gF4UdZddlmZddlmZddlmZmZmZm 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?eeOZPd&S)'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 M/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/polys/rings.pyringr8#s$8 Wfe , ,E 8ej  c6t|||}||jfS)aConstruct a polynomial ring returning ``(ring, (x_1, ..., x_n))``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import xring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, (x, y, z) = xring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) r0r3s r7xringr;Bs"8 Wfe , ,E 5: r9cpt|||}td|jD|j|S)aConstruct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import vring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> vring("x,y,z", ZZ, lex) Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z # noqa: x + y + z >>> type(_) cg|] }|j S)name).0syms r7 zvring..}s 1 1 13ch 1 1 1r9)r1r.rr2r3s r7vringrCas=6 Wfe , ,E 1 1%- 1 1 15:>>> Lr9c( d}t|s|gd}}ttt|}t ||}t ||\}}|j^td|Dg}t||\|_}tt|| fd|D}t|j |j|j }tt|j|} |r || dfS|| fS)adConstruct a ring deriving generators and domain from options and input expressions. Parameters ========== exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable) symbols : sequence of :class:`~.Symbol`/:class:`~.Expr` options : keyword arguments understood by :class:`~.Options` Examples ======== >>> from sympy import sring, symbols >>> x, y, z = symbols("x,y,z") >>> R, f = sring(x + 2*y + 3*z) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> f x + 2*y + 3*z >>> type(_) FTNcPg|]#}t|$Sr>listvalues)r@reps r7rBzsring..s(;;;ctCJJLL));;;r9)optcPg|]"}fd|D#S)c(i|]\}}||Sr>r>)r@mc coeff_maps r7 z$sring...s#999TQIaL999r9)items)r@rIrOs r7rBzsring..s6JJJc9999SYY[[999JJJr9r)r-rGmaprr&r)r4sumrdictzipr1r2r5 from_dict) exprsroptionssinglerJrepscoeffs coeffs_domr6polysrOs @r7sringr^s 4F u  &v We$$ % %E  ) )C)44ID# z;;T;;;R@@!1&c!B!B!B JVZ0011 JJJJTJJJ SXsz39 5 5E U_d++ , ,E uQx  u~r9cHt|tr|rt|dndSt|tr|fSt |rCt d|Drt|St d|Dr|St d)NT)seqr>c3@K|]}t|tVdSN) isinstancestrr@ss r7 z!_parse_symbols..s,33az!S!!333333r9c3@K|]}t|tVdSrb)rcr res r7rgz!_parse_symbols..s,66At$$666666r9zbexpected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions)rcrd_symbolsr r-allr!rs r7_parse_symbolsrls'3.5=xT****2= GT " "z W   337333 3 3 G$$ $ 66g666 6 6 N ~  r9zdict[Any, Any] _ring_cachec.eZdZdZefdZdZdZdZdZ dZ dZ d%d Z d Z ed Zed Zd&dZdZdZdZeZd&dZd&dZdZdZdZdZdZdZdZdZdZ edZ!edZ"dZ#d Z$d!Z%d"Z&d#Z'd$Z(d S)'r1z*Multivariate distributed polynomial ring. ctt|}t|}tj|}t j|j|||f}t|}|v|j r3t|t|j zrtdt|}||_t!||_t%dt&fd|i|_||_ ||_||_|_d|z|_||_t|j|_|j|jfg|_|rt=|}||_ |!|_"|#|_$|%|_&|'|_(|)|_*|+|_,n5d}||_ ||_"d|_$||_&||_(||_*||_,tZur t\|_/n fd|_/ta|j |jD]B\} } tc| tdr(| j3} ti|| stk|| | C|t|<|S)Nz7polynomial ring and it's ground domain share generators PolyElementr8rcdSNr>r>)abs r7z"PolyRing.__new__..srr9cdSrsr>)rtrurNs r7rvz"PolyRing.__new__..sbr9c&t|S)Nkey)max)fr5s r7rvz"PolyRing.__new__..sS->->->r9)6tuplerllen DomainOpt preprocessOrderOpt__name__rmget is_Compositesetrr!object__new__ _hash_tuplehash_hashtyperpdtypengensr4r5 zero_monom_gensr2 _gens_setone_onerr monomial_mulpow monomial_powmulpowmonomial_mulpowldiv monomial_ldivdiv monomial_divlcm monomial_lcmgcd monomial_gcdrr{ leading_expvrUrcrr?hasattrsetattr) clsrr4r5rrobjcodegenmonunitsymbol generatorr?s ` r7rzPolyRing.__new__sw//00G %f--#E**|WeVUC ook** ;" as7||c&.6I6I'I a%&_```..%%C)CO[))CI][NVSMJJCI!CKCICJCI!%ZCNyy{{CHMMCM45CH +%e,,#*;;== #*;;== &-nn&6&6#$+LLNN!#*;;== #*;;== #*;;==  )/#* #* &8&8#$+!#* #* #* ||#&  #>#>#>#> %(ch%?%? 6 6! ff--6!;D"3--6T9555'*K $ r9c|jj}g}t|jD]8}||}|j}|||<||9t|S)z(Return a list of polynomial generators. )r4rrangermonomial_basiszeroappendr})selfrriexpvpolys r7rzPolyRing._genssmkotz""  A&&q))D9DDJ LL    U||r9c*|j|j|jfSrb)rr4r5rs r7__getnewargs__zPolyRing.__getnewargs__s dk4:66r9cx|j}|d=|D]}|dr||=|S)Nr monomial_)__dict__copy startswith)rstaterzs r7 __getstate__zPolyRing.__getstate__sM ""$$ . !  C~~k** #J r9c|jSrb)rrs r7__hash__zPolyRing.__hash__ s zr9ct|to5|j|j|j|jf|j|j|j|jfkSrb)rcr1rr4rr5rothers r7__eq__zPolyRing.