+ i@PRt^RIHt^RIHtHtHtHtHtH t ^RI H t ^RI H t ^RIHt^RIHt^RIHt^RIHtHt^R IHtHt^R IHt^R IHt^R IH t ^R I!H"t"H#t#H$t$^RI%H&t&^RI'H(t(^RI)H*t*^RI+H,t,^RI-H.t.^RI/H0t0H1t1^RI2H3t3H4t4H5t5H6t6^RI7H&t8H9t:H;t;^RIt>H?t?^RI@HAtA^RIBHCtCHDtD^RIEHFtF^RIGHHtH]C]03RRll4tI]C]03Rl4tJ]C]03Rl4tK]CR4tLR tM!R!R"]A]4tN!R#R$](]A]]O4tPR%#)&zSparse polynomial rings. ) annotations)addmulltlegtge)reduce) GeneratorType)cacheit)Expr)igcd)Symbolsymbols) CantSympifysympify)multinomial_coefficients)IPolys)construct_domain)ninf dmp_to_dict dmp_from_dict)Domain) DomainElementPolynomialRingheugcd) MonomialOps)lex MonomialOrder)CoercionFailedGeneratorsErrorExactQuotientFailedMultivariatePolynomialError)rOrder build_options)expr_from_dict _dict_reorder_parallel_dict_from_expr)DefaultPrinting)publicsubsets) is_sequence)pollutecV^8dQhRR/#)orderzMonomialOrder | str)formats"q/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/rings.py __annotate__r5%s!!!2!cB\WV4pV3VP,#)aRConstruct a polynomial ring returning ``(ring, x_1, ..., x_n)``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, x, y, z = ring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) PolyRinggensrdomainr1_rings&&& r4ringr>$s!8 We ,E 8ejj  r6c4\WV4pW3P3#)aXConstruct 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(_) r8r;s&&& r4xringr@Cs8 We ,E :: r6c\WV4p\VPUu.uFqDPNK upVP4V#uupi)a\Construct 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(_) )r9r.rnamer:)rr<r1r=syms&&& r4vringrDbs=6 We ,E %-- 1-3hh- 15::> L 2sAc Rp\V4'gV.Rr0\\\V44p\ W4p\ W4wrTVP f\VUu.uFp\VP44NK up.4p\WtR7wVnp\\Wx44p VUU U u.uF-qfP4U U u/uFwrWV ,bK up p NK/ pp pp \VPVP VP4p \\V P V44p V'd W^,3#W3#uupiuup p iuup p pi)a Construct a ring deriving generators and domain from options and input expressions. Parameters ========== exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable) symbols : sequence of :class:`~.Symbol`/:class:`~.Expr` options : keyword arguments understood by :class:`~.Options` Examples ======== >>> from sympy import sring, symbols >>> x, y, z = symbols("x,y,z") >>> R, f = sring(x + 2*y + 3*z) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> f x + 2*y + 3*z >>> type(_) FT)opt)r-listmaprr&r)r<sumvaluesrdictzipitemsr9r:r1 from_dict)exprsroptionssinglerFrepsrepcoeffs coeffs_dom coeff_mapmcr=polyss&*, r4sringrZs4F u  v We$ %E  )C)4ID zzT;TctCJJL)T;R@!1&!B JV01 EIJTcYY[9[TQaL[9TJ SXXszz399 5E U__d+ ,E Qx  ~< :Js "E8EE#EEc\V\4'dV'd\VRR7#R#\V\4'dV3#\ V4'd\ ;QJdRV4F 'dK RM RM !RV44'd \V4#\ ;QJdRV4F 'dK RM RM !RV44'dV#\ R4h)T)seqc3B"TFp\V\4xK R#5iN) isinstancestr.0ss& r4 !