a RG5d‹*ã@s@dZddlmZddlmZGdd„deƒZGdd„deƒZdS) z-Computations with ideals of polynomial rings.é)ÚCoercionFailed)ÚIntegerPowerablec@seZdZdZdd„Zdd„Zdd„Zdd „Zd d „Zd d „Z dd„Z dd„Z dd„Z dd„Z dd„Zdd„Zdd„Zdd„Zdd„Zd d!„Zd"d#„Zd$d%„Zd&d'„Zd(d)„Zd*d+„Zd,d-„Zd.d/„Zd0d1„Zd2d3„Zd4d5„ZeZd6d7„ZeZ d8d9„Z!d:d;„Z"dd?„Z$d@S)AÚIdealaŠ 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 cCst‚dS)z&Implementation of element containment.N©ÚNotImplementedError©ÚselfÚx©r úS/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/agca/ideals.pyÚ_contains_elem*szIdeal._contains_elemcCst‚dS)z$Implementation of ideal containment.Nr)rÚIr r r Ú_contains_ideal.szIdeal._contains_idealcCst‚dS)z!Implementation of ideal quotient.Nr©rÚJr r r Ú _quotient2szIdeal._quotientcCst‚dS)z%Implementation of ideal intersection.Nrrr r r Ú _intersect6szIdeal._intersectcCst‚dS)z*Return True if ``self`` is the whole ring.Nr©rr r r Ú is_whole_ring:szIdeal.is_whole_ringcCst‚dS)z*Return True if ``self`` is the zero ideal.Nrrr r r Úis_zero>sz Ideal.is_zerocCs| |¡o| |¡S)z!Implementation of ideal equality.)rrr r r Ú_equalsBsz Ideal._equalscCst‚dS)z)Return True if ``self`` is a prime ideal.Nrrr r r Úis_primeFszIdeal.is_primecCst‚dS)z+Return True if ``self`` is a maximal ideal.Nrrr r r Ú is_maximalJszIdeal.is_maximalcCst‚dS)z+Return True if ``self`` is a radical ideal.Nrrr r r Ú is_radicalNszIdeal.is_radicalcCst‚dS)z+Return True if ``self`` is a primary ideal.Nrrr r r Ú is_primaryRszIdeal.is_primarycCst‚dS)z-Return True if ``self`` is a principal ideal.Nrrr r r Ú is_principalVszIdeal.is_principalcCst‚dS)z Compute the radical of ``self``.Nrrr r r ÚradicalZsz Ideal.radicalcCst‚dS)zCompute the depth of ``self``.Nrrr r r Údepth^sz Ideal.depthcCst‚dS)zCompute the height of ``self``.Nrrr r r Úheightbsz Ideal.heightcCs ||_dS©N)Úring)rr r r r Ú__init__jszIdeal.__init__cCs,t|tƒr|j|jkr(td|j|fƒ‚dS)z.Helper to check ``J`` is an ideal of our ring.z J must be an ideal of %s, got %sN)Ú isinstancerr Ú ValueErrorrr r r Ú _check_idealms ÿzIdeal._check_idealcCs| |j |¡¡S)aD 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 )r r Úconvert)rÚelemr r r ÚcontainssszIdeal.containscs*t|tƒrˆ |¡St‡fdd„|DƒƒS)aà 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 c3s|]}ˆ |¡VqdSr)r ©Ú.0r rr r Ú ˜ózIdeal.subset..)r"rrÚall)rÚotherr rr Úsubsetƒs  z Ideal.subsetcKs| |¡|j|fi|¤ŽS)a~ 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)) )r$r©rrÚoptsr r r Úquotientšs zIdeal.quotientcCs| |¡| |¡S)a 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)) )r$rrr r r Ú intersect­s zIdeal.intersectcCst‚dS)zÏ Compute the ideal saturation of ``self`` by ``J``. That is, if ``self`` is the ideal `I`, compute the set `I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}`. Nrrr r r Úsaturate½szIdeal.saturatecCs| |¡| |¡S)aD Compute the ideal generated by the union of ``self`` and ``J``. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x**2 - 1).union(QQ.old_poly_ring(x).ideal((x+1)**2)) == QQ.old_poly_ring(x).ideal(x+1) True )r$Ú_unionrr r r ÚunionÇs z Ideal.unioncCs| |¡| |¡S)a‡ 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)) )r$Ú_productrr r r ÚproductÖs z Ideal.productcCs|S)zâ Reduce the element ``x`` of our ring modulo the ideal ``self``. Here "reduce" has no specific meaning: it could return a unique normal form, simplify the expression a bit, or just do nothing. r rr r r Úreduce_elementèszIdeal.reduce_elementcCsZt|tƒsF|j |¡}t||jƒr&|St||jjƒr<||ƒS| |¡S| |¡| |¡Sr)r"rr Ú quotient_ringÚdtyper%r$r5)rÚeÚRr r r Ú__add__ñs     z Ideal.__add__cCsFt|tƒs2z|j |¡}Wnty0tYS0| |¡| |¡Sr)r"rr ÚidealrÚNotImplementedr$r7©rr;r r r Ú__mul__þs    z Ideal.__mul__cCs |j d¡S©Né)r r>rr r r Ú _zeroth_power szIdeal._zeroth_powercCs|dSrBr rr r r Ú _first_power szIdeal._first_powercCs$t|tƒr|j|jkrdS| |¡S)NF)r"rr rr@r r r Ú__eq__sz Ideal.__eq__cCs ||k Srr r@r r r Ú__ne__sz Ideal.__ne__N)%Ú__name__Ú __module__Ú __qualname__Ú__doc__r rrrrrrrrrrrrrrr!r$r'r.r1r2r3r5r7r8r=Ú__radd__rAÚ__rmul__rDrErFrGr r r r rsD"    rc@s|eZdZdZdd„Zdd„Zdd„Zdd „Zd d „Zd d „Z e dd„ƒZ dd„Z dd„Z dd„Zdd„Zdd„Zdd„ZdS)ÚModuleImplementedIdealzs Ideal implementation relying on the modules code. Attributes: - _module - the underlying module cCst ||¡||_dSr)rr!Ú_module)rr Úmoduler r r r!#s zModuleImplementedIdeal.__init__cCs|j |g¡Sr)rOr'rr r r r 'sz%ModuleImplementedIdeal._contains_elemcCst|tƒst‚|j |j¡Sr)r"rNrrOÚ is_submodulerr r r r*s z&ModuleImplementedIdeal._contains_idealcCs&t|tƒst‚| |j|j |j¡¡Sr)r"rNrÚ __class__r rOr2rr r r r/s z!ModuleImplementedIdeal._intersectcKs$t|tƒst‚|jj|jfi|¤ŽSr)r"rNrrOÚmodule_quotientr/r r r r4s z ModuleImplementedIdeal._quotientcCs&t|tƒst‚| |j|j |j¡¡Sr)r"rNrrRr rOr5rr r r r49s zModuleImplementedIdeal._unioncCsdd„|jjDƒS)zú 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] css|]}|dVqdS)rNr r(r r r r*Kr+z.ModuleImplementedIdeal.gens..©rOÚgensrr r r rU>s zModuleImplementedIdeal.genscCs |j ¡S)a% 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 )rOrrr r r rMszModuleImplementedIdeal.is_zerocCs |j ¡S)a¬ 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 )rOÚis_full_modulerr r r r]sz$ModuleImplementedIdeal.is_whole_ringcs0ddlm‰dd ‡fdd„|jjDƒ¡dS)Nr©Ússtrú<ú,c3s|]\}ˆ|ƒVqdSrr r(rWr r r*qr+z2ModuleImplementedIdeal.__repr__..ú>)Úsympy.printing.strrXÚjoinrOrUrr rWr Ú__repr__os zModuleImplementedIdeal.__repr__cs6tˆtƒst‚| |j|jj‡fdd„|jjDƒŽ¡S)Ncs(g|] \}ˆjjD]\}||g‘qqSr rT)r)r Úy©rr r Ú xr+z3ModuleImplementedIdeal._product..)r"rNrrRr rOÚ submodulerUrr r`r r6ts  ÿzModuleImplementedIdeal._productcCs|j |g¡S)a 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] )rOÚin_terms_of_generatorsr@r r r rczs z-ModuleImplementedIdeal.in_terms_of_generatorscKs|jj|gfi|¤ŽdS)Nr)rOr8)rr Úoptionsr r r r8‰sz%ModuleImplementedIdeal.reduce_elementN)rHrIrJrKr!r rrrr4ÚpropertyrUrrr^r6rcr8r r r r rNs rNN)rKÚsympy.polys.polyerrorsrÚsympy.polys.polyutilsrrrNr r r r Ús