wgFUdZddlmZddlmZddlmZmZmZm Z m Z m Z ddl m Z ddlmZddlmZddlmZdd lmZmZdd lmZmZdd lmZdd lmZdd lm Z ddl!m"Z"m#Z#m$Z$ddl%m&Z&ddl'm(Z(ddl)m*Z*ddl+m,Z,ddl-m.Z.ddl/m0Z0m1Z1m2Z2m3Z3ddl4m5Z6m7Z8m9Z9ddl:m;Z;mm?Z?ddl@mAZAmBZBddlCmDZDddlEmFZFeAe.fdZGeAe.fdZHeAe.fdZIeAdZJdZKiZLd eMd!<Gd"d#e?eZNGd$d%e&e?eeOZPy&)'zSparse polynomial rings. ) annotations)Any)addmulltlegtge)reduce) GeneratorType)Expr)igcd)Symbolsymbols) CantSympifysympify)multinomial_coefficients)IPolys)construct_domain)ninf dmp_to_dict dmp_from_dict) DomainElementPolynomialRingheugcd) MonomialOps)lex)CoercionFailedGeneratorsErrorExactQuotientFailedMultivariatePolynomialError)DomainOrder build_options)expr_from_dict _dict_reorder_parallel_dict_from_expr)DefaultPrinting)publicsubsets) is_sequence)pollutec<t|||}|f|jzS)aConstruct a polynomial ring returning ``(ring, x_1, ..., x_n)``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, x, y, z = ring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) PolyRinggensrdomainorder_rings V/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/polys/rings.pyringr8#s#8 Wfe ,E 8ejj  c8t|||}||jfS)aConstruct a polynomial ring returning ``(ring, (x_1, ..., x_n))``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import xring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, (x, y, z) = xring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) r0r3s r7xringr;Bs!8 Wfe ,E 5:: r9ct|||}t|jDcgc]}|jc}|j|Scc}w)aConstruct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import vring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> vring("x,y,z", ZZ, lex) Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z # noqa: x + y + z >>> type(_) )r1r.rnamer2)rr4r5r6syms r7vringr?as=6 Wfe ,E %-- 13chh 15::> L 2sAc d}t|s|gd}}ttt|}t ||}t ||\}}|j t|Dcgc]}t|jc}g}t||\|_}tt||} |D cgc])}|jD cic] \} } | | |  c} } +}} }} t|j|j |j} tt| j |} |r| | dfS| | fScc}wcc} } wcc} } }w)adConstruct a ring deriving generators and domain from options and input expressions. Parameters ========== exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable) symbols : sequence of :class:`~.Symbol`/:class:`~.Expr` options : keyword arguments understood by :class:`~.Options` Examples ======== >>> from sympy import sring, symbols >>> x, y, z = symbols("x,y,z") >>> R, f = sring(x + 2*y + 3*z) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> f x + 2*y + 3*z >>> type(_) FT)optr)r-listmaprr&r)r4sumvaluesrdictzipitemsr1r2r5 from_dict)exprsroptionssinglerArepsrepcoeffs coeffs_dom coeff_mapmcr6polyss r7sringrUs&4F u v We$ %E  )C)4ID# zzT;ctCJJL);R@!1&c!B JVZ01 EIJJcSYY[9TQIaL9JJ SXXszz399 5E U__d+ ,E uQx  u~< :Js D=4E EEEct|tr|r t|dSdSt|tr|fSt |r1t d|Dr t|St d|Dr|St d)NT)seqc3<K|]}t|tywN) isinstancestr.0ss r7 z!_parse_symbols..s3az!S!3c3<K|]}t|tywrZ)r[r r]s r7r`z!_parse_symbols..s6At$6razbexpected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions)r[r\_symbolsr r-allr!rs r7_parse_symbolsrfsr'3.5xT*=2= GT "z W  373 3G$ $ 6g6 6N ~ r9zdict[Any, Any] _ring_cacheceZdZdZefdZdZdZdZdZ dZ dZ d%d Z d Z ed Zed Zd&dZdZdZdZeZd&dZd&dZdZdZdZdZdZdZdZdZdZ edZ!edZ"dZ#d Z$d!Z%d"Z&d#Z'd$Z(y )'r1z*Multivariate distributed polynomial ring. ctt|}t|}tj|}t j|j |||f}tj|}|L|jr,t|t|jzr tdtj|}||_t!||_t%dt&fd|i|_||_ ||_||_|_d|z|_|j3|_t|j4|_|j0|j8fg|_|rt=|}|j?|_ |jC|_"|jG|_$|jK|_&|jO|_(|jS|_*|jW|_,n5d}||_ ||_"d|_$||_&||_(||_*||_,tZur t\|_/n fd|_/ta|j|j4D]<\} } tc| tds| jf} ti|| r0tk|| | >|t|<|S)Nz7polynomial ring and it's ground domain share generators PolyElementr8rcyNrXrX)abs r7z"PolyRing.__new__..r9cyrmrX)rnrorSs r7rpz"PolyRing.__new__..rqr9ct|S)Nkey)max)fr5s r7rpz"PolyRing.__new__..sS->r9)6tuplerflen DomainOpt preprocessOrderOpt__name__rgget is_Compositesetrr!object__new__ _hash_tuplehash_hashtyperjdtypengensr4r5 zero_monom_gensr2 _gens_setone_onerr monomial_mulpow monomial_powmulpowmonomial_mulpowldiv monomial_ldivdiv monomial_divlcm monomial_lcmgcd monomial_gcdrrv leading_expvrGr[rr=hasattrsetattr) clsrr4r5rrobjcodegenmonunitsymbol generatorr=s ` r7rzPolyRing.__new__s?w/0G %%f-##E*||WeVUC ook* ;""s7|c&..6I'I%&_``..%C)CO[)CI][NVSMJCI!CKCICJCI!%ZCNyy{CHMCM45CH%e,#*;;= #*;;= &-nn&6#$+LLN!#*;;= #*;;= #*;;= )#* #* &8#$+!#* #* #* |#& #> %(chh%? 6! ff-!;;D"3-T95  6(+K $ r9c|jj}g}t|jD]5}|j |}|j }|||<|j |7t|S)z(Return a list of polynomial generators. )r4rrangermonomial_basiszeroappendrx)selfrriexpvpolys r7rzPolyRing._genssfkkootzz" A&&q)D99DDJ LL    U|r9cH|j|j|jfSrZ)rr4r5rs r7__getnewargs__zPolyRing.__getnewargs__s dkk4::66r9cx|jj}|d=|D]}|jds||=|S)Nr monomial_)__dict__copy startswith)rstaterus r7 __getstate__zPolyRing.__getstate__sE ""$ . ! C~~k*#J  r9c|jSrZ)rrs r7__hash__zPolyRing.__hash__ s zzr9ct|txr]|j|j|j|j f|j|j|j|j fk(SrZ)r[r1rr4rr5rothers r7__eq__zPolyRing.__eq__#sV%*D \\4;; DJJ ? ]]ELL%++u{{ C D Dr9c||k( SrZrXrs r7__ne__zPolyRing.