# HG changeset patch # User Simon King # Date 1317313679 -7200 # Node ID d129c0622eb94d3871297ccb39de1edbf44eb729 # Parent 9f95d173ac536e34aa7c6f711843b4aa1632f166 #4539: Fix doc string format; fix doc test errors left over by the previous patches. diff --git a/sage/libs/singular/function.pyx b/sage/libs/singular/function.pyx --- a/sage/libs/singular/function.pyx +++ b/sage/libs/singular/function.pyx @@ -405,20 +405,16 @@ def is_singular_poly_wrapper(p): """ - Checks if p is some data type corresponding to some singular ``poly```. - + Checks if p is some data type corresponding to some singular ``poly``. + EXAMPLE:: - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: from sage.matrix.constructor import Matrix sage: from sage.libs.singular.function import is_singular_poly_wrapper - sage: c=Matrix(2) - sage: c[0,1]=-1 - sage: P = NCPolynomialRing_plural(QQ, 2, 'x,y', c=c, d=Matrix(2)) - sage: (x,y)=P.gens() + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y}) sage: is_singular_poly_wrapper(x+y) True - + """ return PY_TYPE_CHECK(p, MPolynomial_libsingular) or PY_TYPE_CHECK(p, NCPolynomial_plural) @@ -1640,13 +1636,10 @@ sage: singular_list(resolution) [[(-2*y, 2, y + 1, 0), (0, -2, x - 1, 0), (x*y - y, -y + 1, 1, -y), (x^2 + 1, -x - 1, -1, -x)], [(-x - 1, y - 1, 2*x, -2*y)], [(0)]] - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: from sage.matrix.constructor import Matrix - sage: c=Matrix(2) - sage: c[0,1]=-1 - sage: P = NCPolynomialRing_plural(QQ, 2, 'x,y', c=c, d=Matrix(2)) - sage: (x,y)=P.gens() - sage: I= Sequence([x*y,x+y], check=False, immutable=True)#P.ideal(x*y,x+y) + + sage: A. = FreeAlgebra(QQ, 2) + sage: P. = A.g_algebra({y*x:-x*y}) + sage: I= Sequence([x*y,x+y], check=False, immutable=True) sage: twostd = singular_function("twostd") sage: twostd(I) [x + y, y^2] diff --git a/sage/libs/singular/groebner_strategy.pyx b/sage/libs/singular/groebner_strategy.pyx --- a/sage/libs/singular/groebner_strategy.pyx +++ b/sage/libs/singular/groebner_strategy.pyx @@ -299,48 +299,33 @@ EXAMPLES:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(QQ) - sage: I = Ideal([x+z,y+z+1]) - sage: strat = GroebnerStrategy(I); strat - Groebner Strategy for ideal generated by 2 elements - over Multivariate Polynomial Ring in x, y, z over Rational Field + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: NCGroebnerStrategy(I) + Groebner Strategy for ideal generated by 3 elements over Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: x*y - z, z*y: y*z - 2*y, z*x: x*z + 2*x} TESTS:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: strat = GroebnerStrategy(None) + sage: strat = NCGroebnerStrategy(None) Traceback (most recent call last): ... - TypeError: First parameter must be a multivariate polynomial ideal. + TypeError: First parameter must be an ideal in a g-algebra. - sage: P. = PolynomialRing(QQ,order='neglex') + sage: P. = PolynomialRing(CC,order='neglex') sage: I = Ideal([x+z,y+z+1]) - sage: strat = GroebnerStrategy(I) + sage: strat = NCGroebnerStrategy(I) Traceback (most recent call last): ... - NotImplementedError: The local case is not implemented yet. - - sage: P. = PolynomialRing(CC,order='neglex') - sage: I = Ideal([x+z,y+z+1]) - sage: strat = GroebnerStrategy(I) - Traceback (most recent call last): - ... - TypeError: First parameter's ring must be multivariate polynomial ring via libsingular. + TypeError: First parameter must be an ideal in a g-algebra. - sage: P. = PolynomialRing(ZZ) - sage: I = Ideal([x+z,y+z+1]) - sage: strat = GroebnerStrategy(I) - Traceback (most recent call last): - ... - NotImplementedError: Only coefficient fields are implemented so far. - """ if not isinstance(L, NCPolynomialIdeal): raise