B ˜‘[)ã@sPdZddlmZmZddlmZddlmZGdd„deƒZ Gdd„de ƒZ d S) z-Computations with ideals of polynomial rings.é)Úprint_functionÚdivision)ÚCoercionFailed)Úreducec@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$d%„Zd&d'„Zd(d)„Zd*d+„Zd,d-„Zd.d/„Zd0d1„Zd2d3„Zd4d5„ZeZd6d7„ZeZ d8d9„Z!d:d;„Z"dS)?ÚIdealaŠ Abstract base class for ideals. Do not instantiate - use explicit constructors in the ring class instead: >>> from sympy import QQ >>> from sympy.abc import x >>> QQ.old_poly_ring(x).ideal(x+1) Attributes - ring - the ring this ideal belongs to Non-implemented methods: - _contains_elem - _contains_ideal - _quotient - _intersect - _union - _product - is_whole_ring - is_zero - is_prime, is_maximal, is_primary, is_radical - is_principal - height, depth - radical Methods that likely should be overridden in subclasses: - reduce_element cCst‚dS)z&Implementation of element containment.N)ÚNotImplementedError)ÚselfÚx©r ú6lib/python3.7/site-packages/sympy/polys/agca/ideals.pyÚ_contains_elem-szIdeal._contains_elemcCst‚dS)z$Implementation of ideal containment.N)r)rÚIr r r Ú_contains_ideal1szIdeal._contains_idealcCst‚dS)z!Implementation of ideal quotient.N)r)rÚJr r r Ú _quotient5szIdeal._quotientcCst‚dS)z%Implementation of ideal intersection.N)r)rrr r r Ú _intersect9szIdeal._intersectcCst‚dS)z*Return True if ``self`` is the whole ring.N)r)rr r r Ú is_whole_ring=szIdeal.is_whole_ringcCst‚dS)z*Return True if ``self`` is the zero ideal.N)r)rr r r Úis_zeroAsz Ideal.is_zerocCs| |¡o| |¡S)z!Implementation of ideal equality.)r)rrr r r Ú_equalsEsz Ideal._equalscCst‚dS)z)Return True if ``self`` is a prime ideal.N)r)rr r r Úis_primeIszIdeal.is_primecCst‚dS)z+Return True if ``self`` is a maximal ideal.N)r)rr r r Ú is_maximalMszIdeal.is_maximalcCst‚dS)z+Return True if ``self`` is a radical ideal.N)r)rr r r Ú is_radicalQszIdeal.is_radicalcCst‚dS)z+Return True if ``self`` is a primary ideal.N)r)rr r r Ú is_primaryUszIdeal.is_primarycCst‚dS)z-Return True if ``self`` is a principal ideal.N)r)rr r r Ú is_principalYszIdeal.is_principalcCst‚dS)z Compute the radical of ``self``.N)r)rr r r Úradical]sz Ideal.radicalcCst‚dS)zCompute the depth of ``self``.N)r)rr r r Údepthasz Ideal.depthcCst‚dS)zCompute the height of ``self``.N)r)rr r r Úheightesz Ideal.heightcCs ||_dS)N)Úring)rrr r r Ú__init__mszIdeal.__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 isinstancerrÚ ValueError)rrr r r Ú _check_idealpszIdeal._check_idealcCs| |j |¡¡S)a! Return True if ``elem`` is an element of this ideal. >>> 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 rÚconvert)rÚelemr r r Úcontainsvs zIdeal.containscs*t|tƒrˆ |¡St‡fdd„|DƒƒS)a  Returns True if ``other`` is is a subset of ``self``. Here ``other`` may be an ideal. >>> 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 |¡VqdS)N)r )Ú.0r )rr r ú •szIdeal.subset..)rrrÚall)rÚotherr )rr Úsubsetƒs  z Ideal.subsetcKs| |¡|j|f|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 \}`. >>> 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 Úquotient—s zIdeal.quotientcCs| |¡| |¡S)zë Compute the intersection of self with ideal J. >>> 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!r)rrr r r Ú intersect§s 