Otj@P6dZddlmZddlmZmZmZmZmZm Z ddl 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/m0Z0m1Z1ddl2m3Z3m4Z4m5Z5m6Z6ddl7m&Z8m9Z:m;Z;ddlZ>m?Z?ddl@mAZAddlBmCZCmDZDddlEmFZFddlGmHZHeCe0fd(dZIeCe0fdZJeCe0fd ZKeCd!ZLd"ZMGd#d$eAeZNGd%d&e(eAeeOZPd'S))zSparse polynomial rings. ) annotations)addmulltlegtge)reduce) GeneratorType)cacheit)Expr)igcd)Symbolsymbols) CantSympifysympify)multinomial_coefficients)IPolys)construct_domain)ninf dmp_to_dict dmp_from_dict)Domain) DomainElementPolynomialRingheugcd) MonomialOps)lex MonomialOrder)CoercionFailedGeneratorsErrorExactQuotientFailedMultivariatePolynomialError)rOrder build_options)expr_from_dict _dict_reorder_parallel_dict_from_expr)DefaultPrinting)publicsubsets) is_sequence)polluteorderMonomialOrder | strc:t|||}|f|jzS)aConstruct a polynomial ring returning ``(ring, x_1, ..., x_n)``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, x, y, z = ring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) PolyRinggensrdomainr0_rings O/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/rings.pyringr:$s$8 Wfe , ,E 8ej  c6t|||}||jfS)aConstruct a polynomial ring returning ``(ring, (x_1, ..., x_n))``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import xring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, (x, y, z) = xring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) r3r6s r9xringr=Cs"8 Wfe , ,E 5: r;cpt|||}td|jD|j|S)aConstruct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import vring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> vring("x,y,z", ZZ, lex) Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z # noqa: x + y + z >>> type(_) cg|] }|j S)name).0syms r9 zvring..~s 1 1 13ch 1 1 1r;)r4r/rr5r6s r9vringrEbs=6 Wfe , ,E 1 1%- 1 1 15:>>> Lr;c( d}t|s|gd}}ttt|}t ||}t ||\}}|j^td|Dg}t||\|_}tt|| fd|D}t|j |j|j }tt|j|} |r || dfS|| fS)adConstruct a ring deriving generators and domain from options and input expressions. Parameters ========== exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable) symbols : sequence of :class:`~.Symbol`/:class:`~.Expr` options : keyword arguments understood by :class:`~.Options` Examples ======== >>> from sympy import sring, symbols >>> x, y, z = symbols("x,y,z") >>> R, f = sring(x + 2*y + 3*z) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> f x + 2*y + 3*z >>> type(_) FTNcPg|]#}t|$Sr@listvalues)rBreps r9rDzsring..s(;;;ctCJJLL));;;r;)optcPg|]"}fd|D#S)c(i|]\}}||Sr@r@)rBmc coeff_maps r9 z$sring...s#999TQIaL999r;)items)rBrKrQs r9rDzsring..s6JJJc9999SYY[[999JJJr;r)r.rImaprr'r*r7sumrdictzipr4r5r0 from_dict) exprsroptionssinglerLrepscoeffs coeffs_domr8polysrQs @r9sringr`s 4F u  &v We$$ % %E  ) )C)44ID# z;;T;;;R@@!1&c!B!B!B JVZ0011 JJJJTJJJ SXsz39 5 5E U_d++ , ,E uQx  u~r;cHt|tr|rt|dndSt|tr|fSt |rCt d|Drt|St d|Dr|St d)NT)seqr@c3@K|]}t|tVdSN) isinstancestrrBss r9 z!_parse_symbols..s,33az!S!!333333r;c3@K|]}t|tVdSrd)rer rgs r9riz!_parse_symbols..s,66At$$666666r;zbexpected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions)rerf_symbolsr r.allr#rs r9_parse_symbolsrns'3.5=xT****2= GT " "z W   337333 3 3 G$$ $ 66g666 6 6 N ~  r;c~eZdZUdZded<ded<ded<ded <d ed <efd Zd ZdZdZ dZ dZ dZ d1dZ edZdZedZedZdZd2dZdZdZdZeZd2dZd2dZd Zd!Zd"Zd#Zd$Z d%Z!d&Z"d'Z#d(Z$ed)Z%ed*Z&d+Z'd,Z(d-Z)d.Z*d/Z+d0Z,dS)3r4z*Multivariate distributed polynomial ring. ztuple[PolyElement, ...]r5ztuple[Expr, ...]rintngensrr7r!r0ctt|}t|}tj|}t j|j|||f}|jr3t|t|j zrtdt |}||_ t||_||_ ||_||_|_t'|dj|_d|z|_||_t|j|_|j|jfg|_|rt9|}||_||_ |!|_"|#|_$|%|_&|'|_(|)|_*n5d}||_||_ d|_"||_$||_&||_(||_*tVur tX|_-n fd|_-t]|j |jD]B\} } t_| t`r(| j1} te|| stg|| | C|S)Nz7polynomial ring and it's ground domain share generatorsr@rcdSNr@r@)abs r9z"PolyRing.__new__..s2r;cdSrur@)rvrwrPs r9rxz"PolyRing.__new__..s"r;c&t|S)Nkey)max)fr0s r9rxz"PolyRing.