TypeError("First parameter must be an ideal in a g-algebra.") if not isinstance(L.ring(), NCPolynomialRing_plural): - raise TypeError("First parameter's ring must be multivariate polynomial ring via libsingular.") + raise TypeError("First parameter's ring must be a g-algebra.") self._ideal = L @@ -381,11 +366,12 @@ """ TEST:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(32003)) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) - sage: del strat + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) + sage: del strat # indirect doctest """ cdef ring *oldRing = NULL if self._strat: @@ -412,13 +398,13 @@ """ TEST:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(32003)) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) sage: strat # indirect doctest - Groebner Strategy for ideal generated by 2 elements over - Multivariate Polynomial Ring in x, y, z over Finite Field of size 32003 + Groebner Strategy for ideal generated by 3 elements over Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: x*y - z, z*y: y*z - 2*y, z*x: x*z + 2*x} """ return "Groebner Strategy for ideal generated by %d elements over %s"%(self._ideal.ngens(),self._parent) @@ -428,12 +414,14 @@ EXAMPLE:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(32003)) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) - sage: strat.ideal() - Ideal (x + z, y + z) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 32003 + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) + sage: strat.ideal() == I + True + """ return self._ideal @@ -443,12 +431,13 @@ EXAMPLE:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(32003)) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) - sage: strat.ring() - Multivariate Polynomial Ring in x, y, z over Finite Field of size 32003 + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) + sage: strat.ring() is H + True """ return self._parent @@ -456,29 +445,33 @@ """ EXAMPLE:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(19)) - sage: I = Ideal([P(0)]) - sage: strat = GroebnerStrategy(I) - sage: strat == GroebnerStrategy(I) + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) + sage: strat == NCGroebnerStrategy(I) True - sage: I = Ideal([x+1,y+z]) - sage: strat == GroebnerStrategy(I) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()], side='twosided') + sage: strat == NCGroebnerStrategy(I) False """ if not isinstance(other, NCGroebnerStrategy): return cmp(type(self),other(type)) else: - return cmp(self._ideal.gens(),(other)._ideal.gens()) + return cmp((self._ideal.gens(),self._ideal.side()), + ((other)._ideal.gens(), + (other)._ideal.side())) def __reduce__(self): """ EXAMPLE:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(32003)) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) sage: loads(dumps(strat)) == strat True """ @@ -491,25 +484,18 @@ EXAMPLE:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(QQ) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) - sage: strat.normal_form(x*y) # indirect doctest - z^2 - sage: strat.normal_form(x + 1) - -z + 1 + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: JL = H.ideal([x^3, y^3, z^3 - 4*z]) + sage: JT = H.ideal([x^3, y^3, z^3 - 4*z], side='twosided') + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: SL = NCGroebnerStrategy(JL.std()) + sage: ST = NCGroebnerStrategy(JT.std()) + sage: SL.normal_form(x*y^2) + x*y^2 + sage: ST.normal_form(x*y^2) + y*z - TESTS:: - - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(QQ) - sage: I = Ideal([P(0)]) - sage: strat = GroebnerStrategy(I) - sage: strat.normal_form(x) - x - sage: strat.normal_form(P(0)) - 0 """ if unlikely(p._parent is not self._parent): raise TypeError("parent(p) must