U L?hA+@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 cCstdS)z&Implementation of element containment.NNotImplementedErrorselfxr I/var/www/html/venv/lib/python3.8/site-packages/sympy/polys/agca/ideals.py_contains_elem*szIdeal._contains_elemcCstdS)z$Implementation of ideal containment.Nr)rIr r r _contains_ideal.szIdeal._contains_idealcCstdS)z!Implementation of ideal quotient.NrrJr r r _quotient2szIdeal._quotientcCstdS)z%Implementation of ideal intersection.Nrrr r r _intersect6szIdeal._intersectcCstdS)z*Return True if ``self`` is the whole ring.Nrrr r r is_whole_ring:szIdeal.is_whole_ringcCstdS)z*Return True if ``self`` is the zero ideal.Nrrr r r is_zero>sz Ideal.is_zerocCs||o||S)z!Implementation of ideal equality.)rrr r r _equalsBsz Ideal._equalscCstdS)z)Return True if ``self`` is a prime ideal.Nrrr r r is_primeFszIdeal.is_primecCstdS)z+Return True if ``self`` is a maximal ideal.Nrrr r r is_maximalJszIdeal.is_maximalcCstdS)z+Return True if ``self`` is a radical ideal.Nrrr r r is_radicalNszIdeal.is_radicalcCstdS)z+Return True if ``self`` is a primary ideal.Nrrr r r is_primaryRszIdeal.is_primarycCstdS)z-Return True if ``self`` is a principal ideal.Nrrr r r is_principalVszIdeal.is_principalcCstdS)z Compute the radical of ``self``.Nrrr r r radicalZsz Ideal.radicalcCstdS)zCompute the depth of ``self``.Nrrr r r depth^sz Ideal.depthcCstdS)zCompute the height of ``self``.Nrrr r r heightbsz Ideal.heightcCs ||_dSN)ring)rr r r r __init__jszIdeal.__init__cCs,t|tr|j|jkr(td|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)relemr 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)rotherr rr subsets  z Ideal.subsetcKs|||j|f|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$rrroptsr r r quotients 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$rrr r r intersects zIdeal.intersectcCstdS)z Compute the ideal saturation of ``self`` by ``J``. That is, if ``self`` is the ideal `I`, compute the set `I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}`. Nrrr r r saturateszIdeal.saturatecCs||||S)aD Compute the ideal generated by the union of ``self`` and ``J``. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x**2 - 1).union(QQ.old_poly_ring(x).ideal((x+1)**2)) == QQ.old_poly_ring(x).ideal(x+1) True )r$_unionrr r r unions 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$_productrr r r products z Ideal.productcCs|S)z Reduce the element ``x`` of our ring modulo the ideal ``self``. Here "reduce" has no specific meaning: it could return a unique normal form, simplify the expression a bit, or just do nothing. r rr r r reduce_elementszIdeal.reduce_elementcCsZt|tsF|j|}t||jr&|St||jjr<||S||S||||Sr)r"rr Z quotient_ringZdtyper%r$r4)reRr r r __add__s     z Ideal.__add__cCsHt|ts4z|j|}Wntk r2tYSX||||Sr)r"rr idealrNotImplementedr$r6rr8r r r __mul__s   z Ideal.__mul__cCs |jdSN)r r;rr r r _zeroth_power szIdeal._zeroth_powercCs|dSr?r rr r r _first_power szIdeal._first_powercCs$t|tr|j|jkrdS||S)NF)r"rr rr=r r r __eq__sz Ideal.__eq__cCs ||k Srr r=r r r __ne__sz Ideal.__ne__N)%__name__ __module__ __qualname____doc__r rrrrrrrrrrrrrrr!r$r'r-r0r1r2r4r6r7r:__radd__r>__rmul__rArBrCrDr r r r rsD"    rc@s|eZdZdZddZddZddZdd Zd d Zd d Z e ddZ ddZ ddZ ddZddZddZddZdS)ModuleImplementedIdealzs Ideal implementation relying on the modules code. Attributes: - _module - the underlying module cCst||||_dSr)rr!_module)rr moduler r r r!#s zModuleImplementedIdeal.__init__cCs|j|gSr)rLr'rr r r r 'sz%ModuleImplementedIdeal._contains_elemcCst|tst|j|jSr)r"rKrrLZ is_submodulerr r r r*s z&ModuleImplementedIdeal._contains_idealcCs&t|tst||j|j|jSr)r"rKr __class__r rLr1rr r r r/s z!ModuleImplementedIdeal._intersectcKs t|tst|jj|jf|Sr)r"rKrrLZmodule_quotientr.r r r r4s z ModuleImplementedIdeal._quotientcCs&t|tst||j|j|jSr)r"rKrrNr rLr4rr r r r39s zModuleImplementedIdeal._unioncCsdd|jjDS)aB Return generators for ``self``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x, y >>> list(QQ.old_poly_ring(x, y).ideal(x, y, x**2 + y).gens) [DMP_Python([[1], []], QQ), DMP_Python([[1, 0]], QQ), DMP_Python([[1], [], [1, 0]], QQ)] css|]}|dVqdS)rNr r(r r r r*Ksz.ModuleImplementedIdeal.gens..rLgensrr r r rP>s zModuleImplementedIdeal.genscCs |jS)a% Return True if ``self`` is the zero ideal. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ >>> QQ.old_poly_ring(x).ideal(x).is_zero() False >>> QQ.old_poly_ring(x).ideal().is_zero() True )rLrrr r r rMszModuleImplementedIdeal.is_zerocCs |jS)a Return True if ``self`` is the whole ring, i.e. one generator is a unit. Examples ======== >>> from sympy.abc import x >>> from sympy import QQ, ilex >>> QQ.old_poly_ring(x).ideal(x).is_whole_ring() False >>> QQ.old_poly_ring(x).ideal(3).is_whole_ring() True >>> QQ.old_poly_ring(x, order=ilex).ideal(2 + x).is_whole_ring() True )rLZis_full_modulerr r r r]sz$ModuleImplementedIdeal.is_whole_ringcsBddlmfddjjD}ddfdd|Dd S) Nrsstrcsg|]\}j|qSr )r Zto_sympyr(rr r qsz3ModuleImplementedIdeal.__repr__..<,c3s|]}|VqdSrr )r)grQr r r*rsz2ModuleImplementedIdeal.__repr__..>)Zsympy.printing.strrRrLrPjoin)rrPr )rrRr __repr__os zModuleImplementedIdeal.__repr__cs6ttst||j|jjfdd|jjDS)Ncs(g|] \}jjD]\}||gqqSr rO)r)r yrr r rSys z3ModuleImplementedIdeal._product..)r"rKrrNr rL submodulerPrr r[r r5us  zModuleImplementedIdeal._productcCs|j|gS)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)] )rLin_terms_of_generatorsr=r r r r]{s z-ModuleImplementedIdeal.in_terms_of_generatorscKs|jj|gf|dS)Nr)rLr7)rr optionsr r r r7sz%ModuleImplementedIdeal.reduce_elementN)rErFrGrHr!r rrrr3propertyrPrrrYr5r]r7r r r r rKs rKN)rHZsympy.polys.polyerrorsrZsympy.polys.polyutilsrrrKr r r r s