_parse_symbols..s37az!S!!7Fc3B"TFp\V\4xK R#5ir^)r_r ras& r4rdres6gAt$$grfzbexpected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressionsr2)r_r`_symbolsr r-allr"rs&r4_parse_symbolsrks'3.5xT*=2= GT " "z W   3373333373 3 3G$ $ S6g6SSS6g6 6 6N ~ r6cr]tRt^t$RtR]R&R]R&R]R&R]R &R ]R &]3R ltR tRt Rt Rt Rt Rt R2Rlt]R4tRt]R4t]R4tRtR3RltRtRtRt]tR3RltR3RltR tR!tR"tR#t R$t!R%t"R&t#R't$R(t%]R)4t&]R*4t'R+t(R,t)R-t*R.t+R/t,R0t-R1t.R#)4r9z*Multivariate distributed polynomial ring. ztuple[PolyElement, ...]r:ztuple[Expr, ...]rintngensrr<r r1c a\\V44p\V4p\P!V4p\ P!S4oVP WVS3pVP'd7\V4\VP4,'d \R4h\PV4pWVn \V4VnWn WFnW&nSVn\'VR4P(VnRV,VnVP/4Vn\VP04VnVP,VP43.VnV'd\9V4pVP;4VnVP?4Vn VPC4Vn"VPG4Vn$VPK4Vn&VPO4Vn(VPS4Vn*M/RpWnWn RVn"Wn$Wn&Wn(Wn*S\VJd \XVn-M V3RlVn-\]VPVP04FHwr\_V \`4'gKV Pbp \eWk4'dK<\gWkV 4KJ V#)z7polynomial ring and it's ground domain share generatorscR#Nr2r2)abs&&r4"PolyRing.__new__..s2r6cR#rqr2)rrrsrXs&&&r4rtrus"r6c<\VSR7#)key)max)fr1s&r4rtrus QE):r6r2)4tuplerklen DomainOpt preprocessOrderOpt__name__ is_Compositesetrr"object__new__ _hash_tuplehash_hashrnr<r1 PolyElementnewdtype zero_monom_gensr: _gens_setone_onerr monomial_mulpow monomial_powmulpowmonomial_mulpowldiv monomial_ldivdiv monomial_divlcm monomial_lcmgcd monomial_gcdrrz leading_expvrLr_rrBhasattrsetattr) clsrr<r1rnrobjcodegenmonunitsymbol generatorrBs &&&f r4rPolyRing.__new__s w/0G %%f-##E*||WVUC    3w<#fnn2E#E#E!"[\ \nnS!%%     R(,, e99;CHH  ^^VZZ01 !%(G&{{}C &{{}C ").."2C  ' C &{{}C &{{}C &{{}C %G& & "4C  ' & & &  C<"C :C !$S[[#((!; F&&)){{s))Cy1 "< r6c VPPp.p\VP4F5pVP V4pVP pWV&VP V4K7 \V4#)z(Return a list of polynomial generators. )r<rrangernmonomial_basiszeroappendr~)selfrriexpvpolys& r4rPolyRing._gens s_kkootzz"A&&q)D99DJ LL  # U|r6c HVPVPVP3#r^)rr<r1rs&r4__getnewargs__PolyRing.__getnewargs__s dkk4::66r6c VPP4pVRVFpVPR4'gKWK V#)r monomial_)__dict__copy startswith)rstaterys& r4 __getstate__PolyRing.