be the same as this object's parent.") @@ -526,11 +512,12 @@ """ EXAMPLE:: - sage: from sage.libs.singular.groebner_strategy import GroebnerStrategy - sage: P. = PolynomialRing(GF(32003)) - sage: I = Ideal([x + z, y + z]) - sage: strat = GroebnerStrategy(I) - sage: loads(dumps(strat)) == strat # indirect doctest + sage: from sage.libs.singular.groebner_strategy import NCGroebnerStrategy + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: I = H.ideal([y^2, x^2, z^2-H.one_element()]) + sage: strat = NCGroebnerStrategy(I) + sage: loads(dumps(strat)) == strat # indirect doctest True """ return NCGroebnerStrategy(I) diff --git a/sage/rings/ideal_monoid.py b/sage/rings/ideal_monoid.py --- a/sage/rings/ideal_monoid.py +++ b/sage/rings/ideal_monoid.py @@ -81,15 +81,23 @@ sage: R. = QuadraticField(-23) sage: M = sage.rings.ideal_monoid.IdealMonoid(R) - sage: M(a) + sage: M(a) # indirect doctest Fractional ideal (a) sage: M([a-4, 13]) Fractional ideal (13, 1/2*a + 9/2) """ - #print x, type(x) - if isinstance(x, ideal.Ideal_generic): + try: + side = x.side() + except (AttributeError,TypeError): + side = None + try: x = x.gens() - y = self.__R.ideal(x) + except AttributeError: + pass + if side is None: + y = self.__R.ideal(x) + else: + y = self.__R.ideal(x,side=side) y._set_parent(self) return y diff --git a/sage/rings/polynomial/multi_polynomial_ideal.py b/sage/rings/polynomial/multi_polynomial_ideal.py --- a/sage/rings/polynomial/multi_polynomial_ideal.py +++ b/sage/rings/polynomial/multi_polynomial_ideal.py @@ -2957,7 +2957,7 @@ Apparently, ``x*y^2-y*z`` should be in the two-sided, but not in the left ideal:: - sage: x*y^2-y*z in JL + sage: x*y^2-y*z in JL #indirect doctest False sage: x*y^2-y*z in JT True diff --git a/sage/rings/polynomial/plural.pxd b/sage/rings/polynomial/plural.pxd --- a/sage/rings/polynomial/plural.pxd +++ b/sage/rings/polynomial/plural.pxd @@ -21,7 +21,7 @@ # cdef NCPolynomial_plural _one_element # cdef NCPolynomial_plural _zero_element - cdef public object _relations + cdef public object _relations,_relations_commutative pass cdef class ExteriorAlgebra_plural(NCPolynomialRing_plural): diff --git a/sage/rings/polynomial/plural.pyx b/sage/rings/polynomial/plural.pyx --- a/sage/rings/polynomial/plural.pyx +++ b/sage/rings/polynomial/plural.pyx @@ -22,20 +22,23 @@ - Alexander Dreyer (2010-07): noncommutative ring functionality and documentation + - Simon King (2011-09): left and two-sided ideals; normal forms; pickling; + documentation + The underlying libSINGULAR interface was implemented by -- Martin Albrecht (2007-01): initial implementation - -- Joel Mohler (2008-01): misc improvements, polishing - -- Martin Albrecht (2008-08): added `\QQ(a)` and `\ZZ` support - -- Simon King (2009-04): improved coercion - -- Martin Albrecht (2009-05): added `\ZZ/n\ZZ` support, refactoring - -- Martin Albrecht (2009-06): refactored the code to allow better - re-use + - Martin Albrecht (2007-01): initial implementation + + - Joel Mohler (2008-01): misc improvements, polishing + + - Martin Albrecht (2008-08): added `\QQ(a)` and `\ZZ` support + + - Simon King (2009-04): improved coercion + + - Martin Albrecht (2009-05): added `\ZZ/n\ZZ` support, refactoring + + - Martin Albrecht (2009-06): refactored the code to allow better + re-use TODO: @@ -46,9 +49,7 @@ We show how to construct various noncommutative polynomial rings:: sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z + sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex') sage: P Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} @@ -57,10 +58,7 @@ -x*y + 1/2 sage: A. = FreeAlgebra(GF(17), 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z - + sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex') sage: P Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 17, nc-relations: {y*x: -x*y} @@ -68,7 +66,9 @@ -x*y + 7 -Raw use of this class:: +Raw use of this class; *this is not the intended use!