__new__..sQE):):):r;)4tuplernlen DomainOpt preprocessOrderOpt__name__ is_Compositesetrr#object__new__ _hash_tuplehash_hashrqr7r0 PolyElementnewdtype zero_monom_gensr5 _gens_setone_onerr monomial_mulpow monomial_powmulpowmonomial_mulpowldiv monomial_ldivdiv monomial_divlcm monomial_lcmgcd monomial_gcdr r} leading_expvrWrerrAhasattrsetattr) clsrr7r0rqrobjcodegenmonunitsymbol generatorrAs ` r9rzPolyRing.__new__sxw//00G %f--#E**|WeVUC   ]3w<<#fn2E2E#E ]!"[\\ \nnS!!%%%     R((, e99;;CH  ^VZ01  '!%((G&{{}}C &{{}}C ").."2"2C  ' C &{{}}C &{{}}C &{{}}C  %oG&C &C "4"4C  'C &C &C &C  C<<"C  ::::C !$S[#(!;!; 2 2 FI&&)) 2{sD))2Cy111 r;c|jj}g}t|jD]8}||}|j}|||<||9t|S)z(Return a list of polynomial generators. )r7rrangerqmonomial_basiszeroappendr)selfrriexpvpolys r9rzPolyRing._gens smkotz""  A&&q))D9DDJ LL    U||r;c*|j|j|jfSrd)rr7r0rs r9__getnewargs__zPolyRing.__getnewargs__s dk4:66r;cx|j}|d=|D]}|dr||=|S)Nr monomial_)__dict__copy startswith)rstater|s r9 __getstate__zPolyRing.__getstate__sM ""$$ . !  C~~k** #J r;c|jSrd)rrs r9__hash__zPolyRing.__hash__#s zr;ct|to5|j|j|j|jf|j|j|j|jfkSrd)rer4rr7rqr0rothers r9__eq__zPolyRing.__eq__&sI%**D \4; DJ ? ]EL%+u{ C D Dr;c||k Srdr@rs r9__ne__zPolyRing.__ne__+s5=  r;Nc||$t|trt|}||||Srd)rerIr_clonerrr7r0s r9clonezPolyRing.clone.s8  :gt#<#< GnnG{{7FE222r;cZ||p|j|p|j|p|jSrd) __class__rr7r0rs r9rzPolyRing._clone4s/~~g5v7LeNaW[Wabbbr;c@dg|jz}d||<t|S)zReturn the ith-basis element. r)rqr)rrbasiss r9rzPolyRing.monomial_basis8s$DJaU||r;c,|gSrd)rrs r9rz PolyRing.zero>szz"~~r;c6||jSrd)rrrs r9rz PolyRing.oneBszz$)$$$r;cBt|to |j|kS)zATrue if ``element`` is an element of this ring. False otherwise. )rerr:relements r9 is_elementzPolyRing.is_elementFs';//HGLD4HHr;c8|j||Srd)r7convertrr orig_domains r9 domain_newzPolyRing.domain_newJs{""7K888r;c8||j|Srd)term_newr)rcoeffs r9 ground_newzPolyRing.ground_newMs}}T_e444r;cL||}|j}|r|||<|Srd)rr)rmonomrrs r9rzPolyRing.term_newPs0&&y  DK r;ct|tr`||jkr|St|jtr*|jj|jkr||St dt|trt dt|tr| |St|tr; | |S#t$r| |cYSwxYwt|tr||S||S)N conversionparsing)rerr:r7rrNotImplementedErrorrfrVrXrI from_terms ValueError from_listr from_exprrs r9ring_newzPolyRing.ring_newWsC g{ + + ,w|##DK88 8T[=MQXQ]=]=]w///),777  % % ,%i00 0  & & ,>>'** *  & & , /w/// / / /~~g..... /  & & ,>>'** *??7++ +sC//DDc||j}|j}|D]\}}|||}|r|||<|Srd)rrrS)rrrrrrrs r9rXzPolyRing.from_dictosS_ y#MMOO $ $LE5Juk22E $#U  r;cH|t||Srd)rXrVrs r9rzPolyRing.from_termszs~~d7mm[999r;cd|t||jdz |jSNr)rXrrqr7rs r9rzPolyRing.from_list}s(~~k'4:a<MMNNNr;cXjfdt|S)Nc |}||S|jr5ttt t |jS|jr5ttt t |jS| \}}|j r!|dkr|t|zS  |Sr)getis_Addr rrIrTargsis_Mulr as_base_exp is_Integerrprr)exprrbaseexp_rebuildr7mappingrs r9rz(PolyRing._rebuild_expr.._rebuilds D))I$   Ac4Hdi(@(@#A#ABBB Ac4Hdi(@(@#A#ABBB!,,.. c>AcAgg#8D>>3s8833??6>>$+?+?@@@r;)r7r)rrrrr7s` `@@r9 _rebuild_exprzPolyRing._rebuild_exprsU A A A A A A A A$x &&&r;cttt|j|j} |||}||S#t$rtd|d|wxYw)Nz6expected an expression convertible to a polynomial in z, got ) rVrIrWrr5rrr"r)rrrrs r9rzPolyRing.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|z||r< |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) rqrerprrr5indexrfr)rgenrs r9rzPolyRing.indexsg ;z  S ! ! lAAvv!dj..*!!a2ggBF !>!DEEE __S ! ! 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@rBrrhindicess r9rDz!PolyRing.drop..s'OOO$!QQg=M=MA=M=M=Mr;rm)rrTr enumeraterr7rrr5rrs @r9dropz PolyRing.dropscc$*d++,,OOOO)DL"9"9OOO /; ::g:.. .r;cZ|j|}|s|jS||S)Nrm)rr7r)rr|rs r9 __getitem__zPolyRing.__getitem__s2,s# /; ::g:.. .r;c|jjst|jdr ||jjSt d|jz)Nr7r7z%s is not a composite domain)r7rrrrrs r9 to_groundzPolyRing.to_groundsS ; # Kwt{H'E'E K::T[%7:88 8;dkIJJ Jr;c t|Srdrrs r9 to_domainzPolyRing.to_domainsd###r;cFddlm}||j|j|jS)Nr) FracField)sympy.polys.fieldsrrr7r0)rrs r9to_fieldzPolyRing.to_fields.000000yt{DJ???r;c2t|jdkSrrr5rs r9 is_univariatezPolyRing.is_univariates49~~""r;c2t|jdkSrrrs r9is_multivariatezPolyRing.is_multivariates49~~!!r;cp|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 r9rz PolyRing.addsS" I  C3 666 XTXs^#Sr;cp|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 r9rz PolyRing.mulsS" H  C3 666 XTXs^#Sr;ctt|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 r9rDz+PolyRing.drop_to_ground..