ó ¡¼™\c@ sldZddlmZmZddlmZddlmZdefd„ƒYZ de fd„ƒYZ d S( s-Computations with ideals of polynomial rings.iÿÿÿÿ(tprint_functiontdivision(treduce(tCoercionFailedtIdealcB s(eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„ZeZd„ZeZd„Z d„Z!d„Z"RS(sŠ Abstract base class for ideals. Do not instantiate - use explicit constructors in the ring class instead: >>> from sympy import QQ >>> from sympy.abc import x >>> QQ.old_poly_ring(x).ideal(x+1) Attributes - ring - the ring this ideal belongs to Non-implemented methods: - _contains_elem - _contains_ideal - _quotient - _intersect - _union - _product - is_whole_ring - is_zero - is_prime, is_maximal, is_primary, is_radical - is_principal - height, depth - radical Methods that likely should be overridden in subclasses: - reduce_element cC s t‚dS(s&Implementation of element containment.N(tNotImplementedError(tselftx((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt_contains_elem,scC s t‚dS(s$Implementation of ideal containment.N(R(RtI((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt_contains_ideal0scC s t‚dS(s!Implementation of ideal quotient.N(R(RtJ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt _quotient4scC s t‚dS(s%Implementation of ideal intersection.N(R(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt _intersect8scC s t‚dS(s*Return True if ``self`` is the whole ring.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt is_whole_ring<scC s t‚dS(s*Return True if ``self`` is the zero ideal.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytis_zero@scC s|j|ƒo|j|ƒS(s!Implementation of ideal equality.(R (RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt_equalsDscC s t‚dS(s)Return True if ``self`` is a prime ideal.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytis_primeHscC s t‚dS(s+Return True if ``self`` is a maximal ideal.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt is_maximalLscC s t‚dS(s+Return True if ``self`` is a radical ideal.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt is_radicalPscC s t‚dS(s+Return True if ``self`` is a primary ideal.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt is_primaryTscC s t‚dS(s-Return True if ``self`` is a principal ideal.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt is_principalXscC s t‚dS(s Compute the radical of ``self``.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytradical\scC s t‚dS(sCompute the depth of ``self``.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytdepth`scC s t‚dS(sCompute the height of ``self``.N(R(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytheightdscC s ||_dS(N(tring(RR((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__init__lscC sBt|tƒ s"|j|jkr>td|j|fƒ‚ndS(s.Helper to check ``J`` is an ideal of our ring.s J must be an ideal of %s, got %sN(t isinstanceRRt ValueError(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt _check_idealos"cC s|j|jj|ƒƒS(sD Return True if ``elem`` is an element of this ideal. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x+1, x-1).contains(3) True >>> QQ.old_poly_ring(x).ideal(x**2, x**3).contains(x) False (RRtconvert(Rtelem((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytcontainsusc s6t|tƒrˆj|ƒSt‡fd†|DƒƒS(sà Returns True if ``other`` is is a subset of ``self``. Here ``other`` may be an ideal. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> I = QQ.old_poly_ring(x).ideal(x+1) >>> I.subset([x**2 - 1, x**2 + 2*x + 1]) True >>> I.subset([x**2 + 1, x + 1]) False >>> I.subset(QQ.old_poly_ring(x).ideal(x**2 - 1)) True c3 s|]}ˆj|ƒVqdS(N(R(t.0R(R(s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pys šs(RRR tall(Rtother((Rs6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytsubset…s cK s|j|ƒ|j||S(s~ Compute the ideal quotient of ``self`` by ``J``. That is, if ``self`` is the ideal `I`, compute the set `I : J = \{x \in R | xJ \subset I \}`. Examples ======== >>> from sympy.abc import x, y >>> from sympy import QQ >>> R = QQ.old_poly_ring(x, y) >>> R.ideal(x*y).quotient(R.ideal(x)) (RR (RR topts((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytquotientœs cC s|j|ƒ|j|ƒS(s Compute the intersection of self with ideal J. Examples ======== >>> from sympy.abc import x, y >>> from sympy import QQ >>> R = QQ.old_poly_ring(x, y) >>> R.ideal(x).intersect(R.ideal(y)) (RR (RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt intersect¯s cC s t‚dS(sÏ Compute the ideal saturation of ``self`` by ``J``. That is, if ``self`` is the ideal `I`, compute the set `I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}`. N(R(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytsaturate¿scC s|j|ƒ|j|ƒS(sD Compute the ideal generated by the union of ``self`` and ``J``. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x**2 - 1).union(QQ.old_poly_ring(x).ideal((x+1)**2)) == QQ.old_poly_ring(x).ideal(x+1) True (Rt_union(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytunionÉs cC s|j|ƒ|j|ƒS(s‡ Compute the ideal product of ``self`` and ``J``. That is, compute the ideal generated by products `xy`, for `x` an element of ``self`` and `y \in J`. Examples ======== >>> from sympy.abc import x, y >>> from sympy import QQ >>> QQ.old_poly_ring(x, y).ideal(x).product(QQ.old_poly_ring(x, y).ideal(y)) (Rt_product(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytproductØs cC s|S(sâ Reduce the element ``x`` of our ring modulo the ideal ``self``. Here "reduce" has no specific meaning: it could return a unique normal form, simplify the expression a bit, or just do nothing. ((RR((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytreduce_elementêscC s}t|tƒsc|jj|ƒ}t||jƒr7|St||jjƒrV||ƒS|j|ƒS|j|ƒ|j|ƒS(N(RRRt quotient_ringtdtypeRRR*(RtetR((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__add__ós   cC sWt|tƒs=y|jj|ƒ}Wq=tk r9tSXn|j|ƒ|j|ƒS(N(RRRtidealRtNotImplementedRR,(RR0((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__mul__s  cC s;|dkrt‚ntd„|g||jjdƒƒS(NicS s||S(N((Rty((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pytti(RRRR3(Rtexp((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__pow__ s  cC s3t|tƒ s"|j|jkr&tS|j|ƒS(N(RRRtFalseR(RR0((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__eq__s"cC s ||k S(N((RR0((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__ne__s(#t__name__t __module__t__doc__RR R R RRRRRRRRRRRRRR R$R&R'R(R*R,R-R2t__radd__R5t__rmul__R:R<R=(((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR sB!                           tModuleImplementedIdealcB s‰eZdZd„Zd„Zd„Zd„Zd„Zd„Ze d„ƒZ d„Z d „Z d „Z d „Zd „Zd „ZRS(ss Ideal implementation relying on the modules code. Attributes: - _module - the underlying module cC stj||ƒ||_dS(N(RRt_module(RRtmodule((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR#scC s|jj|gƒS(N(RDR (RR((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR'scC s+t|tƒst‚n|jj|jƒS(N(RRCRRDt is_submodule(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR *s cC s:t|tƒst‚n|j|j|jj|jƒƒS(N(RRCRt __class__RRDR'(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR /s cK s.t|tƒst‚n|jj|j|S(N(RRCRRDtmodule_quotient(RR R%((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR 4s cC s:t|tƒst‚n|j|j|jj|jƒƒS(N(RRCRRGRRDR*(RR ((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR)9s cC sd„|jjDƒS(sú Return generators for ``self``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x, y >>> list(QQ.old_poly_ring(x, y).ideal(x, y, x**2 + y).gens) [x, y, x**2 + y] cs s|]}|dVqdS(iN((R!R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pys Ks(RDtgens(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyRI>s cC s |jjƒS(s% Return True if ``self`` is the zero ideal. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x).is_zero() False >>> QQ.old_poly_ring(x).ideal().is_zero() True (RDR(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyRMscC s |jjƒS(s¬ Return True if ``self`` is the whole ring, i.e. one generator is a unit. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ, ilex >>> QQ.old_poly_ring(x).ideal(x).is_whole_ring() False >>> QQ.old_poly_ring(x).ideal(3).is_whole_ring() True >>> QQ.old_poly_ring(x, order=ilex).ideal(2 + x).is_whole_ring() True (RDtis_full_module(R((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR]sc s;ddlm‰ddj‡fd†|jjDƒƒdS(Niÿÿÿÿ(tsstrtqst>(tsympyRKtjoinRDRI(R((RKs6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyt__repr__oscC spt|tƒst‚n|j|j|jjg|jjD],\}|jjD]\}||g^qMq:ŒƒS(N(RRCRRGRRDt submoduleRI(RR RR6((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR+ts cC s|jj|gƒS(s Express ``e`` in terms of the generators of ``self``. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> I = QQ.old_poly_ring(x).ideal(x**2 + 1, x) >>> I.in_terms_of_generators(1) [1, -x] (RDtin_terms_of_generators(RR0((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyRSzs cK s|jj|g|dS(Ni(RDR-(RRtoptions((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyR-‰s(R>R?R@RRR R R R)tpropertyRIRRRQR+RSR-(((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyRCs           N( R@t __future__RRtsympy.core.compatibilityRtsympy.polys.polyerrorsRtobjectRRC(((s6lib/python2.7/site-packages/sympy/polys/agca/ideals.pyts ÿ