Localization#
Localization is an important ring construction tool. Whenever you have to extend a given integral domain such that it contains the inverses of a finite set of elements but should allow non injective homomorphic images this construction will be needed. See the example on Ariki-Koike algebras below for such an application.
EXAMPLES:
sage: # needs sage.modules
sage: LZ = Localization(ZZ, (5,11))
sage: m = matrix(LZ, [[5, 7], [0,11]])
sage: m.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring localized at (5, 11)
sage: ~m # parent of inverse is different: see documentation of m.__invert__
[ 1/5 -7/55]
[ 0 1/11]
sage: _.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: mi = matrix(LZ, ~m)
sage: mi.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring localized at (5, 11)
sage: mi == ~m
True
>>> from sage.all import *
>>> # needs sage.modules
>>> LZ = Localization(ZZ, (Integer(5),Integer(11)))
>>> m = matrix(LZ, [[Integer(5), Integer(7)], [Integer(0),Integer(11)]])
>>> m.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring localized at (5, 11)
>>> ~m # parent of inverse is different: see documentation of m.__invert__
[ 1/5 -7/55]
[ 0 1/11]
>>> _.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
>>> mi = matrix(LZ, ~m)
>>> mi.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring localized at (5, 11)
>>> mi == ~m
True
The next example defines the most general ring containing the coefficients of the irreducible representations of the Ariki-Koike algebra corresponding to the three colored permutations on three elements:
sage: R.<u0, u1, u2, q> = ZZ[]
sage: u = [u0, u1, u2]
sage: S = Set(u)
sage: I = S.cartesian_product(S)
sage: add_units = u + [q, q + 1] + [ui - uj for ui, uj in I if ui != uj]
sage: add_units += [q*ui - uj for ui, uj in I if ui != uj]
sage: L = R.localization(tuple(add_units)); L # needs sage.libs.pari
Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
(q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
>>> from sage.all import *
>>> R = ZZ['u0, u1, u2, q']; (u0, u1, u2, q,) = R._first_ngens(4)
>>> u = [u0, u1, u2]
>>> S = Set(u)
>>> I = S.cartesian_product(S)
>>> add_units = u + [q, q + Integer(1)] + [ui - uj for ui, uj in I if ui != uj]
>>> add_units += [q*ui - uj for ui, uj in I if ui != uj]
>>> L = R.localization(tuple(add_units)); L # needs sage.libs.pari
Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
(q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
Define the representation matrices (of one of the three dimensional irreducible representations):
sage: # needs sage.libs.pari sage.modules
sage: m1 = matrix(L, [[u1, 0, 0], [0, u0, 0], [0, 0, u0]])
sage: m2 = matrix(L, [[(u0*q - u0)/(u0 - u1), (u0*q - u1)/(u0 - u1), 0],
....: [(-u1*q + u0)/(u0 - u1), (-u1*q + u1)/(u0 - u1), 0],
....: [0, 0, -1]])
sage: m3 = matrix(L, [[-1, 0, 0],
....: [0, u0*(1 - q)/(u1*q - u0), q*(u1 - u0)/(u1*q - u0)],
....: [0, (u1*q^2 - u0)/(u1*q - u0), (u1*q^ 2 - u1*q)/(u1*q - u0)]])