__ne__(s5=  r9Nc||j|xs |j|xs |j|xs |jSrZ) __class__rr4r5)rrr4r5s r7clonezPolyRing.clone+s3~~g5v7LeNaW[WaWabbr9cBdg|jz}d||<t|S)zReturn the ith-basis element. r)rrx)rrbasiss r7rzPolyRing.monomial_basis.s$DJJaU|r9c"|jSrZ)rrs r7rz PolyRing.zero4szz|r9c8|j|jSrZ)rrrs r7rz PolyRing.one8szz$))$$r9c:|jj||SrZ)r4convertrelement orig_domains r7 domain_newzPolyRing.domain_new<s{{""7K88r9c:|j|j|SrZ)term_newr)rcoeffs r7 ground_newzPolyRing.ground_new?s}}T__e44r9cN|j|}|j}|r|||<|SrZ)rr)rmonomrrs r7rzPolyRing.term_newBs*&yy DK r9cVt|trj||jk(r|St|jtr4|jj|jk(r|j |St dt|tr t dt|tr|j|St|tr |j|St|tr|j|S|j |S#t$r|j|cYSwxYw)N conversionparsing)r[rjr8r4rrNotImplementedErrorr\rFrIrB from_terms ValueError from_listr from_exprrrs r7ring_newzPolyRing.ring_newIs g{ +w||#DKK8T[[=M=MQXQ]Q]=]w//),77  %%i0 0  &>>'* *  & /w// &>>'* *??7+ +  /~~g.. /sD D('D(c|j}|j}|jD]\}}|||}|s|||<|SrZ)rrrH)rrrrrrrs r7rIzPolyRing.from_dictasL__ yy#MMO $LE5uk2E#U  $  r9c8|jt||SrZ)rIrFrs r7rzPolyRing.from_termsls~~d7m[99r9ch|jt||jdz |jSNr)rIrrr4rs r7rzPolyRing.from_listos&~~k'4::a<MNNr9cTjfdt|S)Nc j|}||S|jr-ttt t |j S|jr-ttt t |j S|j\}}|jr|dkDr|t|zSjj|Sr)r~is_Addr rrBrCargsis_Mulr as_base_exp is_Integerintrr)exprrbaseexp_rebuildr4mappingrs r7rz(PolyRing._rebuild_expr.._rebuildus D)I$  c4Hdii(@#ABBc4Hdii(@#ABB!,,. c>>cAg#D>3s833??6>>$+?@@r9)r4r)rrrrr4s` `@@r7 _rebuild_exprzPolyRing._rebuild_exprrs# A$ &&r9cttt|j|j} |j ||}|j |S#t$rtd|d|wxYw)Nz6expected an expression convertible to a polynomial in z, got ) rFrBrGrr2rrr r)rrrrs r7rzPolyRing.from_exprsmtC dii89: '%%dG4D==& & pcgimno o ps AA2c"||jrd}|Sd}|St|trD|}d|kr||jkr |S|j |kr |dkr| dz }|Std|zt||jr |j j |}|St|tr |jj |}|Std|z#t$rtd|zwxYw#t$rtd|zwxYw)z+Compute index of ``gen`` in ``self.gens``. rrzinvalid generator index: %szinvalid generator: %szEexpected a polynomial generator, an integer, a string or None, got %s) rr[rrrr2indexr\r)rgenrs r7rzPolyRing.indexsD ;zz2/.-S !AAv!djj.$#**!a2gBF !!>!DEE TZZ ( @IIOOC(S ! @LL&&s+ dgjjk k @ !83!>?? @  @ !83!>?? @sC0C6C36Dctt|j|}t|jDcgc] \}}||vs |}}}|s |j S|j |Scc}}w)z,Remove specified generators from this ring. re)rrCr enumeraterr4r)rr2indicesrr_rs r7dropz PolyRing.dropsbc$**d+,"+DLL"9O$!QQg=MAOO;; ::g:. . Ps A,A,c`|j|}|s |jS|j|S)Nre)rr4r)rrurs r7 __getitem__zPolyRing.__getitem__s.,,s#;; ::g:. .r9c|jjst|jdr&|j|jjSt d|jz)Nr4r4z%s is not a composite domain)r4rrrrrs r7 to_groundzPolyRing.to_groundsL ;; # #wt{{H'E::T[[%7%7:8 8;dkkIJ Jr9ct|SrZrrs r7 to_domainzPolyRing.to_domains d##r9c^ddlm}||j|j|jS)Nr) FracField)sympy.polys.fieldsrrr4r5)rrs r7to_fieldzPolyRing.to_fields 0t{{DJJ??r9c2t|jdk(Srryr2rs r7 is_univariatezPolyRing.is_univariates499~""r9c2t|jdkDSrrrs r7is_multivariatezPolyRing.is_multivariates499~!!r9c~|j}|D]+}t|tr||j|z }'||z }-|S)aw Add a sequence of polynomials or containers of polynomials. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> R, x = ring("x", ZZ) >>> R.add([ x**2 + 2*i + 3 for i in range(4) ]) 4*x**2 + 24 >>> _.factor_list() (4, [(x**2 + 6, 1)]) include)rr-r rrobjsprs r7rz PolyRing.addsJ" II C3 6XTXXs^#S   r9c~|j}|D]+}t|tr||j|z}'||z}-|S)a Multiply a sequence of polynomials or containers of polynomials. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> R, x = ring("x", ZZ) >>> R.mul([ x**2 + 2*i + 3 for i in range(4) ]) x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945 >>> _.factor_list() (1, [(x**2 + 3, 1), (x**2 + 5, 1), (x**2 + 7, 1), (x**2 + 9, 1)]) r )rr-r rr s r7rz PolyRing.mulsJ" HH C3 6XTXXs^#S   r9cPtt|j|}t|jDcgc] \}}||vs |}}}t|j Dcgc] \}}||vs |}}}|s|S|j ||j|Scc}}wcc}}w)zd Remove specified generators from the ring and inject them into its domain. rr4)rrCrrrr2rr)rr2rrr_rrs r7drop_to_groundzPolyRing.drop_to_grounds c$**d+,!*4<D::d4j:1 1Kr9ct|jjt|}|jt |S)z9Add the elements of ``symbols`` as generators to ``self``rer)rrrs r7add_genszPolyRing.add_gens&s44<< &&s7|4zz$t*z--r9c|dks||jkDrtd|d|j|s |jS|j}t t |jt|D]Rtfdt |jD}||j||jjz }T|S)zo Return the elementary symmetric polynomial of degree *n* over this ring's generators. rz.Cannot generate symmetric polynomial of order z for c38K|]}t|vywrZ)r)r^rr_s r7r`z*PolyRing.symmetric_poly..7sEac!q&kEs) rrr2rrr,rrrxrr4)rnrrr_s @r7symmetric_polyzPolyRing.symmetric_poly+s q5A NZ[]a]f]fgh h88O99DU4::.A7 >E53DEE eT[[__== >Kr9)NNNrZ))r} __module__ __qualname____doc__rrrrrrrrrrpropertyrrrrrr__call__rIrrrrrrrrrrrrrrrrrrrXr9r7r1r1s4,/?B 7D !c %%95,,H :O'.'>//K$@##""66 H. r9r1c`eZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z d Zd ZdZddZdZdZdZdZdZdZdZdZdZdZdZdZdZedZ edZ!edZ"ed Z#ed!Z$ed"Z%ed#Z&ed$Z'ed%Z(ed&Z)ed'Z*ed(Z+ed)Z,ed*Z-ed+Z.ed,Z/ed-Z0d.Z1d/Z2d0Z3d1Z4d2Z5d3Z6d4Z7d5Z8d6Z9d7Z:d8Z;d9Zd<Z?d=Z@d>ZAd?ZBeAZCeBZDd@ZEdAZFdBZGdCZHdDZIdEZJdFZKddGZLdHZMddIZNdJZOdKZPdLZQdMZRdNZSedOZTedPZUdQZVedRZWdSZXdTZYddUZZddVZ[ddWZ\dXZ]dYZ^dZZ_d[Z`d\Zad]Zbd^Zcd_Zdd`ZedaZfdbZgdcZhddZideZjdfZkdgZlelZmdhZndiZodjZpdkZqdlZrdmZsdnZtdoZudpZvdqZwdrZxdsZydtZzduZ{dvZ|dwZ}dxZ~dyZddzZdd{Zd|Zdd}Zd~ZddZddZddZddZddZdZdZdZdZdZdZdZdZdZdZdZdZddZdZy)rjz5Element of multivariate distributed polynomial ring. c$|j|SrZ)r)rinits r7newzPolyElement.new?s~~d##r9c6|jjSrZ)r8rrs r7parentzPolyElement.parentBsyy""$$r9cL|jt|jfSrZ)r8rB itertermsrs r7rzPolyElement.__getnewargs__Es 4 0122r9Nc|j}|5t|jt|j fx|_}|SrZ)rrr8 frozensetrH)rrs r7rzPolyElement.__hash__Js<   =!%tyy)DJJL2I&J!K KDJ r9c$|j|S)aReturn a copy of polynomial self. Polynomials are mutable; if one is interested in preserving a polynomial, and one plans to use inplace operations, one can copy the polynomial. This method makes a shallow copy. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> R, x, y = ring('x, y', ZZ) >>> p = (x + y)**2 >>> p1 = p.copy() >>> p2 = p >>> p[R.zero_monom] = 3 >>> p x**2 + 2*x*y + y**2 + 3 >>> p1 x**2 + 2*x*y + y**2 >>> p2 x**2 + 2*x*y + y**2 + 3 )r%rs r7rzPolyElement.copyUs4xx~r9c v|j|k(r|S|jj|jk7r`ttt ||jj|j}|j ||jj S|j||jj SrZ)r8rrBrGr(rr4rI)rnew_ringtermss r7set_ringzPolyElement.set_ringqs 99 K YY  ("2"2 2mD$))2C2CXEUEUVWXE&&udii.