__eq__#sI%**D \4; DJ ? ]EL%+u{ C D Dr9c||k Srbr>rs r7__ne__zPolyRing.__ne__(s5=  r9NcZ||p|j|p|j|p|jSrb) __class__rr4r5)rrr4r5s r7clonezPolyRing.clone+s/~~g5v7LeNaW[Wabbbr9c@dg|jz}d||<t|S)zReturn the ith-basis element. r)rr})rrbasiss r7rzPolyRing.monomial_basis.s$DJaU||r9c*|Srb)rrs r7rz PolyRing.zero4szz||r9c6||jSrb)rrrs r7rz PolyRing.one8szz$)$$$r9c8|j||Srb)r4convertrelement orig_domains r7 domain_newzPolyRing.domain_new<s{""7K888r9c8||j|Srb)term_newr)rcoeffs r7 ground_newzPolyRing.ground_new?s}}T_e444r9cL||}|j}|r|||<|Srb)rr)rmonomrrs r7rzPolyRing.term_newBs0&&y  DK r9ct|tr`||jkr|St|jtr*|jj|jkr||St dt|trt dt|tr| |St|tr; | |S#t$r| |cYSwxYwt|tr||S||S)N conversionparsing)rcrpr8r4rrNotImplementedErrorrdrTrVrG from_terms ValueError from_listr from_exprrrs r7ring_newzPolyRing.ring_newIsC g{ + + ,w|##DK88 8T[=MQXQ]=]=]w///),777  % % ,%i00 0  & & ,>>'** *  & & , /w/// / / /~~g..... /  & & ,>>'** *??7++ +sC//DDc||j}|j}|D]\}}|||}|r|||<|Srb)rrrQ)rrrrrrrs r7rVzPolyRing.from_dictasS_ y#MMOO $ $LE5Juk22E $#U  r9cH|t||Srb)rVrTrs r7rzPolyRing.from_termsls~~d7mm[999r9cd|t||jdz |jSNr)rVrrr4rs r7rzPolyRing.from_listos(~~k'4:a<MMNNNr9cXjfdt|S)Nc |}||S|jr5ttt t |jS|jr5ttt t |jS| \}}|j r!|dkr|t|zS  |Sr)ris_Addr rrGrRargsis_Mulr as_base_exp is_Integerintrr)exprrbaseexp_rebuildr4mappingrs r7rz(PolyRing._rebuild_expr.._rebuildus D))I$   Ac4Hdi(@(@#A#ABBB Ac4Hdi(@(@#A#ABBB!,,.. c>AcAgg#8D>>3s8833??6>>$+?+?@@@r9)r4r)rrrrr4s` `@@r7 _rebuild_exprzPolyRing._rebuild_exprrsU A A A A A A A A$x &&&r9cttt|j|j} |||}||S#t$rtd|d|wxYw)Nz6expected an expression convertible to a polynomial in z, got ) rTrGrUrr2rrr r)rrrrs r7rzPolyRing.from_exprstC di8899:: '%%dG44D==&& & p p p*cgcgcgimimnoo o ps A!! Bc8||jrd}n d}nt|tr?|}d|kr ||jkrn|j |kr |dkr| dz }ntd|zt||jr< |j|}n#t$rtd|zwxYwt|tr< |j|}n2#t$rtd|zwxYwtd|z|S)z+Compute index of ``gen`` in ``self.gens``. Nrrzinvalid generator index: %szinvalid generator: %szEexpected a polynomial generator, an integer, a string or None, got %s) rrcrrrr2indexrdr)rgenrs r7rzPolyRing.indexsg ;z  S ! ! lAAvv!dj..*!!a2ggBF !>!DEEE TZ ( ( l @IOOC(( @ @ @ !83!>??? @ S ! ! l @L&&s++ @ @ @ !83!>??? @dgjjkk ks<BB4 C((Dctt|j|fdt|jD}|s|jS||S)z,Remove specified generators from this ring. c"g|] \}}|v | Sr>r>r@rrfindicess r7rBz!PolyRing.drop..s'OOO$!QQg=M=MA=M=M=Mr9rk)rrRr enumeraterr4rrr2rrs @r7dropz PolyRing.dropscc$*d++,,OOOO)DL"9"9OOO /; ::g:.. .r9cZ|j|}|s|jS||S)Nrk)rr4r)rrzrs r7 __getitem__zPolyRing.__getitem__s2,s# /; ::g:.. .r9c|jjst|jdr ||jjSt d|jz)Nr4r4z%s is not a composite domain)r4rrrrrs r7 to_groundzPolyRing.to_groundsS ; # Kwt{H'E'E K::T[%7:88 8;dkIJJ Jr9c t|Srbrrs r7 to_domainzPolyRing.to_domainsd###r9cFddlm}||j|j|jS)Nr) FracField)sympy.polys.fieldsrrr4r5)rrs r7to_fieldzPolyRing.to_fields.000000yt{DJ???r9c2t|jdkSrr~r2rs r7 is_univariatezPolyRing.is_univariates49~~""r9c2t|jdkSrr rs r7is_multivariatezPolyRing.is_multivariates49~~!!r9cp|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.addsS" I  C3 666 XTXs^#Sr9cp|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 rrs r7rz PolyRing.mulsS" H  C3 666 XTXs^#Sr9ctt|j|fdt|jD}fdt|jD}|s|S|||j|S)zd Remove specified generators from the ring and inject them into its domain. c"g|] \}}|v | Sr>r>rs r7rBz+PolyRing.drop_to_ground..s'MMMAAWr>)r@rrrs r7rBz+PolyRing.drop_to_ground..s'KKK3!7:J:J:J:J:Jr9rr4)rrRrrrr2rrrs @r7drop_to_groundzPolyRing.drop_to_grounds c$*d++,,MMMM4>D::d4jj:11 1Kr9ct|jt|}|t |S)z9Add the elements of ``symbols`` as generators to ``self``rkr)rrrs r7add_genszPolyRing.add_gens&s?4<  &&s7||44zz$t**z---r9c|dks ||jkrtd|d|j|s|jS|j}t t |jt|D]Rtfdt |jD}|| ||j jz }S|S)zo Return the elementary symmetric polynomial of degree *n* over this ring's generators. rz.Cannot generate symmetric polynomial of order z for c3:K|]}t|vVdSrb)r)r@rrfs r7rgz*PolyRing.symmetric_poly..7s-EEac!q&kkEEEEEEr9) rrr2rrr,rrr}rr4)rnrrrfs @r7symmetric_polyzPolyRing.symmetric_poly+s q55A NN*Z[Z[Z[]a]f]fghh h 8O9DU4:..A77 > >EEEE53D3DEEEEE eT[_===Kr9)NNNrb))r __module__ __qualname____doc__rrrrrrrrrrpropertyrrrrrr__call__rVrrrrrrrrrr r rrrrr r"r&r>r9r7r1r1s44,/????B   777DDD !!!cccc X%%X%9999555,,,,H    ::::OOO'''.'''