$s'MMMAAW.%s'KKK3!7:J:J:J:J:Jr;rr7)rrTrrrr5rrrs @r9drop_to_groundzPolyRing.drop_to_grounds c$*d++,,MMMM4>D::d4jj:11 1Kr;ct|jt|}|t |S)z9Add the elements of ``symbols`` as generators to ``self``rmr#)rrr%s r9add_genszPolyRing.add_gens4s?4<  &&s7||44zz$t**z---r;c|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|vVdSrd)rp)rBrrhs r9riz*PolyRing.symmetric_poly..Es-EEac!q&kkEEEEEEr;) rqrr5rrr-rrprrr7)rnrrrhs @r9symmetric_polyzPolyRing.symmetric_poly9s q55A NN*Z[Z[Z[]a]f]fghh h 8O9DU4:..A77 > >EEEE53D3DEEEEE eT[_===Kr;)NNNrd)-r __module__ __qualname____doc____annotations__r rrrrrrrrr rrpropertyrrrrrrr__call__rXrrrrrrrr r rrrrrr!r&r(r,r@r;r9r4r4s44!!!!JJJNNN,/<<<<|   777DDD !!!3333  cc Wc X%%X%III9999555,,,,H    ::::OOO'''.'''>//////KKK$$$@@@##X#""X"66 H H H... r;r4ceZdZdZfdZdZdZdZdZdZ dZ d Z d Z d Z d Zd ZdZdZdZddZdZdZdZdZdZdZdZdZdZdZdZdZdZ e!dZ"e!d Z#e!d!Z$e!d"Z%e!d#Z&e!d$Z'e!d%Z(e!d&Z)e!d'Z*e!d(Z+e!d)Z,e!d*Z-e!d+Z.e!d,Z/e!d-Z0e!d.Z1e!d/Z2d0Z3d1Z4d2Z5d3Z6d4Z7d5Z8d6Z9d7Z:d8Z;d9Zd<Z?d=Z@d>ZAd?ZBd@ZCdAZDdBZEdCZFdDZGdEZHdFZIdGZJdHZKdIZLdJZMddKZNdLZOddMZPdNZQdOZRdPZSdQZTdRZUe!dSZVe!dTZWdUZXe!dVZYdWZZdXZ[ddYZ\ddZZ]dd[Z^d\Z_d]Z`d^Zad_Zbd`ZcdaZddbZedcZfddZgdeZhdfZidgZjdhZkdiZldjZmdkZnenZodlZpdmZqdnZrdoZsdpZtdqZudrZvdsZwdtZxduZydvZzdwZ{dxZ|dyZ}dzZ~d{Zd|Zd}Zdd~ZddZdZddZdZddZddZddZddZddZdZdZdZdZdZdZdZdZdZdZdZdZddZdZxZS)rz5Element of multivariate distributed polynomial ring. cXt|||_dSrd)super__init__r:)rr:initrs r9r6zPolyElement.__init__Ms&  r;ct|tsJt|jtsJ|jj}t|t sJ|D]V\}}||sJt||jj ksJtd|DsJWdS)Nc3LK|]}t|to|dkV dS)rN)rerp)rBrs r9riz%PolyElement._check..[s5JJSz#s++8qJJJJJJr;) rerr:r4r7rrSof_typerrqrl)rdomrrs r9_checkzPolyElement._checkSs$ ,,,,,$)X.....i#v&&&&& JJLL K KLE5;;u%% % %%u::0000JJEJJJJJ J JJ J K Kr;c8||j|Srd)rr:)rr7s r9rzPolyElement.new]s~~di...r;c4|jSrd)r:r rs r9parentzPolyElement.parent`sy""$$$r;cR|jt|fSrd)r:rI itertermsrs r9rzPolyElement.__getnewargs__cs! 4 0 01122r;Nc|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 )rrs r9rzPolyElement.copyss4xx~~r;c .|j|kr|S|jj|jkrTttt ||jj|j}|||jjS|||jjSrd)r:rrIrWr)rr7rX)rnew_ringtermss r9set_ringzPolyElement.set_rings 9 K Y ("2 2 2mD$)2CXEUVVWXXE&&udi.>?? ?%%dDI,<== =r;c|s |jj}nIt||jjkr,t d|jjdt|t |g|RS)Nz"Wrong number of symbols, expected z got )r:rrrqrr( as_expr_dict)rrs r9as_exprzPolyElement.as_exprs} i'GG \\TY_ , ,*#g,,,0  d//11.s'LLL<5%xxLLLr;)r:r7rNrA)rrNs @r9rJzPolyElement.as_expr_dicts49#,LLLL4>>;K;KLLLLr;cb|jj}|jr|js |j|fS|}|j|j}|j}|D]}|||| fd| D}|fS)Nc$g|] \}}||zf Sr@r@)rBkvcommons r9rDz,PolyElement.clear_denoms..s%BBBDAq1ah-BBBr;) r:r7is_Fieldhas_assoc_Ringrget_ringrdenomrJrrS)rr7 ground_ringrrWrrrSs @r9 clear_denomszPolyElement.clear_denomss! $f&; $:t# #oo'' o [[]] / /ESu..FFxxBBBBDJJLLBBBCCt|r;c^t|D] \}}|s||= dS)z+Eliminate monomials with zero coefficient. NrIrS)rrQrRs r9 strip_zerozPolyElement.strip_zeros?&&  DAq G  r;c|s| S|j|rt||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)r:rrVrrrrp1p2s r9rzPolyElement.__eq__so" 46M W   # # 4;;r2&& & WWq[[566"',--3 3r;c||k Srdr@r^s r9rzPolyElement.__ne__s8|r;c&|j}||rt|t|krdS|jj}|D]}||||||sdSdSt |dkrdS |j|}|j|||S#t$rYdSwxYw)z+Approximate equality test for polynomials. FTr) r:rrkeysr7almosteqrrconstr")r_r` tolerancer:rdrQs r9rdzPolyElement.almosteqsw ??2   G27799~~RWWYY//u{+HWWYY ! !x1r!ui88! 55!4 WWq[[5 G[((,,{++BHHJJIFFF"   uu s:D DDcHt||fSrd)rrGrs r9sort_keyzPolyElement.sort_keysD 4::<<((r;c|j|r0|||StSrd)r:rrhNotImplemented)r_r`ops r9_cmpzPolyElement._cmpsB 7  b ! ! "2bkkmmR[[]]33 3! !r;c8||tSrd)rlrr^s r9__lt__zPolyElement.__lt__wwr2r;c8||tSrd)rlrr^s r9__le__zPolyElement.__le__ror;c8||tSrd)rlrr^s r9__gt__zPolyElement.__gt__ror;c8||tSrd)rlr r^s r9__ge__zPolyElement.__ge__ror;c|j}||}|jdkr ||jfSt |j}||=|||fS)Nrrm)r:rrqr7rIrrrrr:rrs r9_dropzPolyElement._drops^y JJsOO :??dk> !4<((G djjj111 1r;cx||\}}|jjdkr.|jr|dSt d|z|j}|D]G\}}||dkr%t|}||=||t|<6t d|z|S)NrzCannot drop %sr) rxr:rq is_groundrrrrSrIr)rrrr:rrQrRKs r9rzPolyElement.drops**S//4 9?a  ~ 9zz!