__getstate__sA ""$ . !C~~k**J r6c VP#r^)rrs&r4__hash__PolyRing.__hash__#s zzr6c \V\4;'d^VPVPVPVP 3VPVPVPVP 38H#r^)r_r9rr<rnr1rothers&&r4__eq__PolyRing.__eq__&s\%*DD \\4;; DJJ ? ]]ELL%++u{{ C D Dr6c W8X*#r^r2rs&&r4__ne__PolyRing.__ne__+s   r6Nc pVe"\V\4'd \V4pVPWV4#r^)r_rGr~_clonerrr<r1s&&&&r4clonePolyRing.clone.s.  :gt#<#<GnG{{7E22r6c TPT;'g VPT;'g VPT;'g VP4#r^) __class__rr<r1rs&&&&r4rPolyRing._clone4s<~~g55v7L7LeNaNaW[WaWabbr6c H^.VP,p^W!&\V4#)zReturn the ith-basis element. )rnr~)rrbasiss&& r4rPolyRing.monomial_basis8s"DJJU|r6c $VP.4#r^)rrs&r4r PolyRing.zero>szz"~r6c 8VPVP4#r^)rrrs&r4r PolyRing.oneBszz$))$$r6c P\V\4;'dVPV8H#)zATrue if ``element`` is an element of this ring. False otherwise. )r_rr>relements&&r4 is_elementPolyRing.is_elementFs ';/HHGLLD4HHr6c 8VPPW4#r^)r<convertrr orig_domains&&&r4 domain_newPolyRing.domain_newJs{{""788r6c :VPVPV4#r^)term_newr)rcoeffs&&r4 ground_newPolyRing.ground_newMs}}T__e44r6c XVPV4pVPpV'dW#V&V#r^)rr)rmonomrrs&&& r4rPolyRing.term_newPs(&yy K r6c \V\4'dtWP8XdV#\VP\4'd7VPPVP8XdVP V4#\ R4h\V\4'd \ R4h\V\4'dVPV4#\V\4'dVPV4#\V\4'dVPV4#VP V4# \dTPT4u#i;i) conversionparsing)r_rr>r<rrNotImplementedErrorr`rKrNrG from_terms ValueError from_listr from_exprrs&&r4ring_newPolyRing.ring_newWs g{ + +||#DKK88T[[=M=MQXQ]Q]=]w//),77  % %%i0 0  & &>>'* *  & & /w// & &>>'* *??7+ +  /~~g.. /s)D22EEc VPpVPpVP4FwrVV!Wb4pV'gKWdV&K V#r^)rrrM)rrrrrrrs&&& r4rNPolyRing.from_dictosC__ yy#MMOLEu2Eu#U ,  r6c 8VP\V4V4#r^)rNrKrs&&&r4rPolyRing.from_termszs~~d7m[99r6c nVP\WP^, VP44#)rNrrnr<rs&&r4rPolyRing.from_list}s$~~k'::a<MNNr6c VaaaaSPoVVVV3RloS!\V44#)c <SPV4pVeV#VP'd.\\\ \ SVP 444#VP'd.\\\ \ SVP 444#VP4wr#VP'd V^8dS!V4\V4,#SPSPV44#r^)getis_Addr rrGrHargsis_Mulr as_base_exp is_Integerrmrr)exprrbaseexp_rebuildr<mappingrs& r4r (PolyRing._rebuild_expr.._rebuilds D)I$  c4Hdii(@#ABBc4Hdii(@#ABB!,,. >>>cAg#D>3s833??6>>$+?@@r6)r<r)rrr r r<sf&f@@r4 _rebuild_exprPolyRing._rebuild_exprs) A A$ &&r6c \\\VPVP444pVP W4pVP V4# \d\RT: RT: 24hi;i)z6expected an expression convertible to a polynomial in z, got ) rKrGrLrr:r rr!r)rrr rs&& r4rPolyRing.from_exprsktC dii89: '%%d4D==& & pcgimno o ps AA5c VfVP'd^pV#RpV#\V\4'dTTp^V8:dW P8dV#VP)V8:dVR8:d V)^, pV#\RV,4hVP V4'dVP P V4pV#\V\4'dVPP V4pV#\RV,4h \d\RT,4hi;i \d\RT,4hi;i)z+Compute index of ``gen`` in ``self.gens``. zinvalid generator index: %szinvalid generator: %szEexpected a polynomial generator, an integer, a string or None, got %s) rnr_rmrrr:indexr`r)rgenrs&& r4rPolyRing.indexsK ;zzz2/.