>//////KKK$$$@@@##X#""X"66 H H H... r9r1ceZdZdZdZdZdZdZdZdZ 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~ZddZddZddZddZddZdZdZdZdZdZdZdZdZdZdZdZdZddZdZdS)rpz5Element of multivariate distributed polynomial ring. c,||Srb)r)rinits r7newzPolyElement.new?s~~d###r9c4|jSrb)r8rrs r7parentzPolyElement.parentBsy""$$$r9cR|jt|fSrb)r8rG itertermsrs r7rzPolyElement.__getnewargs__Es! 4 0 01122r9Nc|j}|>> 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 .|j|kr|S|jj|jkrTttt ||jj|j}|||jjS|||jjSrb)r8rrGrUr(rr4rV)rnew_ringtermss r7set_ringzPolyElement.set_ringqs 9 K Y ("2 2 2mD$)2CXEUVVWXXE&&udi.>?? ?%%dDI,<== =r9c|s |jj}nIt||jjkr,t d|jjdt|t |g|RS)Nz"Wrong number of symbols, expected z got )r8rr~rrr' as_expr_dict)rrs r7as_exprzPolyElement.as_exprzs} i'GG \\TY_ , ,*#g,,,0  d//11r>)r@rrto_sympys r7rPz,PolyElement.as_expr_dict..s'LLL<5%xxLLLr9)r8r4r@r3)rr@s @r7r<zPolyElement.as_expr_dicts49#,LLLL4>>;K;KLLLLr9cb|jj}|jr|js |j|fS|}|j|j}|j}|D]}|||| fd| D}|fS)Nc$g|] \}}||zf Sr>r>)r@kvcommons r7rBz,PolyElement.clear_denoms..s%BBBDAq1ah-BBBr9) r8r4is_Fieldhas_assoc_Ringrget_ringrdenomrHr/rQ)rr4 ground_ringrrIrrrEs @r7 clear_denomszPolyElement.clear_denomss! $f&; $:t# #oo'' o [[]] / /ESu..FFxxBBBBDJJLLBBBCCt|r9c^t|D] \}}|s||= dS)z+Eliminate monomials with zero coefficient. NrGrQ)rrCrDs r7 strip_zerozPolyElement.strip_zeros?&&  DAq G  r9c|s| St|tr+|j|jkrt||St |dkrdS||jj|kS)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)rcrpr8rTrr~rrp1p2s r7rzPolyElement.__eq__sw" 46M K ( ( 4RW-?-?;;r2&& & WWq[[566"',--3 3r9c||k Srbr>rPs r7rzPolyElement.__ne__s8|r9c&|j}t||jrt|t|krdS|jj}|D]}||||||sdSdSt|dkrdS |j|}|j| ||S#t$rYdSwxYw)z+Approximate equality test for polynomials. FTr) r8rcrrkeysr4almosteqr~rconstr )rQrR tolerancer8rVrCs r7rVzPolyElement.almosteqsw b$* % % G27799~~RWWYY//u{+HWWYY ! !x1r!ui88! 55!4 WWq[[5 G[((,,{++BHHJJIFFF"   uu s:D DDcHt||fSrb)r~r9rs r7sort_keyzPolyElement.sort_keysD 4::<<((r9ct||jjr0|||StSrb)rcr8rrZNotImplemented)rQrRops r7_cmpzPolyElement._cmps@ b"'- ( ( "2bkkmmR[[]]33 3! !r9c8||tSrb)r^rrPs r7__lt__zPolyElement.__lt__wwr2r9c8||tSrb)r^rrPs r7__le__zPolyElement.__le__rar9c8||tSrb)r^r rPs r7__gt__zPolyElement.__gt__rar9c8||tSrb)r^r rPs r7__ge__zPolyElement.__ge__rar9c|j}||}|jdkr ||jfSt |j}||=|||fS)Nrrk)r8rrr4rGrrrrr8rrs r7_dropzPolyElement._drops^y JJsOO :??dk> !4<((G djjj111 1r9cx||\}}|jjdkr.|jr|dSt d|z|j}|D]G\}}||dkr%t|}||=||t|<6t d|z|S)NrzCannot drop %sr) rjr8r is_groundrrrrQrGr})rrrr8rrCrDKs r7rzPolyElement.drops**S//4 9?a  ~ 9zz!}}$ !1C!78889D  = =1Q4199QA!%&DqNN$%5%;<<<Kr9c|j}||}t|j}||=|||||fS)Nr)r8rrGrrris r7_drop_to_groundzPolyElement._drop_to_groundsMy JJsOOt|$$ AJ$**WT!W*====r9c|jjdkrtd||\}}|j}|jjd}|D]o\}}|d|||dzdz}||vr"|||z|||<C||xx|||z|z cc<p|S)Nrz$Cannot drop only generator to groundr) r8rrrorr4r2r3 mul_ground)rrrr8rrrmons r7rzPolyElement.drop_to_grounds 9?a  CDD D&&s++4ykq! NN,, ? ?LE5)eAaCDDk)C$ %(]66u==S S c58m77>>>  r9cRt||jjdz |jjSr)rr8rr4rs r7to_densezPolyElement.to_dense s"T49?1#4di6FGGGr9c t|Srb)rTrs r7to_dictzPolyElement.to_dict#sDzzr9c|s$||jjjS|d}|d}|j}|j}|j} |j} g} |D]\} } |j| }|rdnd}| || | kr7|| }|r| dr |dd}n5|r| } | |jjj kr| | |d}nd }g}t| D]}| |}|s | |||d}|dkrO|t|ks|d kr| ||d }n|}| |||fz| d |z|r|g|z}| ||| d d vr1| d }|dkr| d dd | S)NMulAtom -  + -rT)strictrFz%s)r{rz)_printr8r4rrrrr9 is_negativerrr parenthesizerrjoinpopinsert)rprinter precedence exp_pattern mul_symbolprec_mul prec_atomr8rrzmsexpvsrrnegativesignscoeffsexpvrrrsexpheads r7rdzPolyElement.str&su 9>>$)"2"788 8e$v& y,  _::<< 2 2KD%{..u55H$/55%D MM$   rzz ..( 1 1# 6 6(#ABBZF#"FEDI,000$11%$1OOFFFE5\\ 0 01g --gaj)D-QQ!88c#hh#''&33C53QQ"LL~!