* +:: + sage: from sage.matrix.constructor import Matrix sage: c = Matrix(3) sage: c[0,1] = -2 @@ -77,9 +77,12 @@ sage: d = Matrix(3) sage: d[0, 1] = 17 + sage: P = QQ['x','y','z'] + sage: c = c.change_ring(P) + sage: d = d.change_ring(P) sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: R. = NCPolynomialRing_plural(QQ, 3, c = c, d = d, order='lex') + sage: R. = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',3),category=Algebras(QQ)) sage: R Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -2*x*y + 17} @@ -90,19 +93,14 @@ sage: f = 57 * a^2*b + 43 * c + 1; f 57*x^2*y + 43*z + 1 - sage: R.term_order() - Lexicographic term order - TESTS:: sage: A. = FreeAlgebra(QQ, 3) sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z - - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural, NCPolynomial_plural - sage: TestSuite(NCPolynomialRing_plural).run() - sage: TestSuite(NCPolynomial_plural).run() + sage: TestSuite(P).run() + sage: loads(dumps(P)) is P + True + """ include "sage/ext/stdsage.pxi" include "sage/ext/interrupt.pxi" @@ -198,6 +196,7 @@ sage: A. = FreeAlgebra(QQ, 3) sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) sage: H is A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) # indirect doctest + True """ if names is None: @@ -241,6 +240,13 @@ Category of algebras over Rational Field sage: TestSuite(H).run() + Note that two variables commute if they are not part of the given + relations:: + + sage: H. = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y}) + sage: x*y == y*x + True + """ def __init__(self, base_ring, names, c, d, order, category, check = True): """ @@ -258,23 +264,31 @@ ``self.gen(j)*self.gen(i) == c[i, j] * self.gen(i)*self.gen(j) + d[i, j],`` - where ``0 <= i < j < self.ngens()``. + where ``0 <= i < j < self.ngens()``. Note that two variables + commute if they are not part of one of these relations. - ``order`` - term order - ``check`` - check the noncommutative conditions (default: ``True``) - EXAMPLES:: + TESTS: + + It is strongly recommended to construct a g-algebra using + :class:`G_AlgFactory`. The following is just for documenting + the arguments of the ``__init__`` method:: sage: from sage.matrix.constructor import Matrix - sage: c = Matrix(3) - sage: c[0,1] = -1 - sage: c[0,2] = 1 - sage: c[1,2] = 1 - - sage: d = Matrix(3) - sage: d[0, 1] = 17 + sage: c0 = Matrix(3) + sage: c0[0,1] = -1 + sage: c0[0,2] = 1 + sage: c0[1,2] = 1 + + sage: d0 = Matrix(3) + sage: d0[0, 1] = 17 + sage: P = QQ['x','y','z'] + sage: c = c0.change_ring(P) + sage: d = d0.change_ring(P) sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: P. = NCPolynomialRing_plural(QQ, c = c, d = d, order='lex') + sage: P. = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',3), category=Algebras(QQ)) sage: P # indirect doctest Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y + 17} @@ -289,7 +303,10 @@ Lexicographic term order sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: P. = NCPolynomialRing_plural(GF(7), 3, c = c, d = d, order='degrevlex') + sage: P = GF(7)['x','y','z'] + sage: c = c0.change_ring(P) + sage: d = d0.change_ring(P) + sage: P. = NCPolynomialRing_plural(GF(7), c = c, d = d, order=TermOrder('degrevlex',3), category=Algebras(GF(7))) sage: P # indirect doctest Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 7, nc-relations: {y*x: -x*y + 3} @@ -323,7 +340,7 @@ self.__ngens = n self.__term_order = order - Ring.