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\}`. N)r)rrr r r Úsaturate´szIdeal.saturatecCs| |¡| |¡S)a! Compute the ideal generated by the union of ``self`` and ``J``. >>> 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_union)rrr r r Úunion¾s z Ideal.unioncCs| |¡| |¡S)ad 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`. >>> 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_product)rrr r r ÚproductÊs 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 r Úreduce_elementÙszIdeal.reduce_elementcCsZt|tƒsF|j |¡}t||jƒr&|St||jjƒr<||ƒS| |¡S| |¡| |¡S)N)rrrZ quotient_ringZdtyper"r!r/)rÚeÚRr r r Ú__add__âs     z Ideal.__add__cCsDt|tƒs0y|j |¡}Wntk r.tSX| |¡| |¡S)N)rrrÚidealrÚNotImplementedr!r1)rr3r r r Ú__mul__ïs  z Ideal.__mul__cCs*|dkr t‚tdd„|g||j d¡ƒS)NrcSs||S)Nr )r Úyr r r ÚþszIdeal.__pow__..é)rrrr6)rZexpr r r Ú__pow__úsz Ideal.__pow__cCs$t|tƒr|j|jkrdS| |¡S)NF)rrrr)rr3r r r Ú__eq__sz Ideal.__eq__cCs ||k S)Nr )rr3r r r Ú__ne__sz Ideal.__ne__N)$Ú__name__Ú __module__Ú __qualname__Ú__doc__r rrrrrrrrrrrrrrrr!r$r)r+r,r-r/r1r2r5Ú__radd__r8Ú__rmul__r<r=r>r r r r r sB!       rc@s|eZdZdZdd„Zdd„Zdd„Zdd „Zd d „Zd d „Z e dd„ƒZ dd„Z dd„Z dd„Zdd„Zdd„Zdd„ZdS)ÚModuleImplementedIdealzs Ideal implementation relying on the modules code. Attributes: - _module - the underlying module cCst ||¡||_dS)N)rrÚ_module)rrÚmoduler r r rs zModuleImplementedIdeal.__init__cCs|j |g¡S)N)rFr$)rr r r r r sz%ModuleImplementedIdeal._contains_elemcCst|tƒst‚|j |j¡S)N)rrErrFZ is_submodule)rrr r r rs z&ModuleImplementedIdeal._contains_idealcCs&t|tƒst‚| |j|j |j¡¡S)N)rrErÚ __class__rrFr,)rrr r r rs z!ModuleImplementedIdeal._intersectcKs t|tƒst‚|jj|jf|ŽS)N)rrErrFZmodule_quotient)rrr*r r r r#s z ModuleImplementedIdeal._quotientcCs&t|tƒst‚| |j|j |j¡¡S)N)rrErrHrrFr/)rrr r r r.(s zModuleImplementedIdeal._unioncCsdd„|jjDƒS)z× Return generators for ``self``. >>> 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 r&7sz.ModuleImplementedIdeal.gens..)rFÚgens)rr r r rI-s zModuleImplementedIdeal.genscCs |j ¡S)a Return True if ``self`` is the zero ideal. >>> 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 )rFr)rr r r r9s zModuleImplementedIdeal.is_zerocCs |j ¡S)a‰ Return True if ``self`` is the whole ring, i.e. one generator is a unit. >>> 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 )rFZis_full_module)rr r r rFs z$ModuleImplementedIdeal.is_whole_ringcs0ddlm‰dd ‡fdd„|jjDƒ¡dS)Nr)Ússtrú<ú,c3s|]\}ˆ|ƒVqdS)Nr )r%r )rJr r r&Wsz2ModuleImplementedIdeal.__repr__..ú>)ZsympyrJÚjoinrFrI)rr )rJr Ú__repr__Us zModuleImplementedIdeal.__repr__cs6tˆtƒst‚| |j|jj‡fdd„|jjDƒŽ¡S)Ncs(g|] \}ˆjjD]\}||g‘qqSr )rFrI)r%r r9)rr r ú ^sz3ModuleImplementedIdeal._product..)rrErrHrrFZ submodulerI)rrr )rr r0Zs zModuleImplementedIdeal._productcCs|j |g¡S)zü Express ``e`` in terms of the generators of ``self``. >>> 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] )rFÚin_terms_of_generators)rr3r r r rQ`s z-ModuleImplementedIdeal.in_terms_of_generatorscKs|jj|gf|ŽdS)Nr)rFr2)rr Zoptionsr r r r2lsz%ModuleImplementedIdeal.reduce_elementN)r?r@rArBrr rrrr.ÚpropertyrIrrrOr0rQr2r r r r rE s   rEN) rBZ __future__rrZsympy.polys.polyerrorsrZsympy.core.compatibilityrÚobjectrrEr r r r Ús