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: 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 *
>>> 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
Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
(q, q + 1, u2, u1 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0, u2*q - u1, u1*q - u0,
u1*q - u2, u0*q - u1, u0*q - u2)
>>> 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
Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
(q, q + 1, u2, u1 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0, u2*q - u1, u1*q - u0,
u1*q - u2, u0*q - u1, u0*q - u2)
Define the representation matrices (of one of the three dimensional irreducible representations):
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 *
>>> 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: 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 *
>>> 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: 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 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0, u2*q - u1,
u1*q - u0, u1*q - u2, u0*q - u1, u0*q - u2)
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 *
>>> 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 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0, u2*q - u1,
u1*q - u0, u1*q - u2, u0*q - u1, u0*q - u2)
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: 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 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0,
u2*q - u1, u1*q - u0, u1*q - u2, u0*q - u1, u0*q - u2)
To: Rational Field
Defn: u0 |--> 3
u1 |--> 5
u2 |--> 7
q |--> 11
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: 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 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0,
u2*q - u1, u1*q - u0, u1*q - u2, u0*q - u1, u0*q - u2)
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
[ ybar 0 0]
[ 0 -ybar - zbar 0]
[ 0 0 -ybar - zbar]
sage: mF1.base_ring() == F
True
>>> from sage.all import *
>>> 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 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0,
u2*q - u1, u1*q - u0, u1*q - u2, u0*q - u1, u0*q - u2)
To: Rational Field
Defn: u0 |--> 3
u1 |--> 5
u2 |--> 7
q |--> 11
>>> 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]
>>> 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 - u2, u1, u0 - u1, u0 - u2, u0, u2*q - u0,
u2*q - u1, u1*q - u0, u1*q - u2, u0*q - u1, u0*q - u2)
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
[ ybar 0 0]
[ 0 -ybar - zbar 0]
[ 0 0 -ybar - zbar]
>>> mF1.base_ring() == F
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:
Parent,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, the base ring must be in the category of
IntegralDomains. 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– a ring in the category ofIntegralDomainsextra_units– tuple of elements ofbase_ringwhich should be turned into unitscategory– (default:None) passed toParentwarning– boolean (default:True); to suppress a warning which is thrown ifselfcannot 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: 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 >>> 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: R.<a> = GF(5)[] sage: L = R.localization((a**2 - 3, a)) sage: L.characteristic() 5
>>> from sage.all import * >>> 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: 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 * >>> 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) 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)) 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() (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() (x, y) >>> Localization(ZZ, Integer(2)).gens() (1,)
- is_field(proof=True)[source]¶
Return
Trueif this ring is a field.INPUT:
proof– boolean (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 returnsTrueif it can establish thatselfis a field andFalseotherwise.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() 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() 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 fraction field of the parent’s base ring
EXAMPLES:
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 * >>> 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: 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 * >>> 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– a ring in the category ofIntegralDomainsadd_units– list of elements from base ringwarning– boolean (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: 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] >>> 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]