sage: m1.base_ring() == L
True
>>> from sage.all import *
>>> # needs sage.libs.pari sage.modules
>>> m1 = matrix(L, [[u1, Integer(0), Integer(0)], [Integer(0), u0, Integer(0)], [Integer(0), Integer(0), u0]])
>>> m2 = matrix(L, [[(u0*q - u0)/(u0 - u1), (u0*q - u1)/(u0 - u1), Integer(0)],
... [(-u1*q + u0)/(u0 - u1), (-u1*q + u1)/(u0 - u1), Integer(0)],
... [Integer(0), Integer(0), -Integer(1)]])
>>> m3 = matrix(L, [[-Integer(1), Integer(0), Integer(0)],
... [Integer(0), u0*(Integer(1) - q)/(u1*q - u0), q*(u1 - u0)/(u1*q - u0)],
... [Integer(0), (u1*q**Integer(2) - u0)/(u1*q - u0), (u1*q** Integer(2) - u1*q)/(u1*q - u0)]])
>>> m1.base_ring() == L
True
Check relations of the Ariki-Koike algebra:
sage: # needs sage.libs.pari sage.modules
sage: m1*m2*m1*m2 == m2*m1*m2*m1
True
sage: m2*m3*m2 == m3*m2*m3
True
sage: m1*m3 == m3*m1
True
sage: m1**3 - (u0+u1+u2)*m1**2 + (u0*u1+u0*u2+u1*u2)*m1 - u0*u1*u2 == 0
True
sage: m2**2 - (q-1)*m2 - q == 0
True
sage: m3**2 - (q-1)*m3 - q == 0
True
sage: ~m1 in m1.parent()
True
sage: ~m2 in m2.parent()
True
sage: ~m3 in m3.parent()
True
>>> from sage.all import *
>>> # needs sage.libs.pari sage.modules
>>> m1*m2*m1*m2 == m2*m1*m2*m1
True
>>> m2*m3*m2 == m3*m2*m3
True
>>> m1*m3 == m3*m1
True
>>> m1**Integer(3) - (u0+u1+u2)*m1**Integer(2) + (u0*u1+u0*u2+u1*u2)*m1 - u0*u1*u2 == Integer(0)
True
>>> m2**Integer(2) - (q-Integer(1))*m2 - q == Integer(0)
True
>>> m3**Integer(2) - (q-Integer(1))*m3 - q == Integer(0)
True
>>> ~m1 in m1.parent()
True
>>> ~m2 in m2.parent()
True
>>> ~m3 in m3.parent()
True
Obtain specializations in positive characteristic:
sage: # needs sage.libs.pari sage.modules
sage: Fp = GF(17)
sage: f = L.hom((3,5,7,11), codomain=Fp); f
Ring morphism:
From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
(q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
To: Finite Field of size 17
Defn: u0 |--> 3
u1 |--> 5
u2 |--> 7
q |--> 11
sage: mFp1 = matrix({k: f(v) for k, v in m1.dict().items()}); mFp1
[5 0 0]
[0 3 0]
[0 0 3]
sage: mFp1.base_ring()
Finite Field of size 17
sage: mFp2 = matrix({k: f(v) for k, v in m2.dict().items()}); mFp2
[ 2 3 0]
[ 9 8 0]
[ 0 0 16]
sage: mFp3 = matrix({k: f(v) for k, v in m3.dict().items()}); mFp3
[16 0 0]
[ 0 4 5]
[ 0 7 6]
>>> from sage.all import *
>>> # needs sage.libs.pari sage.modules
>>> Fp = GF(Integer(17))
>>> f = L.hom((Integer(3),Integer(5),Integer(7),Integer(11)), codomain=Fp); f
Ring morphism:
From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
(q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
To: Finite Field of size 17
Defn: u0 |--> 3
u1 |--> 5
u2 |--> 7
q |--> 11
>>> mFp1 = matrix({k: f(v) for k, v in m1.dict().items()}); mFp1
[5 0 0]
[0 3 0]
[0 0 3]
>>> mFp1.base_ring()
Finite Field of size 17
>>> mFp2 = matrix({k: f(v) for k, v in m2.dict().items()}); mFp2
[ 2 3 0]
[ 9 8 0]
[ 0 0 16]
>>> mFp3 = matrix({k: f(v) for k, v in m3.dict().items()}); mFp3
[16 0 0]
[ 0 4 5]
[ 0 7 6]