-S ! !AAv!jj.$#**!a2gBF !!>!DEE __S ! ! @IIOOC(S ! ! @LL&&s+ dgjjk k @ !83!>?? @  @ !83!>?? @s!DD%D"%Ec \\VPV44p\VP4UUu.uFwr4W29gK VNK pppV'g VP #VP VR7#uuppi)z,Remove specified generators from this ring. rj)rrHr enumeraterr<r)rr:indicesrrcrs&* r4drop PolyRing.dropscc$**d+,"+DLL"9O"9$!Q=MAA"9O;; ::g:. . Ps A6A6c tVPV,pV'g VP#VPVR7#)rj)rr<r)rryrs&& r4 __getitem__PolyRing.__getitem__s.,,s#;; ::g:. .r6c VPP'g\VPR4'd'VPVPPR7#\ RVP,4h)r<r<z%s is not a composite domain)r<rrrrrs&r4 to_groundPolyRing.to_groundsR ;; # # #wt{{H'E'E::T[[%7%7:8 8;dkkIJ Jr6c \V4#r^rrs&r4 to_domainPolyRing.to_domains d##r6c ^^RIHpV!VPVPVP4#)r}) FracField)sympy.polys.fieldsr&rr<r1)rr&s& r4to_fieldPolyRing.to_fields 0t{{DJJ??r6c 2\VP4^8H#rrr:rs&r4 is_univariatePolyRing.is_univariates499~""r6c 2\VP4^8#rr+rs&r4is_multivariatePolyRing.is_multivariates499~!!r6c VPpVF:p\V\R7'dW P!V!, pK2W#, pK< V#)a 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&* r4r PolyRing.addsB" IIC3 66XXs^#  r6c VPpVF:p\V\R7'dW P!V!,pK2W#,pK< V#)ap 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)]) r2)rr-r rr4s&* r4r PolyRing.mulsB" HHC3 66XXs^#  r6c l\\VPV44p\VP4UUu.uFwr4W29gK VNK ppp\VP 4UUu.uFwr6W29gK VNK pppV'gV#VP WPP!V!R7#uuppiuuppi)zL Remove specified generators from the ring and inject them into its domain. rr<)rrHrrrr:rr)rr:rrrcrrs&* r4drop_to_groundPolyRing.drop_to_grounds c$**d+,!*4<D::d4j:1 1Kr6c \VP4P\V44pVP\ V4R7#)z9Add the elements of ``symbols`` as generators to ``self``rjr?)rrrAs&& r4add_gensPolyRing.add_gens4s44<< &&s7|4zz$t*z--r6c aV^8gWP8d\RV: RVP: 24hV'g VP#VPp\ \ VP4\V44Fo\;QJd*.V3Rl\ VP44FNK 5M#!V3Rl\ VP444pW PW0PP4, pK V#)zW Return the elementary symmetric polynomial of degree *n* over this ring's generators. z.Cannot generate symmetric polynomial of order z for c3@<"TFp\VS94xK R#5ir^)rm)rbrrcs& r4rd*PolyRing.symmetric_poly..EsE3Dac!q&kk3Ds) rnrr:rrr,rrmr~rr<)rnrrrcs&& @r4symmetric_polyPolyRing.symmetric_poly9s q5A NZ[]a]f]fgh h88O99DU4::.A7E53DEE53DEE e[[__==8Kr6r2)NNNr^)/r __module__ __qualname____firstlineno____doc____annotations__rrrrrrrrrr rrpropertyrrrrrrr__call__rNrrr rrrrr r#r(r,r/rrr<rBrErK__static_attributes__r2r6r4r9r9s34 !!  