=>>>>LL//// )5( MM*//%00 1 1 1 1 !9 & &::a==Du}} a%%%wwvr9c||jjvSrb)r8rrs r7 is_generatorzPolyElement.is_generatorVsty***r9cJ| p t|dko |jj|vSr)r~r8rrs r7rlzPolyElement.is_groundZs(xLCIINKty/Ct/KLr9cD| pt|dko |jdkSr)r~LCrs r7 is_monomialzPolyElement.is_monomial^s$xr|s r7is_zerozPolyElement.is_zerovs u r9c"||jjkSrb)r8rrs r7is_onezPolyElement.is_onezsAFJr9cJ|jj|jSrb)r8r4rrrs r7is_moniczPolyElement.is_monic~sv}##AD)))r9cd|jj|Srb)r8r4rcontentrs r7 is_primitivezPolyElement.is_primitives"v}##AIIKK000r9cXtd|DS)Nc3<K|]}t|dkVdSrNrSr@rs r7rgz(PolyElement.is_linear..,??u3u::???????r9rj itermonomsrs r7 is_linearzPolyElement.is_linear'?? ??????r9cXtd|DS)Nc3<K|]}t|dkVdS)Nrrs r7rgz+PolyElement.is_quadratic..rr9rrs r7 is_quadraticzPolyElement.is_quadraticrr9cR|jjsdS|j|SNT)r8r dmp_sqf_prs r7 is_squarefreezPolyElement.is_squarefrees)v| 4v"""r9cR|jjsdS|j|Sr)r8rdmp_irreducible_prs r7is_irreduciblezPolyElement.is_irreducibles)v| 4v''***r9cl|jjr|j|Std)Nzcyclotomic polynomial)r8r dup_cyclotomic_pr#rs r7 is_cyclotomiczPolyElement.is_cyclotomics5 6  G6**1-- --.EFF Fr9cd|d|DS)Ncg|] \}}|| f Sr>r>)r@rrs r7rBz'PolyElement.__neg__..s"PPPleU55&/PPPr9)r/r3rs r7__neg__zPolyElement.__neg__s-xxPPdnn>N>NPPPQQQr9c|Srbr>rs r7__pos__zPolyElement.__pos__s r9c`|s|S|j}t||jr]|}|j}|jj}|D]\}}||||z}|r|||<||= |St|trt|jtr|jj|jkrnPt|jjtr*|jjj|kr| |StS | |}|}|s|S|j } | |vr||| <n!|||  kr|| =n|| xx|z cc<|S#t$r tcYSwxYw)aAdd two polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> (x + y)**2 + (x - y)**2 2*x**2 + 2*y**2 )rr8rcrrr4rrQrpr__radd__r\rrrUr ) rQrRr8rrrrCrDcp2rs r7__add__zPolyElement.__add__s 7799 w b$* % % & A%C;#D   1C4LL1$AaDD!H K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% //"%%C A B"""!B%<<"bEEESLEEEH " " "! ! ! ! "s&FF-,F-c&|}|s|S|j} ||}|j}||vr|||<n!||| kr||=n||xx|z cc<|S#t $r t cYSwxYwrb)rr8rrrUr r\)rQr%rr8rs r7rzPolyElement.__radd__s GGII Hw ""AB"""2;;"bEEEQJEEEH " " "! ! ! ! "sA<<BBcX|s|S|j}t||jr]|}|j}|jj}|D]\}}||||z }|r|||<||= |St|trt|jtr|jj|jkrnPt|jjtr*|jjj|kr| |StS | |}|}|j }||vr| ||<n |||kr||=n||xx|zcc<|S#t$r tcYSwxYw)a.Subtract polynomial p2 from p1. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p1 = x + y**2 >>> p2 = x*y + y**2 >>> p1 - p2 -x*y + x )rr8rcrrr4rrQrpr__rsub__r\rrrUr ) rQrRr8rrrrCrDrs r7__sub__zPolyElement.__sub__s  7799 w b$* % % & A%C;#D   1C4LL1$AaDD!H K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% $$B AB"""2;;"bEEERKEEEH " " "! ! ! ! "s&FF)(F)c|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 r\)rQr%r8rrs r7rzPolyElement.__rsub__'sw ""A A $ $d8)$ FAH " " "! ! ! ! "s=AAcP|j}|j}|r|s|St||jr|j}|jj}|j}t|}|D].\}} |D]&\} } ||| } || || | zz|| <'/| |St|trt|jtr|jj|jkrnPt|jjtr*|jjj|kr| |StS ||}|D]\}} | |z} | r| ||<|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 )r8rrcrrr4rrGrQrNrpr__rmul__r\rr )rQrRr8rrrrp2itexp1v1exp2v2rrDs r7__mul__zPolyElement.__mul__Bs w I & &H DJ ' ' &%C;#D,L ##DHHJJ 4 4b $44HD"&,tT22C Sd^^be3AcFF4 LLNNNH K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% $$BHHJJ  brE AdGH " " "! ! ! ! "sFF%$F%c|jj}|s|S |j|}|D]\}}||z}|r|||<|S#t$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 )r8rrrQr r\)rQrRrrrrDs r7rzPolyElement.__rmul__ts GL H ""2&&BHHJJ  brE AdGH " " "! ! ! ! "sAA('A(c|j}|s|r|jStdt|dkryt |d\}}|j}||jjkr|||||<n||z||||<|St|}|dkrtd|dkr| S|dkr| S|dkr|| zSt|dkr| |S| |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)r8rrr~rGrQrr4rrrsquare_pow_multinomial _pow_generic)rr%r8rrrs r7__pow__zPolyElement.__pow__sYy  )x ((( YY!^^ --a0LE5 A ''16$##E1--..16$##E1--.H FF q55011 1 !VV99;;  !VV;;== !VV % % YY!^^((++ +$$Q'' 'r9c||jj}|} |dzr ||z}|dz}|sn|}|dz},|S)NTrr)r8rr)rr%rrNs r7rzPolyElement._pow_generics` IM  1u aCQ AQA r9ctt||}|jj}|jj}|}|jjj}|jj}|D]r\}} |} | } t||D]\} \} }| r|| | | } | || zz} t| } | }| | ||z}|r||| <k| |vr|| =s|Srb) rr~rQr8rrr4rrUr}r)rr% multinomialsrrr9rr multinomialmultinomial_coeff product_monom product_coeffrrrs r7rzPolyElement._pow_multinomials/D 1==CCEE )3Y)  y$y~.:  *K*&M-M'*;'>'> 0 0#^eU0$3OM5#$N$NM!UCZ/M-((E!EHHUD))E1E #U $K r9cN|j}|j}|j}t|}|jj}|j}tt|D]S}||}||} t|D]1} || } ||| } || || || zz|| <2T| d}|j}| D]&\} }|| | } || ||dzz|| <'| |S)asquare of a polynomial Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p.square() x**2 + 2*x*y**2 + y**4 r) r8rrrGrUr4rrr~imul_numrQrN)rr8rrrUrrrk1pkjk2rrCrDs r7rzPolyElement.squares9y IeDIIKK  {( s4yy!! 6 6AaBbB1XX 6 6!W"l2r**S$"T"X+5# 6 JJqMMeJJLL ) )DAqa##BCDMMAqD(AbEE r9c^|j}|stdt||jr||St|t rt|jtr|jj|jkrnPt|jjtr*|jjj|kr||StS | |}| || |fS#t$r tcYSwxYwNpolynomial division)r8ZeroDivisionErrorrcrrrpr4r __rdivmod__r\r quo_ground rem_groundr rQrRr8s r7 __divmod__zPolyElement.