}}$ !1C!78889D  = =1Q4199QA!%&DqNN$%5%;<<<Kr;c|j}||}t|j}||=|||||fS)Nr )r:rrIrrrws r9_drop_to_groundzPolyElement._drop_to_ground%sMy JJsOOt|$$ AJ$**WT!W*====r;c|jjdkrtd||\}}|j}|jjd}|D]o\}}|d|||dzdz}||vr"|||z|||<C||xx|||z|z cc<p|S)Nrz$Cannot drop only generator to groundr) r:rqrr}rr7r5rA mul_ground)rrrr:rrrmons r9r!zPolyElement.drop_to_ground-s 9?a  CDD D&&s++4ykq! NN,, ? ?LE5)eAaCDDk)C$ %(]66u==S S c58m77>>>  r;cRt||jjdz |jjSr)rr:rqr7rs r9to_densezPolyElement.to_dense>s"T49?1#4di6FGGGr;c t|Srd)rVrs r9to_dictzPolyElement.to_dictAsDzzr;c|s$||jjjS|d}|d}|j}|j}|j} |j} g} |D]\} } |j| }|rdnd}| || | kr7|| }|r| dr |dd}n5|r| } | |jjj kr| | |d}nd }g}t| D]}| |}|s | |||d}|dkrO|t|ks|d kr| ||d }n|}| |||fz| d |z|r|g|z}| ||| d d vr1| d }|dkr| d dd | S)NMulAtom -  + -rT)strictrFz%s)rr)_printr:r7rrrqrrG is_negativerrr parenthesizerrpjoinpopinsert)rprinter precedence exp_pattern mul_symbolprec_mul prec_atomr:rrqzmsexpvsrrnegativesignscoeffsexpvrrrsexpheads r9rfzPolyElement.strDsu 9>>$)"2"788 8e$v& y,  _::<< 2 2KD%{..u55H$/55%D MM$   rzz ..( 1 1# 6 6(#ABBZF#"FEDI,000$11%$1OOFFFE5\\ 0 01g --gaj)D-QQ!88c#hh#''&33C53QQ"LL~!=>>>>LL//// )5( MM*//%00 1 1 1 1 !9 & &::a==Du}} a%%%wwvr;c||jjvSrd)r:rrs r9 is_generatorzPolyElement.is_generatortsty***r;cJ| p t|dko |jj|vSr)rr:rrs r9rzzPolyElement.is_groundxs(xLCIINKty/Ct/KLr;cD| pt|dko |jdkSr)rLCrs r9 is_monomialzPolyElement.is_monomial|s$x.,??u3u::???????r;rl itermonomsrs r9 is_linearzPolyElement.is_linear'?? ??????r;cXtd|DS)Nc3<K|]}t|dkVdS)Nrrs r9riz+PolyElement.is_quadratic..rr;rrs r9 is_quadraticzPolyElement.is_quadraticrr;cR|jjsdS|j|SNT)r:rq dmp_sqf_prs r9 is_squarefreezPolyElement.is_squarefrees)v| 4v"""r;cR|jjsdS|j|Sr)r:rqdmp_irreducible_prs r9is_irreduciblezPolyElement.is_irreducibles)v| 4v''***r;cl|jjr|j|Std)Nzcyclotomic polynomial)r:rdup_cyclotomic_pr%rs r9 is_cyclotomiczPolyElement.is_cyclotomics5 6  G6**1-- --.EFF Fr;cd|d|DS)Ncg|] \}}|| f Sr@r@)rBrrs r9rDz'PolyElement.__neg__..s"PPPleU55&/PPPr;)rrArs r9__neg__zPolyElement.__neg__s-xxPPdnn>N>NPPPQQQr;c|Srdr@rs r9__pos__zPolyElement.__pos__s r;c`|s|S|j}||r]|}|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$r tcYSwxYw)aAdd two polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> (x + y)**2 + (x - y)**2 2*x**2 + 2*y**2 )rr:rrr7rrSrerr__radd__rjrrrcr") r_r`r:rrrrQrRcp2rs r9__add__zPolyElement.__add__s 7799 w ??2   & A%C;#D   1C4LL1$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 $r t cYSwxYwrd)rr:rrrcr"rj)r_r+rr:rs r9rzPolyElement.__radd__s GGII Hw ""AB"""2;;"bEEEQJEEEH " " "! ! ! ! "sA<<BBcX|s|S|j}||r]|}|j}|jj}|D]\}}||||z }|r|||<||= |St|trt|jtr|jj|jkrnPt|jjtr*|jjj|kr| |StS | |}|}|j }||vr| ||<n |||kr||=n||xx|zcc<|S#t$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 )rr:rrr7rrSrerr__rsub__rjrrrcr") r_r`r:rrrrQrRrs r9__sub__zPolyElement.__sub__s  7799 w ??2   & A%C;#D   1C4LL1$AaDD!H K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% $$B AB"""2;;"bEEERKEEEH " " "! ! ! ! "s&FF)(F)c|j} ||}|j}|D]}|| ||<||z }|S#t$r tcYSwxYw)a#n - p1 with n convertible to the coefficient domain. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y >>> 4 - p -x - y + 4 )r:rrr"rj)r_r+r:rrs r9rzPolyElement.__rsub__Esw ""A A $ $d8)$ FAH " " "! ! ! ! "s=AAcP|j}|j}|r|s|S||r|j}|jj}|j}t |}|D].\}} |D]&\} } ||| } || || | zz|| <'/||St|trt|jtr|jj|jkrnPt|jjtr*|jjj|kr| |StS ||}|D]\}} | |z} | r| ||<|S#t$r tcYSwxYw)a!Multiply two polynomials. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', QQ) >>> p1 = x + y >>> p2 = x - y >>> p1*p2 x**2 - y**2 )r:rrrr7rrIrSr\rerr__rmul__rjrr")r_r`r:rrrrp2itexp1v1exp2v2rrRs r9__mul__zPolyElement.__mul__as w I & &H __R  &%C;#D,L ##DHHJJ 4 4b $44HD"&,tT22C Sd^^be3AcFF4 LLNNNH K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% $$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 )r:rrrSr"rj)r_r`rrrrRs r9rzPolyElement.