J N ,/<| 7D !3  c c %%I95,,H :O'.'>//K$@##""66 H. r6r9ca]tRtRtRtV3RltRtRtRtRt Rt R t R t R t R tR tRtRtRtRtRRltRtRtRtRtRtRtRtRtRtRtRtRt Rt!]"R 4t#]"R!4t$]"R"4t%]"R#4t&]"R$4t']"R%4t(]"R&4t)]"R'4t*]"R(4t+]"R)4t,]"R*4t-]"R+4t.]"R,4t/]"R-4t0]"R.4t1]"R/4t2]"R04t3R1t4R2t5R3t6R4t7R5t8R6t9R7t:R8t;R9tR<t?R=t@R>tAR?tBR@tCRAtDRBtERCtFRDtGREtHRFtIRGtJRHtKRItLRJtMRKtNRRLltORMtPRRNltQROtRRPtSRQtTRRtURStV]"RT4tW]"RU4tXRVtY]"RW4tZRXt[RYt\RRZlt]RR[lt^RR\lt_R]t`R^taR_tbR`tcRatdRbteRctfRdtgRethRftiRgtjRhtkRitlRjtmRktnRlto]otpRmtqRntrRotsRpttRqtuRrtvRstwRttxRutyRvtzRwt{Rxt|Ryt}Rzt~R{tR|tR}tR~tRRltRRltRtRRltRtRRltRRltRRltRRltRRltRtRtRtRtRtRtRtRtRtRtRtRtRRltRtRtV;t#)riJz5Element of multivariate distributed polynomial ring. c 2<\SV`V4WnR#r^)super__init__r>)rr>initrs&&&r4rXPolyElement.__init__Ms  r6c \V\4'gQh\VP\4'gQhVPPp\V\ 4'gQhVP 4Fwr#VPV4'gQh\V4VPP8XgQh\;QJdRV4F 'dK RM RM !RV44'dKQh R#)c3\"TF"p\V\4;'dV^8xK$ R#5i)r}N)r_rm)rbr s& r4rd%PolyElement._check..[s&JESz#s+88q8Es,,FTN) r_rr>r9r<rrMof_typerrnri)rdomrrs& r4_checkPolyElement._checkSs$ ,,,,$))X....ii#v&&&& JJLLE;;u%% %%u:0 003JEJ333JEJJJ JJ)r6c :VPVPV4#r^)rr>)rrYs&&r4rPolyElement.new]s~~dii..r6c 6VPP4#r^)r>r#rs&r4parentPolyElement.parent`syy""$$r6c LVP\VP443#r^)r>rG itertermsrs&r4rPolyElement.__getnewargs__cs 4 0122r6Nc VPpVf6\VP\VP 4434;VnpV#r^)rrr> frozensetrM)rrs& r4rPolyElement.__hash__hs<   =!%tyy)DJJL2I&J!K KDJ r6c $VPV4#)aReturn a copy of polynomial self. Polynomials are mutable; if one is interested in preserving a polynomial, and one plans to use inplace operations, one can copy the polynomial. This method makes a shallow copy. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> R, x, y = ring('x, y', ZZ) >>> p = (x + y)**2 >>> p1 = p.copy() >>> p2 = p >>> p[R.zero_monom] = 3 >>> p x**2 + 2*x*y + y**2 + 3 >>> p1 x**2 + 2*x*y + y**2 >>> p2 x**2 + 2*x*y + y**2 + 3 )rrs&r4rPolyElement.copyss4xx~r6c  zVPV8XdV#VPPVP8wd`\\\ WPPVP4!4pVP W PP 4#VPWPP 4#r^)r>rrGrLr(rr<rN)rnew_ringtermss&& r4set_ringPolyElement.set_rings 99 K YY  ("2"2 2mD))2C2CXEUEUVWXE&&uii.>.>? ?%%dII,<,<= =r6c V'gVPPpMT\V4VPP8wd1\ RVPP: R\V4: 24h\ VP 4.VO5!