__divmod__s"w &#$9:: : DJ ' ' &66"::  K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[~~b)))%% :$$BMM"%%r}}R'8'89 9 " " "! ! ! ! "sDD,+D,ctSrbr\rPs r7rzPolyElement.__rdivmod__(r9c4|j}|stdt||jr||St|t rt|jtr|jj|jkrnPt|jjtr*|jjj|kr||StS | |}| |S#t$r tcYSwxYwr) r8rrcrremrpr4r__rmod__r\rrr rs r7__mod__zPolyElement.__mod__+sw &#$9:: : DJ ' ' &66"::  K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% %$$B==$$ $ " " "! ! ! ! "sDDDctSrbrrPs r7rzPolyElement.__rmod__Arr9cR|j}|stdt||jr$|jr||dzzS||St|t rt|jtr|jj|jkrnPt|jjtr*|jjj|kr| |StS | |}| |S#t$r tcYSwxYw)Nrr)r8rrcrrquorpr4r __rtruediv__r\rrr rs r7 __truediv__zPolyElement.__truediv__Ds)w &#$9:: : DJ ' ' &~ "28}$vvbzz! K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[r***%% %$$B==$$ $ " " "! ! ! ! "s(DD&%D&ctSrbrrPs r7rzPolyElement.__rtruediv__]rr9c|jj|jj}|j|jj|jrfd}nfd}|S)Ncf|\}}|\}}| kr|}n ||}||||fSdSrbr> a_lm_a_lc b_lm_b_lca_lma_lcb_lmb_lcr domain_quorrs r7term_divz'PolyElement._term_div..term_divlsX& d& d2:: EE(Lt44E$ **T4"8"8884r9cp|\}}|\}}| kr|}n ||}|||zs|||fSdSrbr>r s r7rz'PolyElement._term_div..term_divxs_& d& d2:: EE(Lt44E   **T4"8"8884r9)r8rr4rrrF)rr4rrrrs @@@r7 _term_divzPolyElement._term_dives Y !!Z y- ?   r9c|jd}t|trd}|g}t|st d|s|rjjfSgjfS|D]}|jkrt dt|}fdt|D}| }j}| }d|D} |rd} d} | |kr| dkr| } || || f| | || | | f} | G| \}}||  ||f|| <| || || f}d } n| d z } | |kr| dk| s4| } | | || f}|| =|| jkr||z }|r|s j|fS|d|fS||fS) 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 ringcg|] }j Sr>)rr@rr8s r7rBz#PolyElement.div..s * * *Adi * * *r9c6g|]}|Sr>)r)r@fxs r7rBz#PolyElement.div..s"000r""000r9rNr)r8rcrprjrrrr~rrrr _iadd_monom_iadd_poly_monomr)rfv ret_singler|rfqvrrrexpvsr divoccurredrtermexpv1rNr8s @r7rzPolyElement.divsh8y b+ & & JB2ww ;#$9:: : % %y$)++49}$ G GAv~~ !EFFF GG * * * *q * * * IIKK I>>##00R000 AKa%%K1,,~~''xqw%(BqE%(O1LMM##HE1qE--uaj99BqE**2a551"+>>A"#KKFAa%%K1,, ((MM44/22dG! " 4? " " FA   y!|#!uaxq5Lr9c|}t|tr|g}t|std|j}|j}|j}|j}|j}|}|j } | }|j } |r|D]} || | j } | j| \} }| D].\}}||| }| ||||zz }|s||=)|||</| }| |||f} nC| \}}||vr||xx|z cc<n|||<||=| }| |||f} ||Sr)rcrprjrr8r4rrrLTrrr3r)rGr|r8r4rrrrltfrgtqrMrNmgcgm1c1ltmltcs r7rzPolyElement.rems  a % % A1vv ;#$9:: :v{( I;;==d FFHHe & & &Xc14((>DAq"#++--''B)\"a00 ST]]QrT1!' !"$&AbEE..**C!1S6kE"S!88cFFFcMFFFF AcFcFnn&&?qv+C5 &8r9c8||dSNr)r)r|r%s r7rzPolyElement.quosuuQxx{r9cZ||\}}|s|St||rb)rr")r|r%qrs r7exquozPolyElement.exquos2uuQxx1 ,H%a++ +r9c||jjvr|}n|}|\}}||}||||<n||z }|r|||<n||=|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)rmccpselfrrrNs r7rzPolyElement._iadd_monom sz8 49& & &YY[[FFF e JJt   9 F4LL JA ! t 4L r9c&|}||jjvr|}|\}}|j}|jjj}|jj}|D].\} } || |} || || |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 )r8rrrr4rrrQ) rrRr5rQrMrNrrrrCrDkars r7rzPolyElement._iadd_poly_monom6s* " " "BAfw~"w+ HHJJ  DAqa##BCDMMAaC'E 2rFF r9c|j||stSdkrdStfd|DS)z The leading degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (``float('-inf')``) rc3(K|] }|V dSrbr>r@rrs r7rgz%PolyElement.degree..i'<> >r9c|j||stSdkrdStfd|DS)z The tail degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (``float('-inf')``) rc3(K|] }|V dSrbr>r;s r7rgz*PolyElement.tail_degree..r<r9)r8rrminrr=s @r7 tail_degreezPolyElement.tail_degreewr@r9c |stf|jjzStt t t t|S)z A tuple containing tail degrees in all variables. Note that the degree of 0 is negative infinity (``float('-inf')``) ) rr8rr}rRrFrGrUrrs r7 tail_degreeszPolyElement.tail_degreesrCr9c>|r|j|SdS)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'  9))$// /4r9cL|||jjjSrb)rr8r4rrrs r7 _get_coeffzPolyElement._get_coeffsxxdi.3444r9cv|dkr||jjSt||jjrit |}t|dkr5|d\}}||jjj kr||Std|z)a Returns the coefficient that stands next to the given monomial. Parameters ========== element : PolyElement (with ``is_monomial = True``) or 1 Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring("x,y,z", ZZ) >>> f = 3*x**2*y - x*y*z + 7*z**3 + 23 >>> f.coeff(x**2*y) 3 >>> f.coeff(x*y) 0 >>> f.coeff(1) 23 rrzexpected a monomial, got %s) rMr8rrcrrGr3r~r4rr)rrr9rrs r7rzPolyElement.coeffs4 a<<??49#788 8  1 1 2**,,--E5zzQ$Qx uDI,000??51116@AAAr9c@||jjS)z"Returns the constant coefficient. )rMr8rrs r7rWzPolyElement.conststy3444r9cP||Srb)rMrrs r7rzPolyElement.LCs t0022333r9cJ|}| |jjS|Srb)rr8rrLs r7LMzPolyElement.LMs(  "" <9' 'Kr9cr|jj}|}|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 )r8rrr4rrrrs r7 leading_monomzPolyElement.leading_monoms< IN  ""  +i&*AdGr9c|}||jj|jjjfS|||fSrb)rr8rr4rrMrLs r7r$zPolyElement.LTsG  "" <I($)*:*?