o jgA+@s@dZddlmZddlmZGdddeZGdddeZdS) z-Computations with ideals of polynomial rings.)CoercionFailed)IntegerPowerablec@seZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/Zd0d1Zd2d3Zd4d5ZeZd6d7ZeZ d8d9Z!d:d;Z"dd?Z$d@S)AIdeala 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 O/var/www/html/zoom/venv/lib/python3.10/site-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 _quotient2rzIdeal._quotientcCr)z%Implementation of ideal intersection.rrr r r _intersect6rzIdeal._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 _equalsBsz Ideal._equalscCr)z)Return True if ``self`` is a prime ideal.rrr r r is_primeFrzIdeal.is_primecCr)z+Return True if ``self`` is a maximal ideal.rrr r r is_maximalJrzIdeal.is_maximalcCr)z+Return True if ``self`` is a radical ideal.rrr r r is_radicalNrzIdeal.is_radicalcCr)z+Return True if ``self`` is a primary ideal.rrr r r is_primaryRrzIdeal.is_primarycCr)z-Return True if ``self`` is a principal ideal.rrr r r is_principalVrzIdeal.is_principalcCr)z Compute the radical of ``self``.rrr r r radicalZrz Ideal.radicalcCr)zCompute the depth of ``self``.rrr r r depth^rz Ideal.depthcCr)zCompute the height of ``self``.rrr r r heightbrz Ideal.heightcCs ||_dSN)ring)r r"r r r __init__j 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_idealms  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 containssszIdeal.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 szIdeal.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 roptsr 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" quotient_ringdtyper(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 |jdSN)r"rBrr r r _zeroth_power s zIdeal._zeroth_powercCs|dSrFr rr r r _first_power szIdeal._first_powercCs$t|tr |j|jkr dS||S)NF)r%rr"rrDr r r __eq__s z Ideal.__eq__cCs ||k Sr!r rDr r r __ne__r$z Ideal.__ne__N)%__name__ __module__ __qualname____doc__r rrrrrrrrrrrrrr r#r'r*r0r3r5r6r9r;r<rA__radd__rE__rmul__rHrIrJrKr r r r rsF"     rc@s|eZdZdZddZddZddZdd Zd d Zd d Z e ddZ ddZ ddZ ddZddZddZddZdS)ModuleImplementedIdealzs Ideal implementation relying on the modules code. Attributes: - _module - the underlying module cCst||||_dSr!)rr#_module)r r"moduler r r r##s  zModuleImplementedIdeal.__init__cC|j|gSr!)rSr*rr r r r 'sz%ModuleImplementedIdeal._contains_elemcCst|tst|j|jSr!)r%rRrrS is_submodulerr r r r*s z&ModuleImplementedIdeal._contains_idealcC&t|tst||j|j|jSr!)r%rRr __class__r"rSr5rr r r r/ z!ModuleImplementedIdeal._intersectcKs$t|tst|jj|jfi|Sr!)r%rRrrSmodule_quotientr1r r r r4s z ModuleImplementedIdeal._quotientcCrWr!)r%rRrrXr"rSr9rr r r r89rYzModuleImplementedIdeal._unioncCsdd|jjDS)aB 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) [DMP_Python([[1], []], QQ), DMP_Python([[1, 0]], QQ), DMP_Python([[1], [], [1, 0]], QQ)] css|]}|dVqdS)rNr r+r r r r-Kz.ModuleImplementedIdeal.gens..rSgensrr r r r]>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 )rSrrr r r rMs zModuleImplementedIdeal.is_zerocCr^)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 )rSis_full_modulerr r r r]s z$ModuleImplementedIdeal.is_whole_ringcsBddlmfddjjD}ddfdd|Dd S) Nrsstrcsg|] \}j|qSr )r"to_sympyr+rr r qsz3ModuleImplementedIdeal.__repr__..<,c3s|]}|VqdSr!r )r,gr`r r r-rr[z2ModuleImplementedIdeal.__repr__..>)sympy.printing.strrarSr]join)r r]r )r rar __repr__os  zModuleImplementedIdeal.__repr__cs6ttst||j|jjfdd|jjDS)Ncs(g|]\}jjD]\}||gq qSr r\)r,r yrr r rcys(z3ModuleImplementedIdeal._product..)r%rRrrXr"rS submoduler]rr rlr r:us zModuleImplementedIdeal._productcCrU)aX 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) # doctest: +SKIP [DMP_Python([1], QQ), DMP_Python([-1, 0], QQ)] )rSin_terms_of_generatorsrDr r r rn{s z-ModuleImplementedIdeal.in_terms_of_generatorscKs|jj|gfi|dS)Nr)rSr<)r r optionsr r r r<sz%ModuleImplementedIdeal.reduce_elementN)rLrMrNrOr#r rrrr8propertyr]rrrjr:rnr<r r r r rRs   rRN)rOsympy.polys.polyerrorsrsympy.polys.polyutilsrrrRr r r r s