#)z"Wrong number of symbols, expected z got )r>rrrnrr' as_expr_dict)rrs&*r4as_exprPolyElement.as_exprsfii''G \TYY__ ,#g,0  d//1r<to_sympyrh)rryrrs& r4ruPolyElement.as_expr_dictsC99##,,;?>>;KL;K<5x&;KLLLsA c  VPPpVP'dVP'gVPV3#VP 4pVPpVP pVPpVP4FpV!W5!V44pK TPVP4UUu.uFwrxWxV,3NK upp4p W93#uuppir^) r>r<is_Fieldhas_assoc_Ringrget_ringrdenomrJrrM) rr< ground_ringcommonrrrkvrs & r4 clear_denomsPolyElement.clear_denomss!!f&;&;&;::t# #oo' oo [[]Eu.F#xxDJJLBLDA1h-LBC|CsC c b\VP44FwrV'dKWK R#)z+Eliminate monomials with zero coefficient. NrGrM)rrrs& r4 strip_zeroPolyElement.strip_zeros#&DA1G'r6c V'gV'*#VPPV4'd\PW4#\ V4^8dR#VP VPP 4V8H#)zEquality 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 F)r>rrKrrrrp1p2s&&r4rPolyElement.__eq__s^"6M WW   # #;;r& & Wq[66"'',,-3 3r6c W8X*#r^r2rs&&r4rPolyElement.__ne__s |r6c &VPpVPV4'd\VP44\VP448wdR#VPP pVP4F"pV!W,W,V4'dK!R# R#\ V4^8dR#VPPV4pVPP VP4W4# \dR#i;i)z+Approximate equality test for polynomials. FT) r>rrkeysr<almosteqrrconstr!)rr tolerancer>rrs&&& r4rPolyElement.almosteqsww ??2  2779~RWWY/{{++HWWYrui88  Wq[ G[[((,{{++BHHJFF"  s<D DDc 8\V4VP43#r^)rrqrs&r4sort_keyPolyElement.sort_keysD 4::<((r6c VPPV4'd&V!VP4VP44#\#r^)r>rrNotImplemented)rrops&&&r4_cmpPolyElement._cmps6 77  b ! !bkkmR[[]3 3! !r6c .VPV\4#r^)rrrs&&r4__lt__PolyElement.__lt__wwr2r6c .VPV\4#r^)rrrs&&r4__le__PolyElement.__le__rr6c .VPV\4#r^)rrrs&&r4__gt__PolyElement.__gt__rr6c .VPV\4#r^)rrrs&&r4__ge__PolyElement.__ge__rr6c VPpVPV4pVP^8XdW2P3#\ VP 4pWCW2P VR73#)rrj)r>rrnr<rGrrrrr>rrs&& r4_dropPolyElement._dropsVyy JJsO ::?kk> !4<<(G jjj11 1r6c VPV4wr#VPP^8Xd6VP'dVP ^4#\ RV,4hVP pVP4F>wrVWR,^8Xd\V4pWrWd\V4&K.\ RV,4h V#)rzCannot drop %s) rr>rn is_groundrrrrMrGr~)rrrr>rrrKs&& r4rPolyElement.drops**S/ 99??a ~~~zz!}$ !1C!78899D 419QA%&qN$%5%;<< %Kr6c VPpVPV4p\VP4pWCW2P WBV,R73#)r;)r>rrGrrrs&& r4_drop_to_groundPolyElement._drop_to_ground%sCyy JJsOt||$ J**W!W*===r6c VPP^8Xd \R4hVPV4wr#VPpVP P ^,pVP4FrwrVVRVWR^,R,pWt9d#WV,,PV4WG&KBWG;;,WV,,PV4, uu&Kt V#)rz$Cannot drop only generator to groundN) r>rnrrrr<r:rh mul_ground)rrrr>rrrmons&& r4r<PolyElement.drop_to_ground-s 99??a CD D&&s+yykkq! NN,LE)eaCDk)C (]66u=  c8m77>> - r6c x\WPP^, VPP4#r)rr>rnr<rs&r4to_densePolyElement.to_dense>s&T99??1#4dii6F6FGGr6c \V4#r^)rKrs&r4to_dictPolyElement.to_dictAs Dzr6c V'g0VPVPPP4#VR,pVR,pVPpVPpVP p VP p .p VP4EFwrVPPV 4pV'dRMRpV PV4W8Xd;VPV 4pV'd!VPR4'd VR,pMEV'dV )p WPPP8wdVPWRR7pMRp.p\V 4FpV V,pV'gKVPVV,VRR7pV^8wdKV\V48wgV^8dVPVVR R7pMTpVPVVV3,4KVPR V,4K V'd V.V,pV PVPV44EK V ^,R 9d+V P!