Obtain specializations in characteristic 0:
sage: # needs sage.libs.pari
sage: fQ = L.hom((3,5,7,11), codomain=QQ); fQ
Ring morphism:
From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
u2*q - u1, u2*q - u0, u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
To: Rational Field
Defn: u0 |--> 3
u1 |--> 5
u2 |--> 7
q |--> 11
sage: # needs sage.libs.pari sage.modules sage.rings.finite_rings
sage: mQ1 = matrix({k: fQ(v) for k, v in m1.dict().items()}); mQ1
[5 0 0]
[0 3 0]
[0 0 3]
sage: mQ1.base_ring()
Rational Field
sage: mQ2 = matrix({k: fQ(v) for k, v in m2.dict().items()}); mQ2
[-15 -14 0]
[ 26 25 0]
[ 0 0 -1]
sage: mQ3 = matrix({k: fQ(v) for k, v in m3.dict().items()}); mQ3
[ -1 0 0]
[ 0 -15/26 11/26]
[ 0 301/26 275/26]
sage: # needs sage.libs.pari sage.libs.singular
sage: S.<x, y, z, t> = QQ[]
sage: T = S.quo(x + y + z)
sage: F = T.fraction_field()
sage: fF = L.hom((x, y, z, t), codomain=F); fF
Ring morphism:
From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
u2*q - u1, u2*q - u0, u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
To: Fraction Field of Quotient of Multivariate Polynomial Ring in x, y, z, t
over Rational Field by the ideal (x + y + z)
Defn: u0 |--> -ybar - zbar
u1 |--> ybar
u2 |--> zbar
q |--> tbar
sage: mF1 = matrix({k: fF(v) for k, v in m1.dict().items()}); mF1 # needs sage.modules
[ ybar 0 0]
[ 0 -ybar - zbar 0]
[ 0 0 -ybar - zbar]
sage: mF1.base_ring() == F # needs sage.modules
True
>>> from sage.all import *
>>> # needs sage.libs.pari
>>> fQ = L.hom((Integer(3),Integer(5),Integer(7),Integer(11)), codomain=QQ); fQ
Ring morphism:
From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
u2*q - u1, u2*q - u0, u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
To: Rational Field
Defn: u0 |--> 3
u1 |--> 5
u2 |--> 7
q |--> 11
>>> # needs sage.libs.pari sage.modules sage.rings.finite_rings
>>> mQ1 = matrix({k: fQ(v) for k, v in m1.dict().items()}); mQ1
[5 0 0]
[0 3 0]
[0 0 3]
>>> mQ1.base_ring()
Rational Field
>>> mQ2 = matrix({k: fQ(v) for k, v in m2.dict().items()}); mQ2
[-15 -14 0]
[ 26 25 0]
[ 0 0 -1]
>>> mQ3 = matrix({k: fQ(v) for k, v in m3.dict().items()}); mQ3
[ -1 0 0]
[ 0 -15/26 11/26]
[ 0 301/26 275/26]
>>> # needs sage.libs.pari sage.libs.singular
>>> S = QQ['x, y, z, t']; (x, y, z, t,) = S._first_ngens(4)
>>> T = S.quo(x + y + z)
>>> F = T.fraction_field()
>>> fF = L.hom((x, y, z, t), codomain=F); fF
Ring morphism:
From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
u2*q - u1, u2*q - u0, u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
To: Fraction Field of Quotient of Multivariate Polynomial Ring in x, y, z, t
over Rational Field by the ideal (x + y + z)
Defn: u0 |--> -ybar - zbar
u1 |--> ybar
u2 |--> zbar
q |--> tbar
>>> mF1 = matrix({k: fF(v) for k, v in m1.dict().items()}); mF1 # needs sage.modules
[ ybar 0 0]
[ 0 -ybar - zbar 0]
[ 0 0 -ybar - zbar]
>>> mF1.base_ring() == F # needs sage.modules
True
AUTHORS:
Sebastian Oehms 2019-12-09: initial version.