>.>? ?%%dDII,<,<= =r9c|s|jj}nPt||jjk7r.t d|jjdt|t |j g|S)Nz"Wrong number of symbols, expected z got )r8rryrrr' as_expr_dict)rrs r7as_exprzPolyElement.as_exprzsfii''G \TYY__ ,#g,0  d//1>;KL<5%x&LLLsA c |jj}|jr |js|j|fS|j }|j}|j }|j}|jD]}||||}|j|jDcgc] \}}|||zfc}}} || fScc}}wrZ) r8r4is_Fieldhas_assoc_Ringrget_ringrdenomrEr%rH) rr4 ground_ringcommonrr:rkvrs r7 clear_denomszPolyElement.clear_denomss!!f&;&;::t# #oo' oo [[] /Eu.F /xxDJJLBDAq1ah-BCt|Cs3C cRt|jD] \}}|r ||= y)z+Eliminate monomials with zero coefficient. NrBrH)rr=r>s r7 strip_zerozPolyElement.strip_zeros*& DAqG r9c|s| St|tr/|j|jk(rtj ||St |dkDry|j |jj|k(S)aPEquality test for polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p1 = (x + y)**2 + (x - y)**2 >>> p1 == 4*x*y False >>> p1 == 2*(x**2 + y**2) True rF)r[rjr8rFrryr~rp1p2s r7rzPolyElement.__eq__sd"6M K (RWW-?;;r2& & Wq[66"'',,-3 3r9c||k( SrZrXrDs r7rzPolyElement.__ne__s8|r9c|j}t||jrrt|j t|j k7ry|j j }|j D]}||||||ryyt|dkDry |j j|}|j j |j||S#t$rYywxYw)z+Approximate equality test for polynomials. FTr) r8r[rrkeysr4almosteqryrconstr )rErF tolerancer8rJr=s r7rJzPolyElement.almosteqsww b$** %2779~RWWY/{{++HWWY !1r!ui8  ! Wq[ G[[((,{{++BHHJIFF"  s%C++ C76C7c8t||jfSrZ)ryr/rs r7sort_keyzPolyElement.sort_keysD 4::<((r9ct||jjr%||j|jStSrZ)r[r8rrNNotImplemented)rErFops r7_cmpzPolyElement._cmps3 b"''-- (bkkmR[[]3 3! !r9c.|j|tSrZ)rRrrDs r7__lt__zPolyElement.__lt__wwr2r9c.|j|tSrZ)rRrrDs r7__le__zPolyElement.__le__rUr9c.|j|tSrZ)rRr rDs r7__gt__zPolyElement.__gt__rUr9c.|j|tSrZ)rRr rDs r7__ge__zPolyElement.__ge__rUr9c|j}|j|}|jdk(r||jfSt |j }||=||j |fS)Nrre)r8rrr4rBrrrrr8rrs r7_dropzPolyElement._drops\yy JJsO ::?dkk> !4<<(G djjj11 1r9cb|j|\}}|jjdk(r+|jr|j dSt d|z|j }|jD]7\}}||dk(rt|}||=||t|<+t d|z|S)NrzCannot drop %sr) r^r8r is_groundrrrrHrBrx)rrrr8rr=r>Ks r7rzPolyElement.drops**S/4 99??a ~~zz!}$ !1C!78899D  =1Q419QA!%&DqN$%5%;<<  =Kr9c|j}|j|}t|j}||=||j |||fS)Nr)r8rrBrrr]s r7_drop_to_groundzPolyElement._drop_to_groundsIyy JJsOt||$ AJ$**WT!W*===r9c|jjdk(r td|j|\}}|j}|j j d}|jD]T\}}|d|||dzdz}||vr|||zj|||<3||xx|||zj|z cc<V|S)Nrz$Cannot drop only generator to groundr) r8rrrcrr4r2r) mul_ground)rrrr8rrrmons r7rzPolyElement.drop_to_grounds 99??a CD D&&s+4yykkq! NN, ?LE5)eAaCDk)C$ %(]66u=S S c58m77>>  ? r9crt||jjdz |jjSr)rr8rr4rs r7to_densezPolyElement.to_dense s(T499??1#4dii6F6FGGr9ct|SrZ)rFrs r7to_dictzPolyElement.to_dict#s Dzr9c6|s/|j|jjjS|d}|d}|j}|j}|j } |j } g} |jD]Y\} } |jj| }|rdnd}| j|| | k(r*|j| }|rV|jdrE|dd}n?|r| } | |jjjk7r|j| |d}nd }g}t| D]{}| |}|s |j|||d}|dk7rA|t|k7s|d kr|j||d }n|}|j|||fzh|jd |z}|r|g|z}| j|j|\| d d vr(| j!d }|dk(r| j#d dd j| S)NMulAtom -  + -rT)strictrFz%s)rorn)_printr8r4rrrrr/ is_negativerrr parenthesizerrjoinpopinsert)rprinter precedence exp_pattern mul_symbolprec_mul prec_atomr8rrzmsexpvsrrnegativesignscoeffsexpvrrrsexpheads r7r\zPolyElement.str&s>>$))"2"2"7"78 8e$v& yy,,  __::< 2KD%{{..u5H$5%D MM$ rz . 1 1# 6#ABZF"FEDII,,000$11%$1OFFE5\ 01g --gaj)D-Q!8c#h#'&33C53Q"LL~!=>LL/ 05( MM*//%0 1? 2@ !9 &::a=Du} a%wwvr9c2||jjvSrZ)r8rrs r7 is_generatorzPolyElement.is_generatorVstyy****r9c\| xs(t|dk(xr|jj|vSr)ryr8rrs r7r`zPolyElement.is_groundZs+xLCINKtyy/C/Ct/KLr9cJ| xst|dk(xr|jdk(Sr)ryLCrs r7 is_monomialzPolyElement.is_monomial^s$x.?u3u:??rd itermonomsrs r7 is_linearzPolyElement.is_linear? ???r9cBtd|jDS)Nc38K|]}t|dkyw)Nrrs r7r`z+PolyElement.is_quadratic..rrrrs r7 is_quadraticzPolyElement.is_quadraticrr9cf|jjsy|jj|SNT)r8r dmp_sqf_prs r7 is_squarefreezPolyElement.is_squarefrees%vv||vv""r9cf|jjsy|jj|Sr)r8rdmp_irreducible_prs r7is_irreduciblezPolyElement.is_irreducibles%vv||vv''**r9cz|jjr|jj|Std)Nzcyclotomic polynomial)r8rdup_cyclotomic_pr#rs r7 is_cyclotomiczPolyElement.is_cyclotomics0 66  66**1- --.EF Fr9cx|j|jDcgc] \}}|| f c}}Scc}}wrZ)r%r))rrrs r7__neg__zPolyElement.__neg__s0xxdnn>NPleU55&/PQQPs6 c|SrZrXrs r7__pos__zPolyElement.__pos__s r9c|s|jS|j}t||jrc|j}|j}|j j }|jD]\}}||||z}|r|||<||=|St|trt|j tr$|j j|jk(rn^t|jj tr4|jj j|k(r|j|StS |j|}|j}|s|S|j} | |jvr||| <|S|||  k(r|| =|S|| xx|z cc<|S#t$r tcYSwxYw)aAdd two polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> (x + y)**2 + (x - y)**2 2*x**2 + 2*y**2 )rr8r[rr~r4rrHrjr__radd__rPrrrIr ) rErFr8r r~rr=r>cp2rs r7__add__zPolyElement.__add__s779 ww b$** % A%%C;;##D  14L1$AaD!   H K ($++~64;;;K;Krww;VBGGNNN;@S@SW[@[{{2&%% //"%C AB"" H !B%<"HbESLEH "! ! "sF,,F>=F>c|j}|s|S|j} |j|}|j}||j vr|||<|S||| k(r||=|S||xx|z cc<|S#t $r t cYSwxYwrZ)rr8rrrIr rP)rErr r8rs r7rzPolyElement.__radd__s GGIHww "AB"" H 2;"HbEQJEH "! ! "sA55BBcz|s|jS|j}t||jrc|j}|j}|j j }|jD]\}}||||z }|r|||<||=|St|trt|j tr$|j j|jk(rn^t|jj tr4|jj j|k(r|j|StS |j|}|j}|j}||jvr| ||<|S|||k(r||=|S||xx|zcc<|S#t$r tcYSwxYw)a.Subtract polynomial p2 from p1. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p1 = x + y**2 >>> p2 = x*y + y**2 >>> p1 - p2 -x*y + x )rr8r[rr~r4rrHrjr__rsub__rPrrrIr ) rErFr8r r~rr=r>rs r7__sub__zPolyElement.__sub__s 779 ww b$** % A%%C;;##D  14L1$AaD!   H K ($++~64;;;K;Krww;VBGGNNN;@S@SW[@[{{2&%% $B AB"" H 2;"HbERKEH "! ! "sF((F:9F:c|j} |j|}|j}|D] }|| ||< ||z }|S#t$r tcYSwxYw)a#n - p1 with n convertible to the coefficient domain. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y >>> 4 - p -x - y + 4 )r8rrr rP)rErr8r rs r7rzPolyElement.