__rmul__s GL H ""2&&BHHJJ  brE AdGH " " "! ! ! ! "sAA('A(c>t|tstd|z|dkrtd|z|j}|s|r|jStdt |dkryt|d\}}|j }||j jkr||| ||<n||z|| ||<|St|}|dkrtd|dkr| S|dkr| S|dkr|| zSt |d kr||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 z#exponent must be an integer, got %srz/exponent must be a non-negative integer, got %sz0**0rzNegative exponentr)rerp TypeErrorrr:rrrIrSrr7rrsquare_pow_multinomial _pow_generic)rr+r:rrrs r9__pow__zPolyElement.__pow__s!S!! TAAEFF F UUNQRRSS Sy  )x ((( YY!^^ --a0LE5 A ''16$##E1--..16$##E1--.H FF q55011 1 !VV99;;  !VV;;== !VV % % YY!^^((++ +$$Q'' 'r;c||jj}|} |dzr ||z}|dz}|sn|}|dz},|S)NTrr)r:rr)rr+rrPs r9rzPolyElement._pow_generics` IM  1u aCQ AQA r;ctt||}|jj}|jj}|}|jjj}|jj}|D]r\}} |} | } t||D]\} \} }| r|| | | } | || zz} t| } | }| | ||z}|r||| <k| |vr|| =s|Srd) rrrSr:rrr7rrWrr)rr+ multinomialsrrrGrr multinomialmultinomial_coeff product_monom product_coeffrrrs r9rzPolyElement._pow_multinomials/D 1==CCEE )3Y)  y$y~.:  *K*&M-M'*;'>'> 0 0#^eU0$3OM5#$N$NM!UCZ/M-((E!EHHUD))E1E #U $K r;cN|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) r:rrrIrcr7rrrimul_numrSr\)rr:rrrcrrrk1pkjk2rrQrRs r9rzPolyElement.squares9y IeDIIKK  {( s4yy!! 6 6AaBbB1XX 6 6!W"l2r**S$"T"X+5# 6 JJqMMeJJLL ) )DAqa##BCDMMAqD(AbEE r;c^|j}|std||r||St |t rt |jtr|jj|jkrnPt |jjtr*|jjj|kr||StS | |}| || |fS#t$r tcYSwxYwNpolynomial division)r:ZeroDivisionErrorrrrerr7r __rdivmod__rjr quo_ground rem_groundr"r_r`r:s r9 __divmod__zPolyElement.__divmod__:s"w &#$9:: : __R  &66"::  K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[~~b)))%% :$$BMM"%%r}}R'8'89 9 " " "! ! ! ! "sDD,+D,c|j} ||}||S#t$r tcYSwxYwrd)r:rrr"rjrs r9rzPolyElement.__rdivmod__PZw $$B66"::  " " "! ! ! ! "3AAc4|j}|std||r||St |t rt |jtr|jj|jkrnPt |jjtr*|jjj|kr||StS | |}| |S#t$r tcYSwxYwr) r:rrremrerr7r__rmod__rjrrr"rs r9__mod__zPolyElement.__mod__Ysw &#$9:: : __R  &66"::  K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[{{2&%% %$$B==$$ $ " " "! ! ! ! "DDDc|j} ||}||S#t$r tcYSwxYwrd)r:rr r"rjrs r9rzPolyElement.__rmod__or r c4|j}|std||r||St |t rt |jtr|jj|jkrnPt |jjtr*|jjj|kr||StS | |}| |S#t$r tcYSwxYwr) r:rrquorerr7r __rtruediv__rjrrr"rs r9 __floordiv__zPolyElement.__floordiv__xsw &#$9:: : __R  &66"::  K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[r***%% %$$B==$$ $ " " "! ! ! ! "rc|j} ||}||S#t$r tcYSwxYwrd)r:rrr"rjrs r9 __rfloordiv__zPolyElement.__rfloordiv__r r c4|j}|std||r||St |t rt |jtr|jj|jkrnPt |jjtr*|jjj|kr||StS | |}| |S#t$r tcYSwxYwr) r:rrexquorerr7rrrjrrr"rs r9 __truediv__zPolyElement.__truediv__sw &#$9:: : __R  &88B<<  K ( ( &$+~66 &4;;Krw;V;VBGNN;; &@SW[@[@[r***%% %$$B==$$ $ " " "! ! ! ! "rc|j} ||}||S#t$r tcYSwxYwrd)r:rrr"rjrs r9rzPolyElement.__rtruediv__sZw $$B88B<<  " " "! ! ! ! "r c|jj|jj}|j|jj|jrfd}nfd}|S)Ncf|\}}|\}}| kr|}n ||}||||fSdSrdr@ a_lm_a_lc b_lm_b_lca_lma_lcb_lmb_lcr domain_quorrs r9term_divz'PolyElement._term_div..term_divsX& d& d2:: EE(Lt44E$ **T4"8"8884r;cp|\}}|\}}| kr|}n ||}|||zs|||fSdSrdr@rs r9r&z'PolyElement._term_div..term_divs_& d& d2:: EE(Lt44E   **T4"8"8884r;)r:rr7rrrT)rr7r&r%rrs @@@r9 _term_divzPolyElement._term_divs Y !!Z y- ?   r;c|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@)rrBrr:s r9rDz#PolyElement.div..s * * *Adi * * *r;c6g|]}|Sr@)r)rBfxs r9rDz#PolyElement.div..s"000r""000r;rNr)r:rerrlrrrrrrr(r _iadd_monom_iadd_poly_monomr)rfv ret_singler~rhqvrrr&expvsr divoccurredrtermexpv1rPr:s @r9rzPolyElement.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!|#!uaxq5Lr;c|}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)rerrlrr:r7rrr(LTrrrAr)rGr~r:r7rrr3r&ltfrgtqrOrPmgcgm1c1ltmltcs r9r zPolyElement.rem#s  a % % A1vv ;#$9:: :v{( I;;==d FFHHe & & &Xc14((>DAq"#++--''B)\"a00 ST]]QrT1!' !"$&AbEE..**C!1S6kE"S!88cFFFcMFFFF AcFcFnn&&?qv+C5 &8r;c8||dSNr)r)r~r:s r9rzPolyElement.quoPsuuQxx{r;cZ||\}}|s|St||rd)rr$)r~r:qr3s r9rzPolyElement.exquoSs2uuQxx1 ,H%a++ +r;c||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 )r:rrr)rmccpselfrrrPs r9r.zPolyElement._iadd_monom[sz8 49& & &YY[[FFF e JJt   9 F4LL JA ! t 4L r;c&|}||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 )r:rrrr7rrrS) rr`rIr_rOrPrrrrQrRkars r9r/zPolyElement._iadd_poly_monoms* " " "BAfw~"w+ HHJJ  DAqa##BCDMMAaC'E 2rFF r;c|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 dSrdr@rBrrs r9riz%PolyElement.degree..'