__init__(self, base_ring, names, category) + Ring.__init__(self, base_ring, names, category=category) self._populate_coercion_lists_() #MPolynomialRing_generic.__init__(self, base_ring, n, names, order) @@ -364,8 +381,10 @@ collection). TESTS: + This example caused a segmentation fault with a previous version - of this method: + of this method:: + sage: import gc sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural sage: from sage.algebras.free_algebra import FreeAlgebra @@ -394,8 +413,8 @@ Make sure element is a valid member of self, and return the constructed element. EXAMPLES:: + sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') We can construct elements from the base ring:: @@ -403,25 +422,21 @@ sage: P(1/2) 1/2 - and all kinds of integers:: sage: P(17) 17 - sage: P(int(19)) 19 - sage: P(long(19)) 19 - - TESTS:: + + TESTS: Check conversion from self:: + sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z + sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex') sage: P._element_constructor_(1/2) 1/2 @@ -432,28 +447,8 @@ sage: P._element_constructor_(y*x) -x*y - Raw use of this class:: - sage: from sage.matrix.constructor import Matrix - sage: c = Matrix(3) - sage: c[0,1] = -2 - sage: c[0,2] = 1 - sage: c[1,2] = 1 - - sage: d = Matrix(3) - sage: d[0, 1] = 17 - - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: R. = NCPolynomialRing_plural(QQ, 3, c = c, d = d, order='lex') - sage: R._element_constructor_(x*y) - x*y - - sage: P._element_constructor_(17) - 17 - - sage: P._element_constructor_(int(19)) - 19 - Testing special cases:: + sage: P._element_constructor_(1) 1 @@ -536,13 +531,14 @@ cpdef _coerce_map_from_(self, S): """ The only things that coerce into this ring are: - - the integer ring - - other localizations away from fewer primes - - EXAMPLES:: + + - the integer ring + - other localizations away from fewer primes + + EXAMPLES:: + sage: A. = FreeAlgebra(QQ, 3) sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P._coerce_map_from_(ZZ) True """ @@ -553,96 +549,54 @@ def __hash__(self): - """ - Return a hash for this noncommutative ring, that is, a hash of the string - representation of this polynomial ring. - - EXAMPLES:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: hash(P) # somewhat random output - ... - - TESTS:: - - Check conversion from self:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: from sage.matrix.constructor import Matrix - sage: c = Matrix(3) - sage: c[0,1] = -1 - sage: c[0,2] = 1 - sage: c[1,2] = 1 - - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: R. = NCPolynomialRing_plural(QQ, 3, c = c, d = Matrix(3), order='lex') - sage: hash(R) == hash(P) - True - """ - return hash(str(self.__repr__()) + str(self.term_order()) ) - - - def __cmp__(self, right): - r""" - Non-commutative polynomial rings are said to be equal if: - - - their base rings match, - - their generator names match, - - their term orderings match, and - - their relations match. + """ + Return a hash for this noncommutative ring, that is, a hash of the string + representation of this polynomial ring. + + NOTE: + + G-algebras are unique parents, provided that the g-algebra constructor + is used. Thus, the hash simply is the memory address of the g-algebra + (so, it is a session hash, but no stable hash). It is possible to + destroy uniqueness of g-algebras on purpose, but that's your own + problem if you do those things. EXAMPLES:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - - sage: P == P - True - sage: Q = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: Q == P - True - - sage: from sage.matrix.constructor import Matrix - sage: c = Matrix(3) - sage: c[0,1] = -1 - sage: c[0,2] = 1 - sage: