', 'f1', 'f2', 'f3'), prec=12],
 Multivariate Polynomial Ring in f0, f1, f2, f3 over Rational Field)
>>> c
Completion[('f0', 'f1', 'f2', 'f3'), prec=12]
>>> c(R)
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field
>>> c(R) == M
True

Sage Live

M = PowerSeriesRing(QQ, 4, 'f'); M
(c,R) = M.construction(); (c,R)
c
c(R)
c(R) == M

gen(n=0)[source]¶

Return the \(n\)-th generator of self.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(ZZ, 10, 'v')
sage: M.gen(6)
v6

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(ZZ, Integer(10), 'v')
>>> M.gen(Integer(6))
v6

Sage Live

M = PowerSeriesRing(ZZ, 10, 'v')
M.gen(6)

gens()[source]¶

Return the generators of this ring.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(ZZ, 3, 'v')
sage: M.gens()
(v0, v1, v2)

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(ZZ, Integer(3), 'v')
>>> M.gens()
(v0, v1, v2)

Sage Live

M = PowerSeriesRing(ZZ, 3, 'v')
M.gens()

is_dense()[source]¶

Is self dense? (opposite of sparse)

EXAMPLES:

Sage

sage: M = PowerSeriesRing(ZZ, 3, 's,t,u'); M
Multivariate Power Series Ring in s, t, u over Integer Ring
sage: M.is_dense()
True
sage: N = PowerSeriesRing(ZZ, 3, 's,t,u', sparse=True); N
Sparse Multivariate Power Series Ring in s, t, u over Integer Ring
sage: N.is_dense()
False

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(ZZ, Integer(3), 's,t,u'); M
Multivariate Power Series Ring in s, t, u over Integer Ring
>>> M.is_dense()
True
>>> N = PowerSeriesRing(ZZ, Integer(3), 's,t,u', sparse=True); N
Sparse Multivariate Power Series Ring in s, t, u over Integer Ring
>>> N.is_dense()
False

Sage Live

M = PowerSeriesRing(ZZ, 3, 's,t,u'); M
M.is_dense()
N = PowerSeriesRing(ZZ, 3, 's,t,u', sparse=True); N
N.is_dense()

is_integral_domain(proof=False)[source]¶

Return True if the base ring is an integral domain; otherwise return False.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(QQ,4,'v'); M
Multivariate Power Series Ring in v0, v1, v2, v3 over Rational Field
sage: M.is_integral_domain()
True

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(QQ,Integer(4),'v'); M
Multivariate Power Series Ring in v0, v1, v2, v3 over Rational Field
>>> M.is_integral_domain()
True

Sage Live

M = PowerSeriesRing(QQ,4,'v'); M
M.is_integral_domain()

is_noetherian(proof=False)[source]¶

Power series over a Noetherian ring are Noetherian.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(QQ,4,'v'); M
Multivariate Power Series Ring in v0, v1, v2, v3 over Rational Field
sage: M.is_noetherian()
True

sage: W = PowerSeriesRing(InfinitePolynomialRing(ZZ,'a'),2,'x,y')
sage: W.is_noetherian()
False

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(QQ,Integer(4),'v'); M
Multivariate Power Series Ring in v0, v1, v2, v3 over Rational Field
>>> M.is_noetherian()
True

>>> W = PowerSeriesRing(InfinitePolynomialRing(ZZ,'a'),Integer(2),'x,y')
>>> W.is_noetherian()
False

Sage Live

M = PowerSeriesRing(QQ,4,'v'); M
M.is_noetherian()
W = PowerSeriesRing(InfinitePolynomialRing(ZZ,'a'),2,'x,y')
W.is_noetherian()

is_sparse()[source]¶

Check whether self is sparse.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(ZZ, 3, 's,t,u'); M
Multivariate Power Series Ring in s, t, u over Integer Ring
sage: M.is_sparse()
False
sage: N = PowerSeriesRing(ZZ, 3, 's,t,u', sparse=True); N
Sparse Multivariate Power Series Ring in s, t, u over Integer Ring
sage: N.is_sparse()
True

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(ZZ, Integer(3), 's,t,u'); M
Multivariate Power Series Ring in s, t, u over Integer Ring
>>> M.is_sparse()
False
>>> N = PowerSeriesRing(ZZ, Integer(3), 's,t,u', sparse=True); N
Sparse Multivariate Power Series Ring in s, t, u over Integer Ring
>>> N.is_sparse()
True

Sage Live

M = PowerSeriesRing(ZZ, 3, 's,t,u'); M
M.is_sparse()
N = PowerSeriesRing(ZZ, 3, 's,t,u', sparse=True); N
N.is_sparse()

laurent_series_ring()[source]¶: Laurent series not yet implemented for multivariate power series rings.

ngens()[source]¶

Return number of generators of self.

EXAMPLES:

Sage

sage: M = PowerSeriesRing(ZZ, 10, 'v')
sage: M.ngens()
10

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(ZZ, Integer(10), 'v')
>>> M.ngens()
10

Sage Live

M = PowerSeriesRing(ZZ, 10, 'v')
M.ngens()

prec_ideal()[source]¶

Return the ideal which determines precision; this is the ideal generated by all of the generators of our background polynomial ring.