@ @$//$//0 0r9c`|jj}|}| ||||<|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 )r8rrrTs r7 leading_termzPolyElement.leading_terms6 IN  ""  4jAdGr9c |jjntjturt |ddSt |fddS)Nc|dSr0r>)rs r7rvz%PolyElement._sorted..s qr9T)rzreversec&|dSr0r>)rr5s r7rvz%PolyElement._sorted..suQxr9)r8r5rrrsorted)rr`r5s `r7_sortedzPolyElement._sortedsd =IOEE'..E C<<##9#94HHH H##@#@#@#@$OOO Or9c@d||DS)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] cg|]\}}|Sr>r>)r@_rs r7rBz&PolyElement.coeffs..3s:::81e:::r9r9rr5s r7r[zPolyElement.coeffs$0;:tzz%'8'8::::r9c@d||DS)a Ordered list of polynomial monomials. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.monoms() [(2, 3), (1, 7)] >>> f.monoms(grlex) [(1, 7), (2, 3)] cg|]\}}|Sr>r>)r@rras r7rBz&PolyElement.monoms..Ms:::85!:::r9rbrcs r7monomszPolyElement.monoms5rdr9cl|t||S)aOrdered list of polynomial terms. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.terms() [((2, 3), 2), ((1, 7), 1)] >>> f.terms(grlex) [((1, 7), 1), ((2, 3), 2)] )r^rGrQrcs r7r9zPolyElement.termsOs(0||D..666r9cDt|S)z,Iterator over coefficients of a polynomial. )iterrHrs r7 itercoeffszPolyElement.itercoeffsiDKKMM"""r9cDt|S)z)Iterator over monomials of a polynomial. )rjrUrs r7rzPolyElement.itermonomsmDIIKK   r9cDt|S)z%Iterator over terms of a polynomial. )rjrQrs r7r3zPolyElement.itertermsqDJJLL!!!r9cDt|S)z+Unordered list of polynomial coefficients. rFrs r7 listcoeffszPolyElement.listcoeffsurlr9cDt|S)z(Unordered list of polynomial monomials. )rGrUrs r7 listmonomszPolyElement.listmonomsyrnr9cDt|S)z$Unordered list of polynomial terms. rMrs r7 listtermszPolyElement.listterms}rpr9c||jjvr||zS|s|dS|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)rrNrs r7rzPolyElement.imul_numsb4  Q3J  GGIII F  C cFFFaKFFFFr9c|jj}|j}|j}|D]}|||}|S)z*Returns GCD of polynomial's coefficients. )r8r4rrrk)r|r4contrrs r7rzPolyElement.contentsH{j\\^^ $ $E3tU##DD r9cX|}|||fS)z,Returns content and a primitive polynomial. )rr)r|rzs r7 primitivezPolyElement.primitives&yy{{Q\\$''''r9c>|s|S||jS)z5Divides all coefficients by the leading coefficient. )rrrs r7moniczPolyElement.monics# &H<<%% %r9cs |jjSfd|D}||S)Nc$g|] \}}||zf Sr>r>r@rrr>s r7rBz*PolyElement.mul_ground..s&FFF|ue5%'"FFFr9)r8rr3r/)r|r>r9s ` r7rqzPolyElement.mul_groundsF 6; FFFFq{{}}FFFuuU||r9c|jjfd|D}||S)Nc2g|]\}}||fSr>r>)r@f_monomf_coeffrrs r7rBz)PolyElement.mul_monom..s/]]]>Ngw<<//9]]]r9)r8rrQr/)r|rr9rs ` @r7 mul_monomzPolyElement.mul_monomsGv* ]]]]]RSRYRYR[R[]]]uuU||r9c|\|rs |jjS|jjkr|S|jjfd|D}||S)Nc8g|]\}}||zfSr>r>)r@rrrrrs r7rBz(PolyElement.mul_term..s3cccDTGW<<//?cccr9)r8rrrqrrQr/)r|r!r9rrrs @@@r7mul_termzPolyElement.mul_terms u ' '6;  af' ' '<<&& &v* ccccccXYX_X_XaXacccuuU||r9c(|jj}std|r |jkr|S|jr)|jfd|D}n fd|D}||S)Nrc2g|]\}}||fSr>r>)r@rrrr>s r7rBz*PolyElement.quo_ground..s,PPPucc%mm,PPPr9c.g|]\}}|z ||zfSr>r>rs r7rBz*PolyElement.quo_ground..s2```leUTY\]T]`ueqj)```r9)r8r4rrrFrr3r/)r|r>r4r9rs ` @r7rzPolyElement.quo_grounds ;#$9:: : AOOH ? a*CPPPPPPPPEE````akkmm```EuuU||r9c@\}}|std|s |jjS||jjkr||S|fd|D}|d|DS)Nrc(g|]}|Sr>r>)r@tr!rs r7rBz(PolyElement.quo_term..s%<<<((1d##<<)r@rs r7rBz(PolyElement.quo_term..s:::Q1=q===r9)rr8rrrrr3r/)r|r!rrr9rs ` @r7quo_termzPolyElement.quo_terms u '#$9:: : '6;  af' ' '<<&& &;;==<<<<r>)r@rrrs r7rBz,PolyElement.trunc_ground..s&LLL\UEueai(LLLr9)r8r4is_ZZr3rr/rN)r|rr9rrrs ` r7 trunc_groundzPolyElement.trunc_grounds 6=  ME !  - - u 16>>!AIE eU^,,,,  -MLLLQ[[]]LLLEuuU||  r9c|}|}|}|jj||}||}||}|||fSrb)rr8r4rr)rr'r|fcgcrs r7extract_groundzPolyElement.extract_groundsh  YY[[ YY[[fmB'' LL   LL  Aqyr9c|s|jjjS|jjj|fd|DS)Nc&g|] }|Sr>r>)r@r ground_abss r7rBz%PolyElement._norm..s#NNNUzz%00NNNr9)r8r4rabsrk)r| norm_funcrs @r7_normzPolyElement._normsR P6=% %*J9NNNNallnnNNNOO Or9c6|tSrb)rr{rs r7max_normzPolyElement.max_normwws||r9c6|tSrb)rrSrs r7l1_normzPolyElement.l1_normrr9cJ|j}|gt|z}dg|jz}|D]G}|D]0}t |D]\}}t |||||<1Ht |D] \}} | sd||< t |}td|Dr||fSg} |D]d}|j} | D]1\} } dt| |D}| | t |<2| | e|| fS)Nrrc3"K|] }|dkV dSrr>)r@rus r7rgz&PolyElement.deflate..0s&!!!qAv!!!!!!r9cg|] \}}||z Sr>r>r@rrs r7rBz'PolyElement.deflate..9s 444Aa1f444r9) r8rGrrrrr}rjrr3rUr)r|r%r8r]JrrrrMruHhIrNs r7deflatezPolyElement.deflatesevd1gg  C N ) )A ) )%e,,))DAq!a==AaDD) )aLL  DAq ! !HH !!q!!! ! ! e8O   A AKKMM $ $544Q444#%(( HHQKKKK!t r9c|jj}|D]1\}}dt||D}||t |<2|S)Ncg|] \}}||z Sr>r>rs r7rBz'PolyElement.inflate..Ds ---$!Q!A#---r9)r8rr3rUr})r|rrrrrs r7inflatezPolyElement.inflate@sWv{  # #HAu--#a))---A"DqNN r9cj|}|jj}|jsD|\}}|\}}|||}||z||}|js||S|Srb) r8r4rFr|rrrrqr~)rr'r|r4rrrNrs r7rzPolyElement.lcmIs  #KKMMEBKKMMEB 2r""A qSIIaeeAhh   <<?? "7799 r9c8||dSr0) cofactorsr|r's r7rzPolyElement.gcdYs{{1~~a  r9cT|s|s|jj}|||fS|s||\}}}|||fS|s||\}}}|||fSt|dkr||\}}}|||fSt|dkr||\}}}|||fS||\}\}}||\}}}||||||fSr)r8r _gcd_zeror~ _gcd_monomr_gcdr)r|r'rrcffcfgrs r7rzPolyElement.cofactors\s0  6;Dt# # ++a..KAsCc3;  ++a..KAsCc3;  VVq[[,,q//KAsCc3;  VVq[[,,q//KAsCc3; IIaLL 6AqffQii 3 ! ckk!nnckk!nn==r9cX|jj|jj}}|jr|||fS| || fSrb)r8rrr)r|r'rrs r7rzPolyElement._gcd_zerors:FJ T  "dC< 2tcT> !r9c$ |j}|jj}|jj|j}|jt |d\}}||c |D]\}}| | | | | fg} || | fg} | fd|D} | | | fS)NrcFg|]\}}||fSr>r>)r@r)r*_cgcd_mgcd ground_quors r7rBz*PolyElement._gcd_monom..s:ccc62rmmB.. 2u0E0EFcccr9) r8r4rrrrrGr3r/)r|r'r8 ground_gcdrmfcfr)r*rrrrrrrs @@@@r7rzPolyElement._gcd_monomys(v[_ [_ ( * akkmm$$Q'B2 ukkmm * *FB L++EJub))EE EEE5>" # #eemmB.. 2u0E0EFGHHeecccccccUVU`U`UbUbcccdd#s{r9c|j}|jjr||S|jjr||S|||Srb)r8r4is_QQ_gcd_QQr_gcd_ZZ dmp_inner_gcd)r|r'r8s r7rzPolyElement._gcdsYv ;  ,99Q<<  [  ,99Q<< %%a++ +r9c"t||Srbrrs r7rzPolyElement._gcd_ZZsa||r9c|}|j}||j}|\}}|\}}||}||}||\}}} ||}|j|}} || |j | |}| | |j | |} ||| fS)Nr) r8rr4rHrKr:rrr~rqr) rr'r|r8r8rr*rrrrNs r7rzPolyElement._gcd_QQs v::T[%9%9%;%;:<<  A  A JJx  JJx iill 3 JJt  tQWWYY1ll4  ++DKOOAr,B,BCCll4  ++DKOOAr,B,BCC#s{r9cl|}|j}|s ||jfS|j}|jr|js||\}}}n ||}|\} }|\} }| |}| |}||\}}}|j| | \}} } | |}| |}| | }| | }| } | |jkr||}}n=| |j kr| | }}n*| | }| | }||fS)a Cancel common factors in a rational function ``f/g``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> (2*x**2 - 2).cancel(x**2 - 2*x + 1) (2*x + 2, x - 1) r) r8rr4rFrGrrrHrKr:rqcanonical_unit) rr'r|r8r4rarr2r8cqcpus r7cancelzPolyElement.cancels v dh;  !F$9 !kk!nnGAq!!zz):):z;;HNN$$EBNN$$EB 8$$A 8$$Akk!nnGAq! 11"b99IAr2 4  A 4  A R  A R  A       ??aqAA 6:+  2rqAA QA QA!t r9cN|jj}||jSrb)r8r4rr)r|r4s r7rzPolyElement.canonical_units!$$QT***r9c(|j}||}||}|j}|D]D\}}||r7|||}||||z||<E|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 )r8rrrr3rr) r|r>r8rrMr'rres r7diffzPolyElement.diffsv JJqMM    " " I;;== 6 6KD%Aw 6&&tQ//uT!W}55!r9cdt|cxkr|jjkr=nn:|t t |jj|Std|jjdt|)Nrz expected at least 1 and at most z values, got )r~r8revaluaterGrUr2r)r|rHs r7r+zPolyElement.__call__ s s6{{ * * * *afl * * * * *::d3qv{F#;#;<<== =*TUTZT`T`T`beflbmbmbmnoo or9c |}t|trU|S|d|ddc\ }}| |}|s|S fd|D}||S|j}||}|j|}|jdkr5|jj}| D]\\}}||||zzz }|S| |j} | D]O\} }| || d|| |dzdz} }|||zz}| | vr|| | z}|r|| | <D| | =H|r|| | <P| S)NrrcDg|]\}}||fSr>)r)r@YrtXs r7rBz(PolyElement.evaluate.. s+666!Qqvvayy!n666r9) rcrGrr8rr4rrrr3r) rr>rtr|r8rresultr%rrrrs @r7rzPolyElement.evaluate s  a   %19!aeIFQA 1a  A %66661666zz!}}$v JJqMM K   " " :??[%F {{}} % % e%1*$M99Q<<$D ! , , u 8U2A2Yqstt%<5ad D==!DK/E(&+U  KK,&+U Kr9cf|}t|tr"| |D]\}}|||}|S|j}||}|j|}|jdkrH|jj}| D]\\}} || ||zzz }| |S|j} | D]R\} } | || d|dz| |dzdz} }| ||zz} | | vr| | | z} | r| | | <G| | =K| r| | | <S| S)Nrrq) rcrGsubsr8rr4rrrr3r) rr>rtr|rr8rrr%rrrs r7rzPolyElement.subs2 so  a   19 ! !1FF1aLLHv JJqMM K   " " :??