c[1,2] = 1 - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: R. = NCPolynomialRing_plural(QQ, 3, c = c, d = Matrix(3), order='lex') - sage: R == P - True - - sage: c[0,1] = -2 - sage: R. = NCPolynomialRing_plural(QQ, 3, c = c, d = Matrix(3), order='lex') - sage: P == R - False + sage: A. = FreeAlgebra(QQ, 3) + sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') + sage: {P:2}[P] # indirect doctest + 2 + """ - - if PY_TYPE_CHECK(right, NCPolynomialRing_plural): - - return cmp( (self.base_ring(), map(str, self.gens()), - self.term_order(), self._c, self._d), - (right.base_ring(), map(str, right.gens()), - right.term_order(), - (right)._c, - (right)._d) - ) - else: - return cmp(type(self),type(right)) + return id(self) def __pow__(self, n, _): """ Return the free module of rank `n` over this ring. + NOTE: + + This is not properly implemented yet. Thus, there is + a warning. + EXAMPLES:: + sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z - - sage: f = x^3 + y - sage: f^2 - x^6 + y^2 + sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex') + sage: P^3 + d...: UserWarning: You are constructing a free module + over a noncommutative ring. Sage does not have a concept + of left/right and both sided modules, so be careful. + It's also not guaranteed that all multiplications are + done from the right side. + d...: UserWarning: You are constructing a free module + over a noncommutative ring. Sage does not have a concept + of left/right and both sided modules, so be careful. + It's also not guaranteed that all multiplications are + done from the right side. + Ambient free module of rank 3 over Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} + """ import sage.modules.all return sage.modules.all.FreeModule(self, n) @@ -653,14 +607,14 @@ EXAMPLES:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.term_order() - Lexicographic term order - - sage: P = A.g_algebra(relations={y*x:-x*y}) - sage: P.term_order() - Degree reverse lexicographic term order + sage: A. = FreeAlgebra(QQ, 3) + sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') + sage: P.term_order() + Lexicographic term order + + sage: P = A.g_algebra(relations={y*x:-x*y}) + sage: P.term_order() + Degree reverse lexicographic term order """ return self.__term_order @@ -668,12 +622,14 @@ """ Return False. + .. todo:: Provide a mathematically correct answer. + EXAMPLES:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.is_commutative() - False + sage: A. = FreeAlgebra(QQ, 3) + sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') + sage: P.is_commutative() + False """ return False @@ -683,22 +639,20 @@ EXAMPLES:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.is_field() - False + sage: A. = FreeAlgebra(QQ, 3) + sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') + sage: P.is_field() + False """ return False def _repr_(self): """ - EXAMPLE: - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: from sage.matrix.constructor import Matrix - sage: c=Matrix(2) - sage: c[0,1]=-1 - sage: P. = NCPolynomialRing_plural(QQ, 2, c=c, d=Matrix(2)) - sage: P # indirect doctest + EXAMPLE:: + + sage: A. = FreeAlgebra(QQ, 2) + sage: H. = A.g_algebra({y*x:-x*y}) + sage: H # indirect doctest Noncommutative Multivariate Polynomial Ring in x, y over Rational Field, nc-relations: {y*x: -x*y} sage: x*y x*y @@ -715,39 +669,66 @@ Return an internal list representation of the noncummutative ring. EXAMPLES:: - sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P._ringlist() - [0, ['x', 'y', 'z'], [['lp', (1, 1, 1)], ['C', (0,)]], [0], [ 0 -1 1] - [ 0 0 1] - [ 0 0 0], [0 0 0] - [0 0 0] - [0 0 0]] + + sage: A. = FreeAlgebra(QQ, 3) + sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') + sage: P._ringlist() + [0, ['x', 'y', 'z'], [['lp', (1, 1, 1)], ['C', (0,)]], [0], [ 0 -1 1] + [ 0 0 1] + [ 0 0 0], [0 0 0] + [0 0 0] + [0 0 0]] """ cdef ring* _ring = self._ring if(_ring != currRing): rChangeCurrRing(_ring) from sage.libs.singular.function import singular_function ringlist = singular_function('ringlist') result = ringlist(self, ring=self) - - - - return result def relations(self, add_commutative = False): """ + Return the relations of this g-algebra. + + INPUT: + + ``add_commutative`` (optional bool, default False) + + OUTPUT: + + The defining relations. There are some implicit relations: + Two generators commute if they are not part of any given + relation. The implicit relations are not provided, unless + ``add_commutative==True``. + EXAMPLE:: - sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural - sage: from sage.matrix.constructor import Matrix - sage: c=Matrix(2) - sage: c[0,1]=-1 - sage: P = NCPolynomialRing_plural(QQ, 2, 'x,y', c=c, d=Matrix(2)) - sage: P # indirect doctest - Noncommutative Multivariate Polynomial Ring in x, y over Rational Field, nc-relations: ... + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y}) + sage: x*y == y*x + True + sage: H.relations() + {z*y: y*z - 2*y, z*x: x*z + 2*x} + sage: H.relations(add_commutative=True) + {y*x: x*y, z*y: y*z - 2*y, z*x: x*z + 2*x} + """ + if add_commutative: + if self._relations_commutative is not None: + return self._relations_commutative + + from sage.algebras.free_algebra import FreeAlgebra + A = FreeAlgebra( self.base_ring(), self.ngens(), self.gens() ) + + res = {} + n = self.ngens() + for r in range(0, n-1, 1): + for c in range(r+1, n, 1): + res[ A.gen(c) * A.gen(r) ] = self.gen(c) * self.gen(r) # C[r, c] * P.gen(r) * P.gen(c) + D[r, c] + self._relations_commutative = res + return res + if self._relations is not None: return self._relations @@ -758,10 +739,9 @@ n = self.ngens() for r in range(0, n-1, 1): for c in range(r+1, n, 1): - if (self.gen(c) * self.gen(r) != self.gen(r) * self.gen(c)) or add_commutative: + if (self.gen(c) * self.gen(r) != self.gen(r) * self.gen(c)): res[ A.gen(c) * A.gen(r) ] = self.gen(c) * self.gen(r) # C[r, c] * P.gen(r) * P.gen(c) + D[r, c] - - + self._relations = res return self._relations @@ -772,10 +752,7 @@ EXAMPLES:: sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z - + sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex') sage: P.ngens() 3 """ @@ -797,8 +774,11 @@ sage: P.gen(),P.gen(1) (x, y) - sage: P.gen(1) - y + Note that the generators are not cached:: + + sage: P.gen(1) is P.gen(1) + False + """ cdef poly *_p cdef ring *_ring = self._ring @@ -820,22 +800,24 @@ INPUT: - ``*gens`` - list or tuple of generators (or several input arguments) - - ``coerce`` - bool (default: ``True``); this must be a keyword argument. Only set it to ``False`` if you are certain that each generator is already in the ring. + - ``side`` - string (either "left", which is the default, or "twosided") + Must be a keyword argument. Defines whether the ideal is a left ideal + or a two-sided ideal. Right ideals are not implemented. EXAMPLES:: + sage: A. = FreeAlgebra(QQ, 3) - sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') - sage: P.inject_variables() - Defining x, y, z + sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex') sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x]) - Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} + Left Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} + sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x], side="twosided") + Twosided Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} + """ -# from sage.rings.polynomial.multi_polynomial_ideal