Sebastian Oehms 2022-03-05: fix some corner cases and add
factor()(Issue #33463)
- class sage.rings.localization.Localization(base_ring, extra_units, names=None, normalize=True, category=None, warning=True)[source]#
Bases:
IntegralDomain,UniqueRepresentationThe localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring. Given a (not necessarily commutative) ring \(R\) and a subset \(S\) of \(R\), there exists a ring \(R[S^{-1}]\) together with the ring homomorphism \(R \longrightarrow R[S^{-1}]\) that “inverts” \(S\); that is, the homomorphism maps elements in \(S\) to unit elements in \(R[S^{-1}]\) and, moreover, any ring homomorphism from \(R\) that “inverts” \(S\) uniquely factors through \(R[S^{-1}]\).
The ring \(R[S^{-1}]\) is called the localization of \(R\) with respect to \(S\). For example, if \(R\) is a commutative ring and \(f\) an element in \(R\), then the localization consists of elements of the form \(r/f, r\in R, n \geq 0\) (to be precise, \(R[f^{-1}] = R[t]/(ft-1)\).
The above text is taken from \(Wikipedia\). The construction here used for this class relies on the construction of the field of fraction and is therefore restricted to integral domains.
Accordingly, this class is inherited from
IntegralDomainand can only be used in that context. Furthermore, the base ring should supportsage.structure.element.CommutativeRingElement.divides()and the exact division operator \(//\) (sage.structure.element.Element.__floordiv__()) in order to guarantee a successful application.INPUT:
base_ring– an instance ofRingallowing the construction offraction_field()(that is an integral domain)extra_units– tuple of elements ofbase_ringwhich should be turned into unitsnames– passed toIntegralDomainnormalize– (default:True) passed toIntegralDomaincategory– (default: None) passed toIntegralDomainwarning– (default:True) to suppress a warning which is thrown if self cannot be represented uniquely
REFERENCES:
EXAMPLES:
sage: L = Localization(ZZ, (3,5)) sage: 1/45 in L True sage: 1/43 in L False sage: Localization(L, (7,11)) Integer Ring localized at (3, 5, 7, 11) sage: _.is_subring(QQ) True sage: L(~7) Traceback (most recent call last): ... ValueError: factor 7 of denominator is not a unit sage: Localization(Zp(7), (3, 5)) # needs sage.rings.padics Traceback (most recent call last): ... ValueError: all given elements are invertible in 7-adic Ring with capped relative precision 20 sage: # needs sage.libs.pari sage: R.<x> = ZZ[] sage: L = R.localization(x**2 + 1) sage: s = (x+5)/(x**2+1) sage: s in L True sage: t = (x+5)/(x**2+2) sage: t in L False sage: L(t) Traceback (most recent call last): ... TypeError: fraction must have unit denominator sage: L(s) in R False sage: y = L(x) sage: g = L(s) sage: g.parent() Univariate Polynomial Ring in x over Integer Ring localized at (x^2 + 1,) sage: f = (y+5)/(y**2+1); f (x + 5)/(x^2 + 1) sage: f == g True sage: (y+5)/(y**2+2) Traceback (most recent call last): ... ValueError: factor x^2 + 2 of denominator is not a unit sage: Lau.<u, v> = LaurentPolynomialRing(ZZ) # needs sage.modules sage: LauL = Lau.localization(u + 1) # needs sage.modules sage: LauL(~u).parent() # needs sage.modules Multivariate Polynomial Ring in u, v over Integer Ring localized at (v, u, u + 1)