<> >r;c|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 dSrdr@rOs r9riz*PolyElement.tail_degree..rPr;)r:rrminrrQs @r9 tail_degreezPolyElement.tail_degreerTr;c |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')``) ) rr:rqrrTrZrIrWrrs r9 tail_degreeszPolyElement.tail_degreesrWr;c>|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)r:rrs r9rzPolyElement.leading_expvs'  9))$// /4r;cL|||jjjSrd)rr:r7rrrs r9 _get_coeffzPolyElement._get_coeffsxxdi.3444r;cv|dkr||jjS|j|rit |}t |dkr5|d\}}||jjjkr||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) rar:rrrIrArr7rr)rrrGrrs r9rzPolyElement.coeffs4 a<<??49#788 8 Y ! !' * * 2**,,--E5zzQ$Qx uDI,000??51116@AAAr;c@||jjS)z"Returns the constant coefficient. )rar:rrs r9rezPolyElement.const sty3444r;cP||Srd)rarrs r9rzPolyElement.LC$s t0022333r;cJ|}| |jjS|Srd)rr:rr`s r9LMzPolyElement.LM(s(  "" <9' 'Kr;cr|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 )r:rrr7rrrrs r9 leading_monomzPolyElement.leading_monom0s< IN  ""  +i&*AdGr;c|}||jj|jjjfS|||fSrd)rr:rr7rrar`s r9r9zPolyElement.LTEsG  "" <I($)*:*?@ @$//$//0 0r;c`|jj}|}| ||||<|S)aLeading term as a polynomial element. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> (3*x*y + y**2).leading_term() 3*x*y )r:rrrhs r9 leading_termzPolyElement.leading_termMs6 IN  ""  4jAdGr;c |jjntjturt |ddSt |fddS)Nc|dSrEr@)rs r9rxz%PolyElement._sorted..hs qr;T)r|reversec&|dSrEr@)rr0s r9rxz%PolyElement._sorted..jsuQxr;)r:r0rrr sorted)rrbr0s `r9_sortedzPolyElement._sortedasd =IOEE'..E C<<##9#94HHH H##@#@#@#@$OOO Or;c@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@)rB_rs r9rDz&PolyElement.coeffs..s:::81e:::r;rGrr0s r9r]zPolyElement.coeffsl$0;:tzz%'8'8::::r;c@d||DS)a Ordered list of polynomial monomials. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.monoms() [(2, 3), (1, 7)] >>> f.monoms(grlex) [(1, 7), (2, 3)] cg|]\}}|Sr@r@)rBrrus r9rDz&PolyElement.monoms..s:::85!:::r;rvrws r9monomszPolyElement.monomsrxr;cl|t||S)aOrdered list of polynomial terms. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.terms() [((2, 3), 2), ((1, 7), 1)] >>> f.terms(grlex) [((1, 7), 1), ((2, 3), 2)] )rrrIrSrws r9rGzPolyElement.termss(0||D..666r;cDt|S)z,Iterator over coefficients of a polynomial. )iterrJrs r9 itercoeffszPolyElement.itercoeffsDKKMM"""r;cDt|S)z)Iterator over monomials of a polynomial. )r~rcrs r9rzPolyElement.itermonomsDIIKK   r;cDt|S)z%Iterator over terms of a polynomial. )r~rSrs r9rAzPolyElement.itertermsDJJLL!!!r;cDt|S)z+Unordered list of polynomial coefficients. rHrs r9 listcoeffszPolyElement.listcoeffsrr;cDt|S)z(Unordered list of polynomial monomials. )rIrcrs r9 listmonomszPolyElement.listmonomsrr;cDt|S)z$Unordered list of polynomial terms. r[rs r9 listtermszPolyElement.listtermsrr;c||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)r:rclear)rrPrs r9rzPolyElement.imul_numsb4  Q3J  GGIII F  C cFFFaKFFFFr;c|jj}|j}|j}|D]}|||}|S)z*Returns GCD of polynomial's coefficients. )r:r7rrr)r~r7contrrs r9rzPolyElement.contentsH{j\\^^ $ $E3tU##DD r;c|}||jjjkr||fS|||fS)z,Returns content and a primitive polynomial. )rr:r7rr)r~rs r9 primitivezPolyElement.primitivesByy{{ 16=% % %!9 Q\\$''''r;c>|s|S||jS)z5Divides all coefficients by the leading coefficient. )rrrs r9moniczPolyElement.monics# &H<<%% %r;cs |jjSfd|D}||S)Nc$g|] \}}||zf Sr@r@rBrrrRs r9rDz*PolyElement.mul_ground..s&FFF|ue5%'"FFFr;)r:rrAr)r~rRrGs ` r9rzPolyElement.mul_groundsF 6; FFFFq{{}}FFFuuU||r;c|jjfd|D}||S)Nc2g|]\}}||fSr@r@)rBf_monomf_coeffrrs r9rDz)PolyElement.mul_monom..s/]]]>Ngw<<//9]]]r;)r:rrSr)r~rrGrs ` @r9 mul_monomzPolyElement.mul_monomsGv* ]]]]]RSRYRYR[R[]]]uuU||r;c|\|rs |jjS|jjkr|S|jjfd|D}||S)Nc8g|]\}}||zfSr@r@)rBrrrrrs r9rDz(PolyElement.mul_term..