[%F {{}} % % e%1*$??6** *9D ! , , u 8U2A2Y%5acdd %C5ad D==!DK/E(&+U  KK,&+U Kr9c|}|jj}|s |jgfSfdt |Difd}t t |dz }t t |dd}j}|rd\}}} t |D]V\} \} tfd|Dr3tdt|D} | |kr| | } }}W|dkr|| c} nng} tdd d zD]\}}| ||z | t| | z }| }t | D]\} }||| |z}||z}|t tj}|||fS) 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 c@g|]}|dzS)r)r&rs r7rBz*PolyElement.symmetrize.. s+<<)rr% poly_powersr]s r7get_poly_powerz.PolyElement.symmetrize..get_poly_power s71v[((&+Ahk QF#1v& &r9rrr)rNNc3BK|]}||dzkVdSrr>)r@rrs r7rgz)PolyElement.symmetrize.. s4AAAuQx5Q</AAAAAAr9c3&K|] \}}||zV dSrbr>)r@r%rMs r7rgz)PolyElement.symmetrize.. s* E EA1 E E E E E Er9Nrq)rr8rrrrGrr9rjr{rUrrr}r2)rr|r%rrweights symmetric_height_monom_coeffrrheight exponentsr+m2productrrrr]r8s @@@@r7 symmetrizezPolyElement.symmetrizeY sMv IIKKv J $di# #<<<<588<<<  ' ' ' ' ' ' uQU||$$uQ2''I  &4 #GVV%.qwwyy%9%9 G G!>E5AAAAAAAAAG E EWe1D1D E E EEEF''28%"}}%v uuIeU122Y%566 * *B  b)))) uY'7'7?? ?IG!),, 0 01>>!Q/// LA1 4s49e,,--!W$$r9c |j}|j}tt|jt |j |||fg}npt|trt|}nKt|tr't| fd}ntdt|D](\}\}} || |f||<)|D]d\}} t|}|j} |D]\} }|| dc} || <| r| || zz} | t#|| f} || z }e|S)Nc |dSr0r>)rCgens_maps r7rvz%PolyElement.compose.. sx!~r9ryz9expected a generator, value pair a sequence of such pairsr)r8rrTrUr2rrrcrGr]rQrrrr3rrr})r|r>rtr8r replacementsrCr'rrsubpolyrr%rs @r7r zPolyElement.compose svyDIuTZ'8'899:: =F8LL!T"" ^#Aww At$$ ^%aggii5M5M5M5MNNN  !\]]]"<00 > >IAv1'{DMM!,<,<=LOOKKMM  LE5KKEhG$ $ $1#Ah 58$q!tOG&&e e'<==G GODD r9c:|}|j|fd|D}|s |jjSt |\}}fd|D}|jt t ||S)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 c6g|]\}}|k||fSr>r>)r@rMrNdegrs r7rBz)PolyElement.coeff_wrt.. s*AAADAqQqTS[[!Q[[[r9cFg|]}|ddz|dzdzS)Nrqrr>)r@rMrs r7rBz)PolyElement.coeff_wrt.. s6;;;q!BQB%$,1q566*;;;r9)r8rr3rrUrVrT)rr>rrr9rgr[rs ` @r7 coeff_wrtzPolyElement.coeff_wrt sT  FLLOOAAAAAAKKMMAAA 6; e;;;;F;;;vS%8%8 9 9:::r9c|}|j|}||}||}|dkrtd||}}||kr|S||z dz}|||} |jj|} |||} ||z |dz }} || z} || z| | zz}| |z }||}||krnR| |z}||zS)a Pseudo-remainder of the polynomial ``self`` with respect to ``g``. The pseudo-quotient ``q`` and pseudo-remainder ``r`` with respect to ``z`` when dividing ``f`` by ``g`` 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)``. See :meth:`pdiv` for explanation of pseudo-division. 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-remainder polynomial. 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.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 rrrr8rr?rrr2)rr'r>r|dfdgrdrrlc_gxplc_rrRr%rNs r7premzPolyElement.prem sl  FLLOO XXa[[ XXa[[ 66#$9:: :22 77H GaK{{1b!! V[^ ;;q"%%D7AEqADAD2q5 AAA!BBww  AI1u r9c$|}|j|}||}||}|dkrtd|||}}}||kr||fS||z dz} |||} |jj|} |||} ||z | dz } } || z}|| | | zzz}|| z}|| z| | zz}||z }||}||krnb| | 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)rr'r>r|rrr2rrrrrrrQrr%rNs r7pdivzPolyElement.pdivv sK`  FLLOO XXa[[ XXa[[ 66#$9:: :ab1 77a4K GaK{{1b!! V[^ ;;q"%%D7AEqADAT2q5L ADAD2q5 AAA!BBww% ( !G E E!t r9c>|}|||dS)aW Polynomial pseudo-quotient in multivariate polynomial ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> f.pquo(g) 2*x >>> f.quo(g) # shows the difference between pquo and quo 0 >>> f.pquo(h) 2*x + 2*y - 2 >>> f.quo(h) # shows the difference between pquo and quo 0 See Also ======== prem, pdiv, pexquo, sympy.polys.domains.ring.Ring.quo r)r)rr'r>r|s r7pquozPolyElement.pquo s6 vva||Ar9cj|}|||\}}|jr|St||)a Polynomial exact pseudo-quotient in multivariate polynomial ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> f.pexquo(g) 2*x >>> f.exquo(g) # shows the differnce between pexquo and exquo Traceback (most recent call last): ... ExactQuotientFailed: 2*x + 2*y does not divide x**2 + x*y >>> f.pexquo(h) Traceback (most recent call last): ... ExactQuotientFailed: 2*x + 2 does not divide x**2 + x*y See Also ======== prem, pdiv, pquo, sympy.polys.domains.ring.Ring.exquo )rrr")rr'r>r|r2rs r7pexquozPolyElement.pexquo s=: vva||1 9 ,H%a++ +r9c|}|j|}||}||}||kr||}}||}}|dkrddgS|dkr|dgS||g}||z }d|dzz}|||} | |z} |||} | |z} d| g} | } | r| |} || || | || z f\}}}}| | |zz}|||} | |} ||| } |dkr$| |z}| |dz z}||} n| } | | | |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. 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