^4pVR8XdV P#^R4RPV 4#) MulAtom -  + -rNNT)strictFz%s)rr)_printr>r<rrrnrrq is_negativerrr parenthesizerrmjoinpopinsert)rprinter precedence exp_pattern mul_symbolprec_mul prec_atomr>rrnzmsexpvsrrnegativesignscoeffsexpvrr rsexpheads&&&&& r4r`PolyElement.strDs>>$))"2"2"7"78 8e$v& yy,,  __::LL/"5( MM*//%0 1?(@ !9 &::a=Du} a%wwvr6c 2WPP9#r^)r>rrs&r4 is_generatorPolyElement.is_generatortsyy****r6c ~V'*;'g0\V4^8H;'dVPPV9#r)rr>rrs&r4rPolyElement.is_groundxs1xLLCINKKtyy/C/Ct/KLr6c jV'*;'g&\V4^8H;'dVP^8H#r)rLCrs&r4 is_monomialPolyElement.is_monomial|s*x<r<rrrs&r4rPolyElement.is_negative!yy++DGG44r6c `VPPPVP4#r^)r>r< is_positiverrs&r4rPolyElement.is_positiverr6c `VPPPVP4#r^)r>r<is_nonnegativerrs&r4rPolyElement.is_nonnegative!yy..tww77r6c `VPPPVP4#r^)r>r<is_nonpositiverrs&r4rPolyElement.is_nonpositiverr6c V'*#r^r2r{s&r4is_zeroPolyElement.is_zeros u r6c 2WPP8H#r^)r>rrs&r4is_onePolyElement.is_onesFFJJr6c `VPPPVP4#r^)r>r<rrrs&r4is_monicPolyElement.is_monicsvv}}##ADD))r6c hVPPPVP44#r^)r>r<rcontentrs&r4 is_primitivePolyElement.is_primitives!vv}}##AIIK00r6c \;QJd*RVP44F 'dK R# R#!RVP444#)c3>"TFp\V4^8*xK R#5irNrIrbrs& r4rd(PolyElement.is_linear..?u3u:?FTri itermonomsrs&r4 is_linearPolyElement.is_linear7s? ?ss?s?s? ???r6c \;QJd*RVP44F 'dK R# R#!RVP444#)c3>"TFp\V4^8*xK R#5i)r0Nr r s& r4rd+PolyElement.is_quadratic..r r FTrrs&r4 is_quadraticPolyElement.is_quadraticrr6c tVPP'gR#VPPV4#T)r>rn dmp_sqf_prs&r4 is_squarefreePolyElement.is_squarefrees'vv|||vv""r6c tVPP'gR#VPPV4#r)r>rndmp_irreducible_prs&r4is_irreduciblePolyElement.is_irreducibles'vv|||vv''**r6c VPP'dVPPV4#\R4h)zcyclotomic polynomial)r>r,dup_cyclotomic_pr$rs&r4 is_cyclotomicPolyElement.is_cyclotomics3 66   66**1- --.EF Fr6c xTPVP4UUu.uF wrW)3NK upp4#uuppir^)rrh)rrrs& r4__neg__PolyElement.__neg__s2xxdnn>NP>Nle5&/>NPQQPs6 c V#r^r2rs&r4__pos__PolyElement.