__rsub__'snww "A A $d8)$ $ FAH "! ! "sAAAc|j}|j}|r|s|St||jr|j}|j j}|j }t|j}|jD]*\}} |D] \} } ||| } || || | zz|| <",|j|St|trt|j tr$|j j|jk(rn^t|jj tr4|jj j|k(r|j|StS |j|}|jD]\}} | |z} | s| ||<|S#t$r tcYSwxYw)a!Multiply two polynomials. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', QQ) >>> p1 = x + y >>> p2 = x - y >>> p1*p2 x**2 - y**2 )r8rr[rr~r4rrBrHrBrjr__rmul__rPrr )rErFr8r r~rrp2itexp1v1exp2v2rr>s r7__mul__zPolyElement.__mul__Bs ww IIH DJJ '%%C;;##D,,L #DHHJ 4b $4HD"&tT2C d^be3AcF4 4 LLNH K ($++~64;;;K;Krww;VBGGNNN;@S@SW[@[{{2&%% $BHHJ brEAdG H "! ! "s8F00GGc|jj}|s|S |jj|}|jD]\}}||z}|s|||<|S#t$r t cYSwxYw)ap2 * p1 with p2 in the coefficient domain of p1. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y >>> 4 * p 4*x + 4*y )r8rrrHr rP)rErFr rrr>s r7rzPolyElement.__rmul__ts GGLLH ""2&BHHJ brEAdG H "! ! "sAA0/A0cp|j}|s|r |jStdt|dk(rut |j d\}}|j }||jjk(r|||j||<|S||z||j||<|St|}|dkr td|dk(r|jS|dk(r|jS|dk(r||jzSt|dkr|j|S|j|S)a(raise polynomial to power `n` Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p**3 x**3 + 3*x**2*y**2 + 3*x*y**4 + y**6 z0**0rrzNegative exponentr)r8rrryrBrHrr4rrrsquare_pow_multinomial _pow_generic)rrr8rrr s r7__pow__zPolyElement.__pow__s-yyxx (( Y!^ -a0LE5 A '16$##E1-.H27$##E1-.H F q501 1 !V99;  !V;;= !V % % Y!^((+ +$$Q' 'r9c|jj}|} |dzr||z}|dz}|s |S|j}|dz}*)Nrr)r8rr)rrr rSs r7rzPolyElement._pow_genericsX IIMM 1uaCQ  AQAr9ctt||j}|jj}|jj }|j}|jj j}|jj}|D]g\}} |} | } t||D]\} \} }| s || | | } | || zz} t| } | }|j| ||z}|r||| <`| |vse|| =i|SrZ) rryrHr8rrr4rrGrxr~)rr multinomialsrrr/rr multinomialmultinomial_coeff product_monom product_coeffrrrs r7rzPolyElement._pow_multinomials /D 1=CCE ))33YY))  yy$$yy~~.: *K*&M-M'*;'> 0#^eU$3M5#$NM!UCZ/M 0 -(E!EHHUD)E1E#U $K# & r9c0|j}|j}|j}t|j }|j j}|j }tt|D]?}||}||} t|D]%} || } ||| } || || || zz|| <'A|jd}|j}|jD] \} }|| | } || ||dzz|| <"|j|S)asquare of a polynomial Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p.square() x**2 + 2*x*y**2 + y**4 r) r8rr~rBrIr4rrryimul_numrHrB)rr8r r~rIrrrk1pkjk2rr=r>s r7rzPolyElement.squaresyy IIeeDIIK {{(( s4y! 6AaBbB1X 6!W"2r*S$"T"X+5# 6 6 JJqMeeJJL )DAqa#BDMAqD(AbE ) r9cl|j}|s tdt||jr|j |St|t rt|j tr$|j j|jk(rn^t|jj tr4|jj j|k(r|j|StS |j|}|j||j|fS#t$r tcYSwxYwNpolynomial division)r8ZeroDivisionErrorr[rrrjr4r __rdivmod__rPr quo_ground rem_groundr rErFr8s r7 __divmod__zPolyElement.__divmod__sww#$9: : DJJ '66":  K ($++~64;;;K;Krww;VBGGNNN;@S@SW[@[~~b))%% :$BMM"%r}}R'89 9 "! ! "s.D!!D32D3ctSrZrPrDs r7rzPolyElement.__rdivmod__(r9cJ|j}|s tdt||jr|j |St|t rt|j tr$|j j|jk(rn^t|jj tr4|jj j|k(r|j|StS |j|}|j|S#t$r tcYSwxYwr) r8rr[rremrjr4r__rmod__rPrrr rs r7__mod__zPolyElement.__mod__+sww#$9: : DJJ '66":  K ($++~64;;;K;Krww;VBGGNNN;@S@SW[@[{{2&%% %$B==$ $ "! ! "s.DD"!D"ctSrZrrDs r7rzPolyElement.__rmod__Arr9cr|j}|s tdt||jr%|jr||dzzS|j |St|t rt|jtr$|jj|jk(rn^t|jjtr4|jjj|k(r|j|StS |j|}|j|S#t$r tcYSwxYw)Nrr)r8rr[rrquorjr4r __rtruediv__rPrrr rs r7 __truediv__zPolyElement.__truediv__Dsww#$9: : DJJ '~~28}$vvbz! K ($++~64;;;K;Krww;VBGGNNN;@S@SW[@[r**%% %$B==$ $ "! ! "sD$$D65D6ctSrZrrDs r7rzPolyElement.__rtruediv__]rr9c|jj|jj}|j|jj|j r fd}|Sfd}|S)NcV|\}}|\}}| k(r|}n ||}| |||fSyrZrX a_lm_a_lc b_lm_b_lca_lma_lcb_lmb_lcr domain_quorrs r7term_divz'PolyElement._term_div..term_divlsH& d& d2: E(t4E$ *T4"888r9c`|\}}|\}}| k(r|}n ||}|||zs |||fSyrZrXrs r7rz'PolyElement._term_div..term_divxsM& d& d2: E(t4E  *T4"888r9)r8rr4rrr7)rr4rrrrs @@@r7 _term_divzPolyElement._term_divesX YY ! !!!ZZ yy-- ?? 0 r9c|j}d}t|trd}|g}t|s t d|s(|r|j |j fSg|j fS|D]}|j|k7st dt|}t|Dcgc]}|j }}|j}|j } |j} |D cgc]} | j} } |rd}d} ||kr|| dk(rw|j}| |||f| |||| |f}|9|\}}||j||f||<|j|||| f}d} n|dz }||kr| dk(rw| s)|j}| j|||f} ||=|r|jk(r| |z } |r|s|j | fS|d| fS|| fScc}wcc} w)aUDivision algorithm, see [CLO] p64. fv array of polynomials return qv, r such that self = sum(fv[i]*qv[i]) + r All polynomials are required not to be Laurent polynomials. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> f = x**3 >>> f0 = x - y**2 >>> f1 = x - y >>> qv, r = f.div((f0, f1)) >>> qv[0] x**2 + x*y**2 + y**4 >>> qv[1] 0 >>> r y**6 FTrz"self and f must have the same ringrr)r8r[rjrdrrrryrrrr _iadd_monom_iadd_poly_monomr)rfvr8 ret_singlerwr_rqvr rrfxexpvs divoccurredrtermexpv1rSs r7rzPolyElement.divs8yy b+ &JB2w#$9: :yy$))++499}$ GAvv~ !EFF G G!&q *Adii * * IIK II>>#-/0r"00AKa%K1,~~'qw%(BqE%(O1LM##HE1qE--uaj9BqE**2a551"+>A"#KFAa%K1,(MM44/2dG!" 4?? " FA yy!|#!uaxq5L=+1s G!"G&c|}t|tr|g}t|s td|j}|j }|j }|j}|j }|j}|j} |j}|j} |r|D]r} || | j} | | \} }| jD](\}}||| }| ||||zz }|s||=$|||<*|j}||||f} n9| \}}||vr||xx|z cc<n|||<||=|j}||||f} |r|Sr)r[rjrdrr8r4rrrLTrr~r)r)rGrwr8r4rrrrltfr~gtqrRrSmgcgm1c1ltmltcs r7rzPolyElement.remsz  a %A1v#$9: :vv{{(( II;;=dd FFHee &c144(>DAq"#++-'B)"a0 T]QrT1! !"$&AbE '..*C!1S6k &"S!8cFcMF AcFcFnn&?qv+C58r9c*|j|dSNr)r)rwrs r7rzPolyElement.quosuuQx{r9cJ|j|\}}|s|St||rZ)rr")rwrqrs r7exquozPolyElement.exquos(uuQx1H%a+ +r9c||jjvr|j}n|}|\}}|j|}||||<|S||z }|r|||<|S||=|S)aadd to self the monomial coeff*x0**i0*x1**i1*... unless self is a generator -- then just return the sum of the two. mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...) Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x**4 + 2*y >>> m = (1, 2) >>> p1 = p._iadd_monom((m, 5)) >>> p1 x**4 + 5*x*y**2 + 2*y >>> p1 is p True >>> p = x >>> p1 = p._iadd_monom((m, 5)) >>> p1 5*x*y**2 + x >>> p1 is p False )r8rrr~)rmccpselfrrrSs r7r zPolyElement._iadd_monom s~8 499&& &YY[FF e JJt  9 F4L JA t  4L r9c^|}||jjvr|j}|\}}|j}|jjj }|jj }|jD](\} } || |} || || |zz} | r| || <&|| =*|S)aEadd to self the product of (p)*(coeff*x0**i0*x1**i1*...) unless self is a generator -- then just return the sum of the two. mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...) Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring('x, y, z', ZZ) >>> p1 = x**4 + 2*y >>> p2 = y + z >>> m = (1, 2, 3) >>> p1 = p1._iadd_poly_monom(p2, (m, 3)) >>> p1 x**4 + 3*x*y**3*z**3 + 3*x*y**2*z**4 + 2*y )r8rrr~r4rrrH) rrFr&rErRrSr~rrr=r>kars r7r zPolyElement._iadd_poly_monom6s* "" "BAffww~~""ww++ HHJ DAqa#BDMAaC'E2rF   r9c|jj||stSdkrytfd|j DS)z The leading degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (``float('-inf')``) rc3(K|] }| ywrZrXr^rrs r7r`z%PolyElement.degree..i7166<<' 'S$sALLN';"<=> >r9c|jj||stSdkrytfd|j DS)z The tail degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (``float('-inf')``) rc3(K|] }| ywrZrXr,s r7r`z*PolyElement.tail_degree..r-r.)r8rrminrr/s @r7 tail_degreezPolyElement.tail_degreewr2r9c |stf|jjzStt t t t|jS)z A tuple containing tail degrees in all variables. Note that the degree of 0 is negative infinity (``float('-inf')``) ) rr8rrxrCr8rBrGrrs r7 tail_degreeszPolyElement.tail_degreesr5r9c>|r|jj|Sy)aTLeading monomial tuple according to the monomial ordering. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring('x, y, z', ZZ) >>> p = x**4 + x**3*y + x**2*z**2 + z**7 >>> p.leading_expv() (4, 0, 0) N)r8rrs r7rzPolyElement.leading_expvs 99))$/ /r9cb|j||jjjSrZ)r~r8r4rrrs r7 _get_coeffzPolyElement._get_coeffs#xxdii..3344r9cx|dk(r%|j|jjSt||jjrct |j }t|dk(r<|d\}}||jjjk(r|j|Std|z)a Returns the coefficient that stands next to the given monomial. Parameters ========== element : PolyElement (with ``is_monomial = True``) or 1 Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring("x,y,z", ZZ) >>> f = 3*x**2*y - x*y*z + 7*z**3 + 23 >>> f.coeff(x**2*y) 3 >>> f.coeff(x*y) 0 >>> f.coeff(1) 23 rrzexpected a monomial, got %s) r?r8rr[rrBr)ryr4rr)rrr/rrs r7rzPolyElement.coeffs4 a<??499#7#78 8  1**,-E5zQ$Qx uDII,,000??5116@AAr9cL|j|jjS)z"Returns the constant coefficient. )r?r8rrs r7rKzPolyElement.conststyy3344r9c@|j|jSrZ)r?rrs r7rzPolyElement.LCst00233r9cV|j}||jjS|SrZ)rr8rr>s r7LMzPolyElement.LMs*  " <99'' 'Kr9c|jj}|j}|r#|jjj||<|S)a Leading monomial as a polynomial element. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> (3*x*y + y**2).leading_monom() x*y )r8rrr4rrr rs r7 leading_monomzPolyElement.leading_monoms@ IINN  " ii&&**AdGr9c|j}|6|jj|jjjfS||j |fSrZ)rr8rr4rr?r>s r7rzPolyElement.LTsN  " <II(($))*:*:*?*?@ @$//$/0 0r9cf|jj}|j}|||||<|S)aLeading term as a polynomial element. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> (3*x*y + y**2).leading_term() 3*x*y )r8rrrFs r7 leading_termzPolyElement.leading_terms7 IINN  "  4jAdGr9c|jjntjturt |ddSt |fddS)Nc |dSr!rX)rs r7rpz%PolyElement._sorted..s qr9T)rureversec|dSr!rX)rr5s r7rpz%PolyElement._sorted..suQxr9)r8r5r|r{rsorted)rrWr5s `r7_sortedzPolyElement._sortedsL =IIOOE''.E C<##94H H##@$O Or9cV|j|Dcgc]\}}| c}}Scc}}w)aOrdered list of polynomial coefficients. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.coeffs() [2, 1] >>> f.coeffs(grlex) [1, 2] r/)rr5_rs r7rOzPolyElement.coeffss%0(,zz%'8:81e::: %cV|j|Dcgc]\}}| c}}Scc}}w)a Ordered list of polynomial monomials. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.monoms() [(2, 3), (1, 7)] >>> f.monoms(grlex) [(1, 7), (2, 3)] rR)rr5rrSs r7monomszPolyElement.monoms5s%0(,zz%'8:85!:::rTcT|jt|j|S)aOrdered list of polynomial terms. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.terms() [((2, 3), 2), ((1, 7), 1)] >>> f.terms(grlex) [((1, 7), 1), ((2, 3), 2)] )rPrBrH)rr5s r7r/zPolyElement.termsOs 0||D.66r9c4t|jS)z,Iterator over coefficients of a polynomial. )iterrErs r7 itercoeffszPolyElement.itercoeffsiDKKM""r9c4t|jS)z)Iterator over monomials of a polynomial. )rYrIrs r7rzPolyElement.itermonomsmDIIK  r9c4t|jS)z%Iterator over terms of a polynomial. )rYrHrs r7r)zPolyElement.itertermsqDJJL!!r9c4t|jS)z+Unordered list of polynomial coefficients. )rBrErs r7 listcoeffszPolyElement.listcoeffsur[r9c4t|jS)z(Unordered list of polynomial monomials. )rBrIrs r7 listmonomszPolyElement.listmonomsyr]r9c4t|jS)z$Unordered list of polynomial terms. rArs r7 listtermszPolyElement.listterms}r_r9c||jjvr||zS|s|jy|D]}||xx|zcc<|S)a:multiply inplace the polynomial p by an element in the coefficient ring, provided p is not one of the generators; else multiply not inplace Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p1 = p.imul_num(3) >>> p1 3*x + 3*y**2 >>> p1 is p True >>> p = x >>> p1 = p.imul_num(3) >>> p1 3*x >>> p1 is p False N)r8rclear)r rSrs r7rzPolyElement.imul_numsO4   Q3J GGI  C cFaKF r9c|jj}|j}|j}|j D] }|||} |S)z*Returns GCD of polynomial's coefficients. )r8r4rrrZ)rwr4contrrs r7rzPolyElement.contentsI{{jj\\^ $EtU#D $ r9cH|j}||j|fS)z,Returns content and a primitive polynomial. )rr)rwris r7 primitivezPolyElement.primitives!yy{Q\\$'''r9c@|s|S|j|jS)z5Divides all coefficients by the leading coefficient. )rrrs r7moniczPolyElement.monicsH<<% %r9c|s|jjS|jDcgc] \}}|||zf}}}|j|Scc}}wrZ)r8rr)r%)rwr0rrr/s r7rezPolyElement.mul_groundsM66;; 78{{}F|ue5%'"FFuuU|GsAc|jj}|jDcgc]\}}||||f}}}|j|Scc}}wrZ)r8rrHr%)rwrrf_monomf_coeffr/s r7 mul_monomzPolyElement.mul_monomsRvv** RSRYRYR[]>Ngw</9]]uuU|^sAcJ|\}}|r|s|jjS||jjk(r|j|S|jj}|j Dcgc]\}}|||||zf}}}|j |Scc}}wrZ)r8rrrerrHr%)rwrrrrrprqr/s r7mul_termzPolyElement.mul_terms u66;;  aff'' '<<& &vv** XYX_X_XacDTGW</?ccuuU|ds3Bc |jj}|s td|r||jk(r|S|jr8|j }|j Dcgc]\}}||||f}}}n-|j Dcgc]\}}||zr |||zf}}}|j|Scc}}wcc}}wr)r8r4rrr7rr)r%)rwr0r4rrrr/s r7rzPolyElement.quo_grounds#$9: :AOH ??