>>> from sage.all import * >>> L = Localization(ZZ, (Integer(3),Integer(5))) >>> Integer(1)/Integer(45) in L True >>> Integer(1)/Integer(43) in L False >>> Localization(L, (Integer(7),Integer(11))) Integer Ring localized at (3, 5, 7, 11) >>> _.is_subring(QQ) True >>> L(~Integer(7)) Traceback (most recent call last): ... ValueError: factor 7 of denominator is not a unit >>> Localization(Zp(Integer(7)), (Integer(3), Integer(5))) # needs sage.rings.padics Traceback (most recent call last): ... ValueError: all given elements are invertible in 7-adic Ring with capped relative precision 20 >>> # needs sage.libs.pari >>> R = ZZ['x']; (x,) = R._first_ngens(1) >>> L = R.localization(x**Integer(2) + Integer(1)) >>> s = (x+Integer(5))/(x**Integer(2)+Integer(1)) >>> s in L True >>> t = (x+Integer(5))/(x**Integer(2)+Integer(2)) >>> t in L False >>> L(t) Traceback (most recent call last): ... TypeError: fraction must have unit denominator >>> L(s) in R False >>> y = L(x) >>> g = L(s) >>> g.parent() Univariate Polynomial Ring in x over Integer Ring localized at (x^2 + 1,) >>> f = (y+Integer(5))/(y**Integer(2)+Integer(1)); f (x + 5)/(x^2 + 1) >>> f == g True >>> (y+Integer(5))/(y**Integer(2)+Integer(2)) Traceback (most recent call last): ... ValueError: factor x^2 + 2 of denominator is not a unit >>> Lau = LaurentPolynomialRing(ZZ, names=('u', 'v',)); (u, v,) = Lau._first_ngens(2)# needs sage.modules >>> LauL = Lau.localization(u + Integer(1)) # needs sage.modules >>> LauL(~u).parent() # needs sage.modules Multivariate Polynomial Ring in u, v over Integer Ring localized at (v, u, u + 1)
More examples will be shown typing
sage.rings.localization?- Element[source]#
alias of
LocalizationElement
- characteristic()[source]#
Return the characteristic of
self.EXAMPLES:
sage: # needs sage.libs.pari sage: R.<a> = GF(5)[] sage: L = R.localization((a**2 - 3, a)) sage: L.characteristic() 5
>>> from sage.all import * >>> # needs sage.libs.pari >>> R = GF(Integer(5))['a']; (a,) = R._first_ngens(1) >>> L = R.localization((a**Integer(2) - Integer(3), a)) >>> L.characteristic() 5
- fraction_field()[source]#
Return the fraction field of
self.EXAMPLES:
sage: # needs sage.libs.pari sage: R.<a> = GF(5)[] sage: L = Localization(R, (a**2 - 3, a)) sage: L.fraction_field() Fraction Field of Univariate Polynomial Ring in a over Finite Field of size 5 sage: L.is_subring(_) True
>>> from sage.all import * >>> # needs sage.libs.pari >>> R = GF(Integer(5))['a']; (a,) = R._first_ngens(1) >>> L = Localization(R, (a**Integer(2) - Integer(3), a)) >>> L.fraction_field() Fraction Field of Univariate Polynomial Ring in a over Finite Field of size 5 >>> L.is_subring(_) True
- gen(i)[source]#
Return the
i-th generator ofselfwhich is thei-th generator of the base ring.EXAMPLES:
sage: R.<x, y> = ZZ[] sage: R.localization((x**2 + 1, y - 1)).gen(0) # needs sage.libs.pari x sage: ZZ.localization(2).gen(0) 1
>>> from sage.all import * >>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2) >>> R.localization((x**Integer(2) + Integer(1), y - Integer(1))).gen(Integer(0)) # needs sage.libs.pari x >>> ZZ.localization(Integer(2)).gen(Integer(0)) 1
- gens()[source]#
Return a tuple whose entries are the generators for this object, in order.
EXAMPLES:
sage: R.<x, y> = ZZ[] sage: Localization(R, (x**2 + 1, y - 1)).gens() # needs sage.libs.pari (x, y) sage: Localization(ZZ, 2).gens() (1,)
>>> from sage.all import * >>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2) >>> Localization(R, (x**Integer(2) + Integer(1), y - Integer(1))).gens() # needs sage.libs.pari (x, y) >>> Localization(ZZ, Integer(2)).gens() (1,)
- is_field(proof=True)[source]#
Return
Trueif this ring is a field.INPUT:
proof– (default:True) Determines what to do in unknown cases
ALGORITHM:
If the parameter
proofis set toTrue, the returned value is correct but the method might throw an error. Otherwise, if it is set toFalse, the method returns True if it can establish that self is a field and False otherwise.EXAMPLES:
sage: R = ZZ.localization((2, 3)) sage: R.is_field() False
>>> from sage.all import * >>> R = ZZ.localization((Integer(2), Integer(3))) >>> R.is_field() False
- krull_dimension()[source]#
Return the Krull dimension of this localization.