EXAMPLES:

Sage

sage: A.<s,t,u> = PowerSeriesRing(ZZ)
sage: A.prec_ideal()
Ideal (s, t, u) of
 Multivariate Polynomial Ring in s, t, u over Integer Ring

Python

>>> from sage.all import *
>>> A = PowerSeriesRing(ZZ, names=('s', 't', 'u',)); (s, t, u,) = A._first_ngens(3)
>>> A.prec_ideal()
Ideal (s, t, u) of
 Multivariate Polynomial Ring in s, t, u over Integer Ring

Sage Live

A.<s,t,u> = PowerSeriesRing(ZZ)
A.prec_ideal()

remove_var(*var)[source]¶

Remove given variable or sequence of variables from self.

EXAMPLES:

Sage

sage: A.<s,t,u> = PowerSeriesRing(ZZ)
sage: A.remove_var(t)
Multivariate Power Series Ring in s, u over Integer Ring
sage: A.remove_var(s,t)
Power Series Ring in u over Integer Ring


sage: M = PowerSeriesRing(GF(5),5,'t'); M
Multivariate Power Series Ring in t0, t1, t2, t3, t4
 over Finite Field of size 5
sage: M.remove_var(M.gens()[3])
Multivariate Power Series Ring in t0, t1, t2, t4
 over Finite Field of size 5

Python

>>> from sage.all import *
>>> A = PowerSeriesRing(ZZ, names=('s', 't', 'u',)); (s, t, u,) = A._first_ngens(3)
>>> A.remove_var(t)
Multivariate Power Series Ring in s, u over Integer Ring
>>> A.remove_var(s,t)
Power Series Ring in u over Integer Ring


>>> M = PowerSeriesRing(GF(Integer(5)),Integer(5),'t'); M
Multivariate Power Series Ring in t0, t1, t2, t3, t4
 over Finite Field of size 5
>>> M.remove_var(M.gens()[Integer(3)])
Multivariate Power Series Ring in t0, t1, t2, t4
 over Finite Field of size 5

Sage Live

A.<s,t,u> = PowerSeriesRing(ZZ)
A.remove_var(t)
A.remove_var(s,t)
M = PowerSeriesRing(GF(5),5,'t'); M
M.remove_var(M.gens()[3])

Removing all variables results in the base ring:

Sage

sage: M.remove_var(*M.gens())
Finite Field of size 5

Python

>>> from sage.all import *
>>> M.remove_var(*M.gens())
Finite Field of size 5

Sage Live

M.remove_var(*M.gens())

term_order()[source]¶

Print term ordering of self. Term orderings are implemented by the TermOrder class.

EXAMPLES:

Sage

sage: M.<x,y,z> = PowerSeriesRing(ZZ,3)
sage: M.term_order()
Negative degree lexicographic term order
sage: m = y*z^12 - y^6*z^8 - x^7*y^5*z^2 + x*y^2*z + M.O(15); m
x*y^2*z + y*z^12 - x^7*y^5*z^2 - y^6*z^8 + O(x, y, z)^15

sage: N = PowerSeriesRing(ZZ,3,'x,y,z', order='deglex')
sage: N.term_order()
Degree lexicographic term order
sage: N(m)
-x^7*y^5*z^2 - y^6*z^8 + y*z^12 + x*y^2*z + O(x, y, z)^15

Python

>>> from sage.all import *
>>> M = PowerSeriesRing(ZZ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = M._first_ngens(3)
>>> M.term_order()
Negative degree lexicographic term order
>>> m = y*z**Integer(12) - y**Integer(6)*z**Integer(8) - x**Integer(7)*y**Integer(5)*z**Integer(2) + x*y**Integer(2)*z + M.O(Integer(15)); m
x*y^2*z + y*z^12 - x^7*y^5*z^2 - y^6*z^8 + O(x, y, z)^15

>>> N = PowerSeriesRing(ZZ,Integer(3),'x,y,z', order='deglex')
>>> N.term_order()
Degree lexicographic term order
>>> N(m)
-x^7*y^5*z^2 - y^6*z^8 + y*z^12 + x*y^2*z + O(x, y, z)^15

Sage Live

M.<x,y,z> = PowerSeriesRing(ZZ,3)
M.term_order()
m = y*z^12 - y^6*z^8 - x^7*y^5*z^2 + x*y^2*z + M.O(15); m
N = PowerSeriesRing(ZZ,3,'x,y,z', order='deglex')
N.term_order()
N(m)

sage.rings.multi_power_series_ring.is_MPowerSeriesRing(x)[source]¶: Return True if input is a multivariate power series ring.

sage.rings.multi_power_series_ring.unpickle_multi_power_series_ring_v0(base_ring, num_gens, names, order, default_prec, sparse)[source]¶

Unpickle (deserialize) a multivariate power series ring according to the given inputs.

EXAMPLES:

Sage

sage: P.<x,y> = PowerSeriesRing(QQ)
sage: loads(dumps(P)) == P # indirect doctest
True

Python

>>> from sage.all import *
>>> P = PowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> loads(dumps(P)) == P # indirect doctest
True

Sage Live

P.<x,y> = PowerSeriesRing(QQ)
loads(dumps(P)) == P # indirect doctest