**CABPuc%m,PEP>?kkm`leUTY\]T]ueqj)`E`uuU| Q`s"B9 B? B?cj|\}}|s td|s|jjS||jjk(r|j |S|j }|j Dcgc] }||| }}|j|Dcgc]}|| c}Scc}wcc}wr)rr8rrrrr)r%)rwrrrrtr/s r7quo_termzPolyElement.quo_terms u#$9: :66;;  aff'' '<<& &;;=-.[[]<(1d#<<uu%:Q1=q:;;=:s7B+B0 B0cb|jjjr@g}|jD]*\}}||z}||dzkDr||z }|j ||f,n'|jDcgc] \}}|||zf}}}|j |}|j |Scc}}w)Nr)r8r4is_ZZr)rr%rB)rwr r/rrrs r7 trunc_groundzPolyElement.trunc_grounds 66==  E !  - u 16>!AIE eU^,  ->?[[]L\UEueai(LELuuU|  Ms4B+c|}|j}|j}|jjj||}|j |}|j |}|||fSrZ)rr8r4rr)rrrwfcgcrs r7extract_groundzPolyElement.extract_grounds_  YY[ YY[ffmmB' LL  LL Aqyr9c|s |jjjS|jjj}||j Dcgc] }|| c}Scc}wrZ)r8r4rabsrZ)rw norm_func ground_absrs r7_normzPolyElement._normsP66==%% %**JallnNUz%0NO ONsA-c,|jtSrZ)rrvrs r7max_normzPolyElement.max_normwws|r9c,|jtSrZ)rrDrs r7l1_normzPolyElement.l1_normrr9c@|j}|gt|z}dg|jz}|D]<}|jD]'}t |D]\}}t |||||<)>t |D] \}} | r d||<t |}td|Dr||fSg} |D]f}|j} |jD]4\} } t| |Dcgc] \}}||z }}}| | t |<6| j| h|| fScc}}w)Nrrc3&K|] }|dk( ywrrX)r^ros r7r`z&PolyElement.deflate..0s!!qAv!s) r8rBrrrrrxrdrr)rGr)rwrr8rTJr rrrRroHhIrrNs r7deflatezPolyElement.deflates@vvd1g  C N )A )%e,)DAq!a=AaD) ) ) aL DAq!  !H !q! !e8O  A AKKM $5),Q4Aa1f44#%(  $ HHQK !t 5s!D c|jj}|jD]4\}}t||Dcgc] \}}||z }}}||t |<6|Scc}}wrZ)r8rr)rGrx)rwrrrrrrrs r7inflatezPolyElement.inflate@sbvv{{  #HAu"%a)-$!Q!A#-A-"DqN # .sA cZ|}|jj}|js8|j\}}|j\}}|j ||}||zj |j |}|js|jS|jSrZ) r8r4r7rkrrrrerm)rrrwr4r}r~rSrs r7rzPolyElement.lcmIs KKMEBKKMEB 2r"A qSIIaeeAh <<? "779 r9c*|j|dSr!) cofactorsrwrs r7rzPolyElement.gcdYs{{1~a  r9c |s|s|jj}|||fS|s|j|\}}}|||fS|s|j|\}}}|||fSt|dk(r|j |\}}}|||fSt|dk(r|j |\}}}|||fS|j |\}\}}|j |\}}}|j||j||j|fSr)r8r _gcd_zerory _gcd_monomr_gcdr)rwrrrcffcfgrs r7rzPolyElement.cofactors\s66;;Dt# #++a.KAsCc3; ++a.KAsCc3;  Vq[,,q/KAsCc3;  Vq[,,q/KAsCc3; IIaL 6AqffQi 3 ! ckk!nckk!n==r9c|jj|jj}}|jr|||fS| || fSrZ)r8rrr)rwrrrs r7rzPolyElement._gcd_zerors@FFJJ T  dC< 2tcT> !r9c B|j}|jj}|jj}|j}|j }t |jd\}}||} } |jD]\} } || | } || | } |j| | fg} |j||| ||| fg}|j|jD cgc]\} } || | || | fc} } }| ||fScc} } wr!) r8r4rrrrrBr)r%)rwrr8 ground_gcd ground_quorrmfcf_mgcd_cgcdrrrrrs r7rzPolyElement._gcd_monomysvv[[__ [[__ (( ** akkm$Q'B2ukkm *FB +Eub)E * EEE5>" #eemB. 2u0EFGHeeUVU`U`Ubc62rmB. 2u0EFcd#s{ds2D c|j}|jjr|j|S|jjr|j |S|j ||SrZ)r8r4is_QQ_gcd_QQrz_gcd_ZZ dmp_inner_gcd)rwrr8s r7rzPolyElement._gcdsTvv ;;  99Q<  [[  99Q< %%a+ +r9ct||SrZrrs r7rzPolyElement._gcd_ZZsa|r9cx|}|j}|j|jj}|j \}}|j \}}|j |}|j |}|j |\}}} |j |}|j|j}} |j |j|jj| |}| j |j|jj| |} ||| fS)Nr) r8rr4r9r?r0rrrmrer) rrrwr8r.rrrrrrSs r7rzPolyElement._gcd_QQs vv::T[[%9%9%;:< A A JJx  JJx iil 3 JJt ttQWWY1ll4 ++DKKOOAr,BCll4 ++DKKOOAr,BC#s{r9cB|}|j}|s||jfS|j}|jr |js|j |\}}}n|j |j}|j\} }|j\} }|j|}|j|}|j |\}}}|jj | | \}} } |j|}|j|}|j| }|j| }|j} | |jk(r||}}||fS| |j k(r | | }}||fS|j| }|j| }||fS)a Cancel common factors in a rational function ``f/g``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> (2*x**2 - 2).cancel(x**2 - 2*x + 1) (2*x + 2, x - 1) r) r8rr4r7r8rrr9r?r0recanonical_unit) rrrwr8r4rSr r#r.cqcpus r7cancelzPolyElement.cancels vvdhh; F$9$9kk!nGAq!zz):z;HNN$EBNN$EB 8$A 8$Akk!nGAq! 11"b9IAr2 4 A 4 A R A R A      ?aqA!t 6::+ 2rqA !t  QA QA!t r9cd|jj}|j|jSrZ)r8r4rr)rwr4s r7rzPolyElement.canonical_units$$$QTT**r9c|j}|j|}|j|}|j}|j D]7\}}||s |j ||}|j |||z||<9|S)a!Computes partial derivative in ``x``. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring("x,y", ZZ) >>> p = x + x**2*y**3 >>> p.diff(x) 2*x*y**3 + 1 )r8rrrr)rr) rwr0r8rrRrrres r7diffzPolyElement.diffsvv JJqM    " II;;= 6KD%Aw&&tQ/uT!W}5! 6r9c$dt|cxkr|jjkr;nn8|jt t |jj |Std|jjdt|)Nrz expected at least 1 and at most z values, got )ryr8revaluaterBrGr2r)rwrEs r7r!zPolyElement.__call__ sb s6{ *affll *::d3qvv{{F#;<= =TUTZTZT`T`beflbmno or9c|}t|tr\|Z|d|ddc\}}}|j||}|s|S|Dcgc]\}}|j||f}}}|j|S|j}|j |}|j j|}|jdk(r=|j j}|jD]\\} } || || zzz }|S|j|j} |jD]@\} } | || d|| |dzdz} } | || zz} | | vr| | | z} | r| | | <5| | =9| s<| | | <B| Scc}}w)Nrr) r[rBrrr8rr4rrrr)) rr0rnrwXYr8rresultrrrrs r7rzPolyElement.evaluate s  a 19!aeIFQA 1a A346!Qqvvay!n66zz!}$vv JJqM KK   " ::?[[%%F {{} % e%1*$ %M99Q<$$D ! , u 8U2AYqst%<5ad D=!DK/E&+U  K&+U  ,KA7sE#cT|}t|tr ||D]\}}|j||}|S|j}|j |}|j j |}|jdk(rL|j j}|jD]\\}} || ||zzz }|j|S|j} |jD]C\} } | || d|dz| |dzdz} }| ||zz} | | vr| | | z} | r| | | <8| | =<| s?| | | <E| S)Nrrk) r[rBsubsr8rr4rrrr)r) rr0rnrwrr8rrrrrrs r7rzPolyElement.subs2 sO  a 19 !1FF1aL !Hvv JJqM KK   " ::?[[%%F {{} % e%1*$ %??6* *99D ! , u 8U2AY%5acd %C5ad D=!DK/E&+U  K&+U  ,Kr9c||j}|j}|j}|s||jgfSt |Dcgc]}|j |dzc}ifd}t t |dz }t t |dd}|j}|rd\} } } t|jD]E\}\} tfd|Dstdt|D} | | kDs@| | } } } G| dk7r| | c} nnwg}tddd zD]\}}|j||z ||jt|| z }| }t|D]\}}||||z}||z}|rt t|j}|||fScc}w) aX Rewrite *self* in terms of elementary symmetric polynomials. Explanation =========== If this :py:class:`~.PolyElement` belongs to a ring of $n$ variables, we can try to write it as a function of the elementary symmetric polynomials on $n$ variables. We compute a symmetric part, and a remainder for any part we were not able to symmetrize. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> R, x, y = ring("x,y", ZZ) >>> f = x**2 + y**2 >>> f.symmetrize() (x**2 - 2*y, 0, [(x, x + y), (y, x*y)]) >>> f = x**2 - y**2 >>> f.symmetrize() (x**2 - 2*y, -2*y**2, [(x, x + y), (y, x*y)]) Returns ======= Triple ``(p, r, m)`` ``p`` is a :py:class:`~.PolyElement` that represents our attempt to express *self* as a function of elementary symmetric polynomials. Each variable in ``p`` stands for one of the elementary symmetric polynomials. The correspondence is given by ``m``. ``r`` is the remainder. ``m`` is a list of pairs, giving the mapping from variables in ``p`` to elementary symmetric polynomials. The triple satisfies the equation ``p.compose(m) + r == self``. If the remainder ``r`` is zero, *self* is symmetric. If it is nonzero, we were not able to represent *self* as symmetric. See Also ======== sympy.polys.polyfuncs.symmetrize References ========== .. [1] Lauer, E. Algorithms for symmetrical polynomials, Proc. 1976 ACM Symp. on Symbolic and Algebraic Computing, NY 242-247. https://dl.acm.org/doi/pdf/10.1145/800205.806342 rc8||fvr ||z||f<||fSrZrX)rr poly_powersrTs r7get_poly_powerz.PolyElement.symmetrize..get_poly_power s41v[(&+Ahk QF#1v& &r9rr)rNNc3:K|]}||dzk\ywrrX)r^rrs r7r`z)PolyElement.symmetrize.. s"AAuQx5Q</Asc3,K|] \}}||zywrZrX)r^rrRs r7r`z)PolyElement.symmetrize.. s EA1 EsNrk)rr8rrrrrBrr/rdrvrGrrrxr2)rrwr8rrrrweights symmetric_height_monom_coeffrheight exponentsrm2productrrrrTs @@@r7 symmetrizezPolyElement.symmetrizeY sv IIKvv JJdii# #388E5AAA EWe1D EEF'28%  G"}%v uIeU12Y%56 *B  b) * uY'7? ?IG!), 01>!Q// 0 LA14s499e,-!W$$S=s F9c |j}|j}tt|jt |j  |||fg}nVt|tr t|}n:t|trt|j fd}n tdt|D]!\}\}} ||j|f||<#|jD]]\}} t|}|j} |D]\} }|| dc} || <| s| || zz} | j!t#|| f} || z }_|S)Nc|dSr!rX)r=gens_maps r7rpz%PolyElement.compose.. sx!~r9rtz9expected a generator, value pair a sequence of such pairsr)r8rrFrGr2rrr[rBrOrHrrrr)rrtrx)rwr0rnr8r replacementsr=rrrsubpolyrrrs @r7rzPolyElement.compose sDvvyyDIIuTZZ'89: =F8L!T"#Aw At$%aggi5MN  !\]]"<0 >IAv1'{DMM!,<=LO >KKM LE5KEhhG$ $1#Ah 58q!tOG $ &&e e'<=G GOD  r9c|}|jj|}|jDcgc]\}}|||k(s||f}}}|s|jjSt |\}} |Dcgc]}|d|dz||dzdz}}|jj t t || Scc}}wcc}w)aU Coefficient of ``self`` with respect to ``x**deg``. Treating ``self`` as a univariate polynomial in ``x`` this finds the coefficient of ``x**deg`` as a polynomial in the other generators. Parameters ========== x : generator or generator index The generator or generator index to compute the expression for. deg : int The degree of the monomial to compute the expression for. Returns ======= :py:class:`~.PolyElement` The coefficient of ``x**deg`` as a polynomial in the same ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x, y, z = ring("x, y, z", ZZ) >>> p = 2*x**4 + 3*y**4 + 10*z**2 + 10*x*z**2 >>> deg = 2 >>> p.coeff_wrt(2, deg) # Using the generator index 10*x + 10 >>> p.coeff_wrt(z, deg) # Using the generator 10*x + 10 >>> p.coeff(z**2) # shows the difference between coeff and coeff_wrt 10 See Also ======== coeff, coeffs Nrkr)r8rr)rrGrIrF) rr0degr rrRrSr/rVrOs r7 coeff_wrtzPolyElement.coeff_wrt sT  FFLLO$%KKMADAqQqTS[!QAA66;; e4:;q!BQ%$,1q56*;;vvS%8 9::B >> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2 >>> f.prem(g) # first generator is chosen by default if it is not given -4*y + 4 >>> f.rem(g) # shows the differnce between prem and rem x**2 + x*y >>> f.prem(g, y) # generator is given 0 >>> f.prem(g, 1) # generator index is given 0 See Also ======== pdiv, pquo, pexquo, sympy.polys.domains.ring.Ring.rem rrrr8rr1rrr2)rrr0rwdfdgrdrrlc_gxplc_rrRrrSs r7premzPolyElement.prem sl  FFLLO XXa[ XXa[ 6#$9: :22 7H GaK{{1b! VV[[^;;q"%D7AEqADAD2q5 AAA!BBw AI1u r9c|}|jj|}|j|}|j|}|dkr td|||}}}||kr||fS||z dz} |j ||} |jj |} |j ||} ||z | dz } } || z}|| | | zzz}|| z}|| z| | zz}||z }|j|}||krnY| | z}||z}||z}||fS)a| Computes the pseudo-division of the polynomial ``self`` with respect to ``g``. The pseudo-division algorithm is used to find the pseudo-quotient ``q`` and pseudo-remainder ``r`` such that ``m*f = g*q + r``, where ``m`` represents the multiplier and ``f`` is the dividend polynomial. The pseudo-quotient ``q`` and pseudo-remainder ``r`` are polynomials in the variable ``x``, with the degree of ``r`` with respect to ``x`` being strictly less than the degree of ``g`` with respect to ``x``. The multiplier ``m`` is defined as ``LC(g, x) ^ (deg(f, x) - deg(g, x) + 1)``, where ``LC(g, x)`` represents the leading coefficient of ``g``. It is important to note that in the context of the ``prem`` method, multivariate polynomials in a ring, such as ``R[x,y,z]``, are treated as univariate polynomials with coefficients that are polynomials, such as ``R[x,y][z]``. When dividing ``f`` by ``g`` with respect to the variable ``z``, the pseudo-quotient ``q`` and pseudo-remainder ``r`` satisfy ``m*f = g*q + r``, where ``deg(r, z) < deg(g, z)`` and ``m = LC(g, z)^(deg(f, z) - deg(g, z) + 1)``. In this function, the pseudo-remainder ``r`` can be obtained using the ``prem`` method, the pseudo-quotient ``q`` can be obtained using the ``pquo`` method, and the function ``pdiv`` itself returns a tuple ``(q, r)``. Parameters ========== g : :py:class:`~.PolyElement` The polynomial to divide ``self`` by. x : generator or generator index, optional The main variable of the polynomials and default is first generator. Returns ======= :py:class:`~.PolyElement` The pseudo-division polynomial (tuple of ``q`` and ``r``). Raises ====== ZeroDivisionError : If ``g`` is the zero polynomial. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2 >>> f.pdiv(g) # first generator is chosen by default if it is not given (2*x + 2*y - 2, -4*y + 4) >>> f.div(g) # shows the difference between pdiv and div (0, x**2 + x*y) >>> f.pdiv(g, y) # generator is given (2*x**3 + 2*x**2*y + 6*x**2 + 2*x*y + 8*x + 4, 0) >>> f.pdiv(g, 1) # generator index is given (2*x**3 + 2*x**2*y + 6*x**2 + 2*x*y + 8*x + 4, 0) See Also ======== prem Computes only the pseudo-remainder more efficiently than `f.pdiv(g)[1]`. pquo Returns only the pseudo-quotient. pexquo Returns only an exact pseudo-quotient having no remainder. div Returns quotient and remainder of f and g polynomials. rrrr)rrr0rwrrr#rrrrrrrQrrrSs r7pdivzPolyElement.pdivv s8`  FFLLO XXa[ XXa[ 6#$9: :ab1 7a4K GaK{{1b! VV[[^;;q"%D7AEqADAT2q5L ADAD2q5 AAA!BBw%( !G E E!t r9c0|}|j||dS)aW Polynomial pseudo-quotient in multivariate polynomial ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> f.pquo(g) 2*x >>> f.quo(g) # shows the difference between pquo and quo 0 >>> f.pquo(h) 2*x + 2*y - 2 >>> f.quo(h) # shows the difference between pquo and quo 0 See Also ======== prem, pdiv, pexquo, sympy.polys.domains.ring.Ring.quo r)r)rrr0rws r7pquozPolyElement.pquo s6 vva|Ar9cd|}|j||\}}|jr|St||)a Polynomial exact pseudo-quotient in multivariate polynomial ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> f.pexquo(g) 2*x >>> f.exquo(g) # shows the differnce between pexquo and exquo Traceback (most recent call last): ... ExactQuotientFailed: 2*x + 2*y does not divide x**2 + x*y >>> f.pexquo(h) Traceback (most recent call last): ... ExactQuotientFailed: 2*x + 2 does not divide x**2 + x*y See Also ======== prem, pdiv, pquo, sympy.polys.domains.ring.Ring.exquo )rrr")rrr0rwr#rs r7pexquozPolyElement.pexquo s5: vva|1 99H%a+ +r9c|}|jj|}|j|}|j|}||kr||}}||}}|dk(rddgS|dk(r|dgS||g}||z }d|dzz}|j||} | |z} |j ||} | |z} d| g} | } | r| j|} |j | || | || z f\}}}}| | |zz}|j||} | j |} |j || } |dkDr | |z}| |dz z}|j |} n| } | j | | r|S)a Computes the subresultant PRS of two polynomials ``self`` and ``g``. Parameters ========== g : :py:class:`~.PolyElement` The second polynomial. x : generator or generator index The variable with respect to which the subresultant sequence is computed. Returns ======= R : list Returns a list polynomials representing the subresultant PRS. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2*y + x*y >>> g = x + y >>> f.subresultants(g) # first generator is chosen by default if not given [x**2*y + x*y, x + y, y**3 - y**2] >>> f.subresultants(g, 0) # generator index is given [x**2*y + x*y, x + y, y**3 - y**2] >>> f.subresultants(g, y) # generator is given [x**2*y + x*y, x + y, x**3 + x**2] rrr)r8rr1rrrr$)rrr0rwrrRrdrorlcrSSr=r r#s r7 subresultantszPolyElement.subresultants7 sD  FFLLO HHQK HHQK q5aqAaqA 6q6M 6q6M F E QUO FF1aL E[[A  !G F B A HHQKAq!a%JAq!Qa1f Aq! A AQ"B1uSQJ!a%LGGAJC HHaRL'*r9c:|jj||SrZ)r8dmp_half_gcdexrs r7 half_gcdexzPolyElement.half_gcdex svv$$Q**r9c:|jj||SrZ)r8 dmp_gcdexrs r7gcdexzPolyElement.gcdex vv1%%r9c:|jj||SrZ)r8 dmp_resultantrs r7 resultantzPolyElement.resultant svv##Aq))r9c8|jj|SrZ)r8dmp_discriminantrs r7 discriminantzPolyElement.discriminant svv&&q))r9cz|jjr|jj|Std)Nzpolynomial decomposition)r8r dup_decomposer#rs r7 decomposezPolyElement.decompose s0 66  66''* *-.HI Ir9c||jjr|jj||Std)Nzshift: use shift_list instead)r8r dup_shiftr#rwrns r7shiftzPolyElement.shift s2 66  66##Aq) )-.MN Nr9c:|jj||SrZ)r8 dmp_shiftrs r7 shift_listzPolyElement.shift_list rr9cz|jjr|jj|Std)Nzsturm sequence)r8r dup_sturmr#rs r7sturmzPolyElement.sturm s0 66  66##A& &-.>? ?r9c8|jj|SrZ)r8 dmp_gff_listrs r7gff_listzPolyElement.gff_list vv""1%%r9c8|jj|SrZ)r8dmp_normrs r7normzPolyElement.norm svvq!!r9c8|jj|SrZ)r8 dmp_sqf_normrs r7sqf_normzPolyElement.sqf_norm r r9c8|jj|SrZ)r8 dmp_sqf_partrs r7sqf_partzPolyElement.sqf_part r r9c<|jj||S)N)rd)r8 dmp_sqf_list)rwrds r7sqf_listzPolyElement.sqf_list svv""1#"..r9c8|jj|SrZ)r8dmp_factor_listrs r7 factor_listzPolyElement.factor_list svv%%a((r9rZ)F)r}rrrr%r'rrrrr0r3r2r?rBrrrJrNrRrTrWrYr[r^rrcrrhrjr\r rr`rrrtrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr __floordiv__ __rfloordiv__rrrrr$r r r1r4r9r;rr?rrKrrDrGrrJrPrOrVr/rZrr)rarcrerrrkrmrerrrtrrxr{rrrrrrrrrrrrrrrrrrr!rrrrrrrrrrrrrrrrrrr r rrrrrXr9r7rjrj<sF?$%3 E 8> =M" 44G0)"  2*>"H.`++MM==55558888**11@@@@## ++ GG R4l(4l60d:.(` :"H:,%,%2L MBJX+Z,*X#J= ?= ?(5#BJ544*11( P;4;474#!"#!"!F ( &  <$J PB !>," ,*6p+2p *X%Nk%Z@3;jYv||<#,JXz+&**J O &@ &"&&/)r9rjN)Qr __future__rtypingroperatorrrrrr r functoolsr typesr sympy.core.exprr sympy.core.intfuncrsympy.core.symbolrrrcsympy.core.sympifyrrsympy.ntheory.multinomialrsympy.polys.compatibilityrsympy.polys.constructorrsympy.polys.densebasicrrr!sympy.polys.domains.domainelementr"sympy.polys.domains.polynomialringrsympy.polys.heuristicgcdrsympy.polys.monomialsrsympy.polys.orderingsrsympy.polys.polyerrorsr r!r"r#sympy.polys.polyoptionsr$rzr%r|r&sympy.polys.polyutilsr'r(r)sympy.printing.defaultsr*sympy.utilitiesr+r,sympy.utilities.iterablesr-sympy.utilities.magicr.r8r;r?rUrfrg__annotations__r1rFrjrXr9r7r6s"-- #93>,4CC;=+-%66GG==3+1) #!!<!$<!$<22h @! ^ uup I&)-+tI&)r9