Since the current implementation just allows integral domains as base ring and localization at a finite set of elements the spectrum of
selfis open in the irreducible spectrum of its base ring. Therefore, by density we may take the dimension from there.EXAMPLES:
sage: R = ZZ.localization((2, 3)) sage: R.krull_dimension() 1
>>> from sage.all import * >>> R = ZZ.localization((Integer(2), Integer(3))) >>> R.krull_dimension() 1
- ngens()[source]#
Return the number of generators of
selfaccording to the same method for the base ring.EXAMPLES:
sage: R.<x, y> = ZZ[] sage: Localization(R, (x**2 + 1, y - 1)).ngens() # needs sage.libs.pari 2 sage: Localization(ZZ, 2).ngens() 1
>>> from sage.all import * >>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2) >>> Localization(R, (x**Integer(2) + Integer(1), y - Integer(1))).ngens() # needs sage.libs.pari 2 >>> Localization(ZZ, Integer(2)).ngens() 1
- class sage.rings.localization.LocalizationElement(parent, x)[source]#
Bases:
IntegralDomainElementElement class for localizations of integral domains
INPUT:
parent– instance ofLocalizationx– instance ofFractionFieldElementwhose parent is the fractionfield of the parent’s base ring
EXAMPLES:
sage: # needs sage.libs.pari sage: from sage.rings.localization import LocalizationElement sage: P.<x,y,z> = GF(5)[] sage: L = P.localization((x, y*z - x)) sage: LocalizationElement(L, 4/(y*z-x)**2) (-1)/(y^2*z^2 - 2*x*y*z + x^2) sage: _.parent() Multivariate Polynomial Ring in x, y, z over Finite Field of size 5 localized at (x, y*z - x)
>>> from sage.all import * >>> # needs sage.libs.pari >>> from sage.rings.localization import LocalizationElement >>> P = GF(Integer(5))['x, y, z']; (x, y, z,) = P._first_ngens(3) >>> L = P.localization((x, y*z - x)) >>> LocalizationElement(L, Integer(4)/(y*z-x)**Integer(2)) (-1)/(y^2*z^2 - 2*x*y*z + x^2) >>> _.parent() Multivariate Polynomial Ring in x, y, z over Finite Field of size 5 localized at (x, y*z - x)
- denominator()[source]#
Return the denominator of
self.EXAMPLES:
sage: L = Localization(ZZ, (3,5)) sage: L(7/15).denominator() 15
>>> from sage.all import * >>> L = Localization(ZZ, (Integer(3),Integer(5))) >>> L(Integer(7)/Integer(15)).denominator() 15
- factor(proof=None)[source]#
Return the factorization of this polynomial.
INPUT:
proof– (optional) if given it is passed to the corresponding method of the numerator ofself
EXAMPLES:
sage: P.<X, Y> = QQ['x, y'] sage: L = P.localization(X - Y) sage: x, y = L.gens() sage: p = (x^2 - y^2)/(x-y)^2 # needs sage.libs.singular sage: p.factor() # needs sage.libs.singular (1/(x - y)) * (x + y)
>>> from sage.all import * >>> P = QQ['x, y']; (X, Y,) = P._first_ngens(2) >>> L = P.localization(X - Y) >>> x, y = L.gens() >>> p = (x**Integer(2) - y**Integer(2))/(x-y)**Integer(2) # needs sage.libs.singular >>> p.factor() # needs sage.libs.singular (1/(x - y)) * (x + y)
- inverse_of_unit()[source]#
Return the inverse of
self.EXAMPLES:
sage: P.<x,y,z> = ZZ[] sage: L = Localization(P, x*y*z) sage: L(x*y*z).inverse_of_unit() # needs sage.libs.singular 1/(x*y*z) sage: L(z).inverse_of_unit() # needs sage.libs.singular 1/z
>>> from sage.all import * >>> P = ZZ['x, y, z']; (x, y, z,) = P._first_ngens(3) >>> L = Localization(P, x*y*z) >>> L(x*y*z).inverse_of_unit() # needs sage.libs.singular 1/(x*y*z) >>> L(z).inverse_of_unit() # needs sage.libs.singular 1/z
- is_unit()[source]#
Return
Trueifselfis a unit.EXAMPLES:
sage: # needs sage.libs.pari sage.singular sage: P.<x,y,z> = QQ[] sage: L = P.localization((x, y*z)) sage: L(y*z).is_unit() True sage: L(z).is_unit() True sage: L(x*y*z).is_unit() True
>>> from sage.all import * >>> # needs sage.libs.pari sage.singular >>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3) >>> L = P.localization((x, y*z)) >>> L(y*z).is_unit() True >>> L(z).is_unit() True >>> L(x*y*z).is_unit() True
- sage.rings.localization.normalize_extra_units(base_ring, add_units, warning=True)[source]#
Function to normalize input data.