#s3cccDTGW<<//?cccr;)r:rrrrrSr)r~r6rGrrrs @@@r9mul_termzPolyElement.mul_terms u ' '6;  af' ' '<<&& &v* ccccccXYX_X_XaXacccuuU||r;c(|jj}std|r |jkr|S|jr)|jfd|D}n fd|D}||S)Nrc2g|]\}}||fSr@r@)rBrrrrRs r9rDz*PolyElement.quo_ground..0s,PPPucc%mm,PPPr;c.g|]\}}|z ||zfSr@r@rs r9rDz*PolyElement.quo_ground..2s2```leUTY\]T]`ueqj)```r;)r:r7rrrTrrAr)r~rRr7rGrs ` @r9rzPolyElement.quo_ground&s ;#$9:: : AOOH ? a*CPPPPPPPPEE````akkmm```EuuU||r;c@\}}|std|s |jjS||jjkr||S|fd|D}|d|DS)Nrc(g|]}|Sr@r@)rBtr6r&s r9rDz(PolyElement.quo_term..Bs%<<<((1d##<<.Cs:::Q1=q===r;)rr:rrrr(rAr)r~r6rrrGr&s ` @r9quo_termzPolyElement.quo_term6s u '#$9:: : '6;  af' ' '<<&& &;;==<<<<.Qs&LLL\UEueai(LLLr;)r:r7is_ZZrArrr\)r~rrGrrrs ` r9 trunc_groundzPolyElement.trunc_groundEs 6=  ME !  - - u 16>>!AIE eU^,,,,  -MLLLQ[[]]LLLEuuU||  r;c|}|}|}|jj||}||}||}|||fSrd)rr:r7rr)rr<r~fcgcrs r9extract_groundzPolyElement.extract_groundYsh  YY[[ YY[[fmB'' LL   LL  Aqyr;c|s|jjjS|jjj|fd|DS)Nc&g|] }|Sr@r@)rBr ground_abss r9rDz%PolyElement._norm..js#NNNUzz%00NNNr;)r:r7rabsr)r~ norm_funcrs @r9_normzPolyElement._normesR P6=% %*J9NNNNallnnNNNOO Or;c6|tSrd)rr}rs r9max_normzPolyElement.max_normlwws||r;c6|tSrd)rrUrs r9l1_normzPolyElement.l1_normorr;cJ|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@)rBrws r9riz&PolyElement.deflate..s&!!!qAv!!!!!!r;cg|] \}}||z Sr@r@rBrrs r9rDz'PolyElement.deflate..s 444Aa1f444r;) r:rIrqrrrrrlrrArWr)r~r:r:r_JrrrrOrwHhIrNs r9deflatezPolyElement.deflatersevd1gg  C N ) )A ) )%e,,))DAq!a==AaDD) )aLL  DAq ! !HH !!q!!! ! ! e8O   A AKKMM $ $544Q444#%(( HHQKKKK!t r;c|jj}|D]1\}}dt||D}||t |<2|S)Ncg|] \}}||z Sr@r@rs r9rDz'PolyElement.inflate..s ---$!Q!A#---r;)r:rrArWr)r~rrrrrs r9inflatezPolyElement.inflatesWv{  # #HAu--#a))---A"DqNN r;cj|}|jj}|jsD|\}}|\}}|||}||z||}|js||S|Srd) r:r7rTrrrrrr)rr<r~r7rrrPrs r9rzPolyElement.lcms  #KKMMEBKKMMEB 2r""A qSIIaeeAhh   <<?? "7799 r;c8||dSrE) cofactorsr~r<s r9rzPolyElement.gcds{{1~~a  r;cT|s|s|jj}|||fS|s||\}}}|||fS|s||\}}}|||fSt|dkr||\}}}|||fSt|dkr||\}}}|||fS||\}\}}||\}}}||||||fSr)r:r _gcd_zeror _gcd_monomr_gcdr)r~r<rrcffcfgrs r9rzPolyElement.cofactorss0  6;Dt# # ++a..KAsCc3;  ++a..KAsCc3;  VVq[[,,q//KAsCc3;  VVq[[,,q//KAsCc3; IIaLL 6AqffQii 3 ! ckk!nnckk!nn==r;cX|jj|jj}}|jr|||fS| || fSrd)r:rrr)r~r<rrs r9rzPolyElement._gcd_zeros:FJ T  "dC< 2tcT> !r;c$ |j}|jj}|jj|j}|jt |d\}}||c |D]\}}| | | | | fg} || | fg} | fd|D} | | | fS)NrcFg|]\}}||fSr@r@)rBr>r?_cgcd_mgcd ground_quors r9rDz*PolyElement._gcd_monom..s:ccc62rmmB.. 2u0E0EFcccr;) r:r7rrrrrIrAr)r~r<r: ground_gcdrmfcfr>r?rrrrrrrs @@@@r9rzPolyElement._gcd_monoms(v[_ [_ ( * akkmm$$Q'B2 ukkmm * *FB L++EJub))EE EEE5>" # #eemmB.. 2u0E0EFGHHeecccccccUVU`U`UbUbcccdd#s{r;c|j}|jjr||S|jjr||S|||Srd)r:r7is_QQ_gcd_QQr_gcd_ZZ dmp_inner_gcd)r~r<r:s r9rzPolyElement._gcdsYv ;  ,99Q<<  [  ,99Q<< %%a++ +r;c"t||Srdrrs r9rzPolyElement._gcd_ZZsa||r;c|}|j}||j}|\}}|\}}||}||}||\}}} ||}|j|}} || |j | |}| | |j | |} ||| fS)Nr ) r:rr7rVrYrHrrrrr) rr<r~r:rFrr?rrrrPs r9rzPolyElement._gcd_QQs v::T[%9%9%;%;:<<  A  A JJx  JJx iill 3 JJt  tQWWYY1ll4  ++DKOOAr,B,BCCll4  ++DKOOAr,B,BCC#s{r;cd|}|j}|s ||jfS|j}|jr|js||\}}}n ||}|\} }|\} }| |}| |}||\}}}|j| | \}} } | |}| |}| | }| | }| } | |jkrn=| |j kr| | }}n*| | }| | }||fS)a Cancel common factors in a rational function ``f/g``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> (2*x**2 - 2).cancel(x**2 - 2*x + 1) (2*x + 2, x - 1) r ) r:rr7rTrUrrrVrYrHrcanonical_unit) rr<r~r:r7rurrGrFcqcpus r9cancelzPolyElement.cancels v dh;  !F$9 !kk!nnGAq!!zz):):z;;HNN$$EBNN$$EB 8$$A 8$$Akk!nnGAq! 11"b99IAr2 4  A 4  A R  A R  A       ??  6:+  2rqAA QA QA!t r;cN|jj}||jSrd)r:r7rr)r~r7s r9rzPolyElement.canonical_unit6 s!$$QT***r;c(|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 )r:rrrrArr) r~rRr:rrOr<rres r9diffzPolyElement.diff: sv JJqMM    " " I;;== 6 6KD%Aw 6&&tQ//uT!W}55!r;cdt|cxkr|jjkr=nn:|t t |jj|Std|jjdt|)Nrz expected at least 1 and at most z values, got )rr:rqevaluaterIrWr5r)r~rJs r9r2zPolyElement.