import \ -# NCPolynomialIdeal coerce = kwds.get('coerce', True) if len(gens) == 1: gens = gens[0] @@ -872,6 +854,7 @@ ring = singular_function('ring') return ring(L, ring=self) +# TODO: Implement this properly! # def quotient(self, I): # """ # Construct quotient ring of ``self`` and the two-sided Groebner basis of `ideal` @@ -943,6 +926,7 @@ 2/3 TESTS:: + sage: A. = FreeAlgebra(QQ, 3) sage: R = A.g_algebra(relations={y*x:-x*y}, order='lex') sage: R.inject_variables() @@ -1037,6 +1021,7 @@ False TESTS:: + sage: A. = FreeAlgebra(QQ, 3) sage: Q = A.g_algebra(relations={y*x:-x*y}, order='lex') sage: Q.inject_variables() @@ -1164,6 +1149,7 @@ (y, 1/4*x*y + 2/7) TESTS:: + sage: A. = FreeAlgebra(QQ, 3) sage: Q = A.g_algebra(relations={y*x:-x*y}, order='lex') sage: Q.inject_variables() @@ -1314,6 +1300,18 @@ return M def unpickle_NCPolynomial_plural(NCPolynomialRing_plural R, d): + """ + Auxiliary function to unpickle a non-commutative polynomial. + + TEST:: + + sage: A. = FreeAlgebra(QQ, 3) + sage: H. = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) + sage: p = x*y+2*z+4*x*y*z*x + sage: loads(dumps(p)) == p # indirect doctest + True + + """ cdef ring *r = R._ring cdef poly *m, *p cdef int _i, _e @@ -1703,7 +1701,10 @@ sage: (x*z).reduce(I) -x - sage: I.twostd() + + The Groebner basis shows that the result is correct:: + + sage: I.std() Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: x*y - z, z*y: y*z - 2*y, z*x: x*z + 2*x} @@ -1833,7 +1834,8 @@ degree, which is the maximum degree of any monomial. OUTPUT: - integer + + integer EXAMPLES:: @@ -1970,7 +1972,8 @@ - a monomial (very fast, but not as flexible) OUTPUT: - element of the parent of this element. + + element of the parent of this element. .. note:: @@ -2089,10 +2092,12 @@ - ``mon`` - a monomial OUTPUT: - coefficient in base ring + + coefficient in base ring SEE ALSO: - For coefficients in a base ring of fewer variables, look at ``coefficient``. + + For coefficients in a base ring of fewer variables, look at ``coefficient``. EXAMPLES:: @@ -2212,7 +2217,7 @@ INPUT: - ``x`` - a tuple or, in case of a single-variable MPolynomial - ring x can also be an integer. + ring x can also be an integer. EXAMPLES:: @@ -2276,7 +2281,7 @@ INPUT: - ``as_ETuples`` - (default: ``True``) if true returns the result as an list of ETuples - otherwise returns a list of tuples + otherwise returns a list of tuples EXAMPLES:: @@ -2660,7 +2665,11 @@ EXAMPLES:: - + sage: A. = FreeAlgebra(QQ, 3) + sage: H = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y}) + sage: H.gen(2) # indirect doctest + z + """ cdef NCPolynomial_plural p = PY_NEW(NCPolynomial_plural) p._parent = parent @@ -2676,6 +2685,7 @@ Construct MPolynomialRing_libsingular from ringWrap, assumming the ground field to be base_ring EXAMPLES:: + sage: H. = PolynomialRing(QQ, 3) sage: from sage.libs.singular.function import singular_function @@ -2720,7 +2730,6 @@ Construct NCPolynomialRing_plural from ringWrap, assumming the ground field to be base_ring EXAMPLES:: - EXAMPLES:: sage: A. = FreeAlgebra(QQ, 3) sage: H = A.g_algebra({y*x:x*y-1}) @@ -2872,9 +2881,12 @@ if (r in alt_vars) and (c in alt_vars): relations[ A.gen(c) * A.gen(r) ] = - A.gen(r) * A.gen(c) - H = A.g_algebra(relations=relations, order=order) + cdef NCPolynomialRing_plural H = A.g_algebra(relations=relations, order=order) I = H.ideal([H.gen(i) *H.gen(i) for i in alt_vars]).twostd() - return H.quotient(I) + L = H._ringlist() + L[3] = I + W = H._list_to_ring(L) + return new_NRing(W, H.base_ring()) cdef poly *addwithcarry(poly *tempvector, poly *maxvector, int pos, ring *_ring): if p_GetExp(tempvector, pos, _ring) < p_GetExp(maxvector, pos, _ring):