The given list will be replaced by a list of the involved prime factors (if possible).
INPUT:
base_ring– an instance ofIntegralDomainadd_units– list of elements from base ringwarning– (default:True) to suppress a warning which is thrown if no normalization was possible
OUTPUT:
List of all prime factors of the elements of the given list.
EXAMPLES:
sage: from sage.rings.localization import normalize_extra_units sage: normalize_extra_units(ZZ, [3, -15, 45, 9, 2, 50]) [2, 3, 5] sage: P.<x,y,z> = ZZ[] sage: normalize_extra_units(P, # needs sage.libs.pari ....: [3*x, z*y**2, 2*z, 18*(x*y*z)**2, x*z, 6*x*z, 5]) [2, 3, 5, z, y, x] sage: P.<x,y,z> = QQ[] sage: normalize_extra_units(P, # needs sage.libs.pari ....: [3*x, z*y**2, 2*z, 18*(x*y*z)**2, x*z, 6*x*z, 5]) [z, y, x] sage: # needs sage.libs.singular sage: R.<x, y> = ZZ[] sage: Q.<a, b> = R.quo(x**2 - 5) sage: p = b**2 - 5 sage: p == (b-a)*(b+a) True sage: normalize_extra_units(Q, [p]) # needs sage.libs.pari doctest:...: UserWarning: Localization may not be represented uniquely [b^2 - 5] sage: normalize_extra_units(Q, [p], warning=False) # needs sage.libs.pari [b^2 - 5]
>>> from sage.all import * >>> from sage.rings.localization import normalize_extra_units >>> normalize_extra_units(ZZ, [Integer(3), -Integer(15), Integer(45), Integer(9), Integer(2), Integer(50)]) [2, 3, 5] >>> P = ZZ['x, y, z']; (x, y, z,) = P._first_ngens(3) >>> normalize_extra_units(P, # needs sage.libs.pari ... [Integer(3)*x, z*y**Integer(2), Integer(2)*z, Integer(18)*(x*y*z)**Integer(2), x*z, Integer(6)*x*z, Integer(5)]) [2, 3, 5, z, y, x] >>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3) >>> normalize_extra_units(P, # needs sage.libs.pari ... [Integer(3)*x, z*y**Integer(2), Integer(2)*z, Integer(18)*(x*y*z)**Integer(2), x*z, Integer(6)*x*z, Integer(5)]) [z, y, x] >>> # needs sage.libs.singular >>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2) >>> Q = R.quo(x**Integer(2) - Integer(5), names=('a', 'b',)); (a, b,) = Q._first_ngens(2) >>> p = b**Integer(2) - Integer(5) >>> p == (b-a)*(b+a) True >>> normalize_extra_units(Q, [p]) # needs sage.libs.pari doctest:...: UserWarning: Localization may not be represented uniquely [b^2 - 5] >>> normalize_extra_units(Q, [p], warning=False) # needs sage.libs.pari [b^2 - 5]