__call__S s s6{{ * * * *afl * * * * *::d3qv{F#;#;<<== =*TUTZT`T`T`beflbmbmbmnoo or;c |}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)rBYrvXs r9rDz(PolyElement.evaluate..c s+666!Qqvvayy!n666r;) rerIrr:rr7rrqrrAr) rrRrvr~r:rresultr+rrrrs @r9rzPolyElement.evaluateY s  a   %19!aeIFQA 1a  A %66661666zz!}}$v JJqMM K   " " :??[%F {{}} % % e%1*$M99Q<<$D ! , , u 8U2A2Yqstt%<5ad D==!DK/E(&+U  KK,&+U Kr;cf|}t|tr"| |D]\}}|||}|S|j}||}|j|}|jdkrH|jj}| D]\\}} || ||zzz }| |S|j} | D]R\} } | || d|dz| |dzdz} }| ||zz} | | vr| | | z} | r| | | <G| | =K| r| | | <S| S)Nrrs) rerIsubsr:rr7rrqrrAr) rrRrvr~rr:rrr+rrrs r9rzPolyElement.subs so  a   19 ! !1FF1aLLHv JJqMM K   " " :??[%F {{}} % % e%1*$??6** *9D ! , , u 8U2A2Y%5acdd %C5ad D==!DK/E(&+U  KK,&+U Kr;c|}|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,r+s r9rDz*PolyElement.symmetrize.. s+<<.get_poly_power s71v[((&+Ahk QF#1v& &r;rrr)rNNc3BK|]}||dzkVdSrr@)rBrrs r9riz)PolyElement.symmetrize.. s4AAAuQx5Q</AAAAAAr;c3&K|] \}}||zV dSrdr@)rBr+rOs r9riz)PolyElement.symmetrize.. s* E EA1 E E E E E Er;Nrs)rr:rqrrrIrrGrlr}rWrrrr5)rr~r+rrweights symmetric_height_monom_coeffrrheight exponentsr@m2productrrrr_r:s @@@@r9 symmetrizezPolyElement.symmetrize sMv IIKKv J $di# #<<<<588<<<  ' ' ' ' ' ' uQU||$$uQ2''I  &4 #GVV%.qwwyy%9%9 G G!>E5AAAAAAAAAG E EWe1D1D E E EEEF''28%"}}%v uuIeU122Y%566 * *B  b)))) uY'7'7?? ?IG!),, 0 01>>!Q/// LA1 4s49e,,--!W$$r;c |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 |dSrEr@)rQgens_maps r9rxz%PolyElement.compose..$ sx!~r;r{z9expected a generator, value pair a sequence of such pairsr)r:rrVrWr5rrqrerIrqrSrrrrArrr)r~rRrvr:r replacementsrQr<rrsubpolyrr+rs @r9r&zPolyElement.compose svyDIuTZ'8'899:: =F8LL!T"" ^#Aww At$$ ^%aggii5M5M5M5MNNN  !\]]]"<00 > >IAv1'{DMM!,<,<=LOOKKMM  LE5KKEhG$ $ $1#Ah 58$q!tOG&&e e'<==G GODD r;c:|}|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@)rBrOrPdegrs r9rDz)PolyElement.coeff_wrt..e s*AAADAqQqTS[[!Q[[[r;cFg|]}|ddz|dzdzS)Nrsrr@)rBrOrs r9rDz)PolyElement.coeff_wrt..k s6;;;q!BQB%$,1q566*;;;r;)r:rrArrWrXrV)rrRr rrGr{r]rs ` @r9 coeff_wrtzPolyElement.coeff_wrt9 sT  FLLOOAAAAAAKKMMAAA 6; e;;;;F;;;vS%8%8 9 9:::r;c|}|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 difference 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 rrrr:rrSrr r5)rr<rRr~dfdgr3drrlc_gxplc_rrRr:rPs r9premzPolyElement.premn sl  FLLOO XXa[[ XXa[[ 66#$9:: :22 77H GaK{{1b!! V[^ ;;q"%%D7AEqADAD2q5 AAA!BBww  AI1u r;c$|}|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<rRr~rrrGr3rrrrrrQrr:rPs r9pdivzPolyElement.pdiv sK`  FLLOO XXa[[ XXa[[ 66#$9:: :ab1 77a4K GaK{{1b!! V[^ ;;q"%%D7AEqADAT2q5L ADAD2q5 AAA!BBww% ( !G E E!t r;c>|}|||dS)aW Polynomial pseudo-quotient in multivariate polynomial ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> f.pquo(g) 2*x >>> f.quo(g) # shows the difference between pquo and quo 0 >>> f.pquo(h) 2*x + 2*y - 2 >>> f.quo(h) # shows the difference between pquo and quo 0 See Also ======== prem, pdiv, pexquo, sympy.polys.domains.ring.Ring.quo r)r)rr<rRr~s r9pquozPolyElement.pquoG s6 vva||Ar;cj|}|||\}}|jr|St||)a Polynomial exact pseudo-quotient in multivariate polynomial ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> f.pexquo(g) 2*x >>> f.exquo(g) # shows the difference between pexquo and exquo Traceback (most recent call last): ... ExactQuotientFailed: 2*x + 2*y does not divide x**2 + x*y >>> f.pexquo(h) Traceback (most recent call last): ... ExactQuotientFailed: 2*x + 2 does not divide x**2 + x*y See Also ======== prem, pdiv, pquo, sympy.polys.domains.ring.Ring.exquo )rrr$)rr<rRr~rGr3s r9pexquozPolyElement.pexquoe s=: vva||1 9 ,H%a++ +r;c|}|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. 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)r:rrSrr rr)rr<rRr~r+rOrdrwrlcrPSrQrrGs r9 subresultantszPolyElement.subresultants sD  FLLOO HHQKK HHQKK q55aqAaqA 66q6M 66q6M F E QUO FF1aLL E[[A   !G F B  A HHQKKKAq!a%JAq!Qa1f Aq! 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