o 8Va$*@s>dZddlmZddlmZGdddZGdddeZdS) z-Computations with ideals of polynomial rings.)reduce)CoercionFailedc@seZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/Zd0d1Zd2d3Zd4d5ZeZd6d7ZeZ d8d9Z!d:d;Z"dS)?Ideala 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 cCt)z&Implementation of element containment.NotImplementedErrorselfxr 9/usr/lib/python3/dist-packages/sympy/polys/agca/ideals.py_contains_elem+zIdeal._contains_elemcCr)z$Implementation of ideal containment.r)r Ir r r _contains_ideal/rzIdeal._contains_idealcCr)z!Implementation of ideal quotient.rr Jr r r _quotient3rzIdeal._quotientcCr)z%Implementation of ideal intersection.rrr r r _intersect7rzIdeal._intersectcCr)z*Return True if ``self`` is the whole ring.rr r r r is_whole_ring;rzIdeal.is_whole_ringcCr)z*Return True if ``self`` is the zero ideal.rrr r r is_zero?rz Ideal.is_zerocCs||o ||S)z!Implementation of ideal equality.)rrr r r _equalsCsz Ideal._equalscCr)z)Return True if ``self`` is a prime ideal.rrr r r is_primeGrzIdeal.is_primecCr)z+Return True if ``self`` is a maximal ideal.rrr r r is_maximalKrzIdeal.is_maximalcCr)z+Return True if ``self`` is a radical ideal.rrr r r is_radicalOrzIdeal.is_radicalcCr)z+Return True if ``self`` is a primary ideal.rrr r r is_primarySrzIdeal.is_primarycCr)z-Return True if ``self`` is a principal ideal.rrr r r is_principalWrzIdeal.is_principalcCr)z Compute the radical of ``self``.rrr r r radical[rz Ideal.radicalcCr)zCompute the depth of ``self``.rrr r r depth_rz Ideal.depthcCr)zCompute the height of ``self``.rrr r r heightcrz Ideal.heightcCs ||_dSN)ring)r r"r r r __init__k zIdeal.__init__cCs,t|tr |j|jkrtd|j|fdS)z.Helper to check ``J`` is an ideal of our ring.z J must be an ideal of %s, got %sN) isinstancerr" ValueErrorrr r r _check_idealns  zIdeal._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"convert)r elemr r r containstszIdeal.containscs*t|tr |Stfdd|DS)a 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|]}|VqdSr!)r .0r rr r zIdeal.subset..)r%rrall)r otherr rr subsets  z Ideal.subsetcKs|||j|fi|S)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'rr rZoptsr r r quotients zIdeal.quotientcC||||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 intersects zIdeal.intersectcCr)z 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\}`. rrr r r saturatezIdeal.saturatecCr4)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'_unionrr r r unions z Ideal.unioncCr4)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'_productrr r r products  z Ideal.productcCs|S)z 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 reduce_elementr7zIdeal.reduce_elementcCsZt|ts#|j|}t||jr|St||jjr||S||S||||Sr!)r%rr"Z quotient_ringZdtyper(r'r9)r eRr r r __add__s      z Ideal.__add__cCsFt|tsz|j|}Wn tytYSw||||Sr!)r%rr"idealrNotImplementedr'r;r r=r r r __mul__s    z Ideal.__mul__cCs*|dkrttdd|g||jdS)NrcSs||Sr!r )r yr r r szIdeal.__pow__..)rrr"r@)r Zexpr r r __pow__ sz Ideal.__pow__cCs$t|tr |j|jkr dS||S)NF)r%rr"rrBr r r __eq__s z Ideal.__eq__cCs ||k Sr!r rBr r r __ne__r$z Ideal.__ne__N)$__name__ __module__ __qualname____doc__r rrrrrrrrrrrrrr r#r'r*r1r3r5r6r9r;r<r?__radd__rC__rmul__rGrHrIr r r r rsD"     rc@s|eZdZdZddZddZddZdd Zd d Zd d Z e ddZ ddZ ddZ ddZddZddZddZdS)ModuleImplementedIdealzs Ideal implementation relying on the modules code. Attributes: - _module - the underlying module cCst||||_dSr!)rr#_module)r r"moduler r r r#"s  zModuleImplementedIdeal.__init__cC|j|gSr!)rQr*rr r r r &sz%ModuleImplementedIdeal._contains_elemcCst|tst|j|jSr!)r%rPrrQZ is_submodulerr r r r)s z&ModuleImplementedIdeal._contains_idealcC&t|tst||j|j|jSr!)r%rPr __class__r"rQr5rr r r r. z!ModuleImplementedIdeal._intersectcKs$t|tst|jj|jfi|Sr!)r%rPrrQZmodule_quotientr2r r r r3s z ModuleImplementedIdeal._quotientcCrTr!)r%rPrrUr"rQr9rr r r r88rVzModuleImplementedIdeal._unioncCsdd|jjDS)z Return generators for ``self``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x, y >>> list(QQ.old_poly_ring(x, y).ideal(x, y, x**2 + y).gens) [x, y, x**2 + y] css|]}|dVqdS)rNr r+r r r r-Jsz.ModuleImplementedIdeal.gens..rQgensrr r r rX=s zModuleImplementedIdeal.genscC |jS)a% Return True if ``self`` is the zero ideal. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x).is_zero() False >>> QQ.old_poly_ring(x).ideal().is_zero() True )rQrrr r r rLs zModuleImplementedIdeal.is_zerocCrY)a Return True if ``self`` is the whole ring, i.e. one generator is a unit. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ, ilex >>> QQ.old_poly_ring(x).ideal(x).is_whole_ring() False >>> QQ.old_poly_ring(x).ideal(3).is_whole_ring() True >>> QQ.old_poly_ring(x, order=ilex).ideal(2 + x).is_whole_ring() True )rQZis_full_modulerr r r r\s z$ModuleImplementedIdeal.is_whole_ringcs0ddlmddfdd|jjDdS)Nrsstr<,c3s|]\}|VqdSr!r r+rZr r r-pr.z2ModuleImplementedIdeal.__repr__..>)Zsympyr[joinrQrXrr rZr __repr__ns $zModuleImplementedIdeal.__repr__cs6ttst||j|jjfdd|jjDS)Ncs(g|]\}jjD]\}||gq qSr rW)r,r rDrr r ws(z3ModuleImplementedIdeal._product..)r%rPrrUr"rQZ submodulerXrr rar r:ss zModuleImplementedIdeal._productcCrS)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] )rQin_terms_of_generatorsrBr r r rcys z-ModuleImplementedIdeal.in_terms_of_generatorscKs|jj|gfi|dS)Nr)rQr<)r r optionsr r r r<sz%ModuleImplementedIdeal.reduce_elementN)rJrKrLrMr#r rrrr8propertyrXrrr`r:rcr<r r r r rPs   rPN)rM functoolsrZsympy.polys.polyerrorsrrrPr r r r s