__pos__s r6c V'gVP4#VPpVPV4'dmVP4pVPpVPP pVP 4F$wrgV!We4V,pV'dWsV&K"W6K& V#\V\4'd\VP\4'd'VPPVP8XdMf\VPP\4'd7VPPPV8XdVPV4#\#VPV4pVP4pV'gV#VPp WP49dWV &V#WV ,)8XdW9V#W9;;,V, uu&V# \d \u#i;i)zAdd two polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> (x + y)**2 + (x - y)**2 2*x**2 + 2*y**2 )rr>rrr<rrMr_rr__radd__rrrrr!) rrr>r6rrrrcp2rs && r4__add__PolyElement.__add__s779 ww ??2   A%%C;;##D L1$aD # H K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[{{2&%% //"%C AB"" H B%<HESLEH "! ! "s+G$$G87G8c <VP4pV'gV#VPpVPV4pVPpW@P 49dWV&V#WV,)8XdW$V#W$;;,V, uu&V# \ d \ u#i;ir^)rr>rrrr!r)rrJr6r>rs&& r4r,PolyElement.__radd__s GGIHww "AB"" H 2;HEQJEH "! ! "sBBBc V'gVP4#VPpVPV4'dmVP4pVPpVPP pVP 4F$wrgV!We4V, pV'dWsV&K"W6K& V#\V\4'd\VP\4'd'VPPVP8XdMf\VPP\4'd7VPPPV8XdVPV4#\#VPV4pVP4pVPpWP49dV)W8&V#WV,8XdW8V#W8;;,V,uu&V# \d \u#i;i)zSubtract 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>rrr<rrMr_rr__rsub__rrrrr!) rrr>r6rrrrrs && r4__sub__PolyElement.__sub__s~ 779 ww ??2   A%%C;;##D L1$aD # H K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[{{2&%% $B AB" H 2;HERKEH "! ! "s+GG.-G.c VPpVPV4pVPpVFpW,)W4&K W1, pV# \d \u#i;i)zn - 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!r)rrJr>r6rs&& r4r3PolyElement.__rsub__Escww "A A8) FAH "! ! "sA AAc VPpVPpV'd V'gV#VPV4'dVPpVPPpVP p\ VP44pVP4F/wrVF$wrV!W4p V!W4W,,W<&K& K1 VP4V#\V\4'd\VP\4'd'VPPVP8XdMf\VPP\4'd7VPPPV8XdVPV4#\#VPV4pVP4FwrW,p V 'gKWV&K V# \d \u#i;i)zMultiply 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>rrrr<rrGrMrr_rr__rmul__rrr!)rrr>r6rrrp2itexp1v1exp2v2r rs&& r4__mul__PolyElement.__mul__asy ww IIH __R %%C;;##D,,L #DHHJ $HD&t2C ^be3AF!%' LLNH K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[{{2&%% $BHHJE1dG' H "! ! "s!G##G76G7c VPPpV'gV#VPPV4pVP4Fwr4W,pV'gKWRV&K V# \d \ u#i;i)zp2 * 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>rrrMr!r)rrr6r;r<rs&& r4r9PolyElement.__rmul__sw GGLLH ""2&BHHJE1dG'H "! ! "sA..BBc .\V\4'g\RV,4hV^8d\RV,4hVPpV'g V'd VP #\R4h\ V4^8Xdy\VP44^,wr4VPpWBPP 8XdWEVPW14&V#WA,WRPW14&V#\V4pV^8d \R4hV^8XdVP4#V^8XdVP4#V^8XdWP4,#\ V4^8:dVPV4#VPV4#)zraise 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 %sz/exponent must be a non-negative integer, got %sz0**0zNegative exponent)r_rm TypeErrorrr>rrrGrMrr<rrsquare_pow_multinomial _pow_generic)rrJr>rrr6s&& r4__pow__PolyElement.__pow__sT!S!!AAEF F UNQRRS Syyxx (( Y!^ -a0LE A '16$##E-.H27##E-.H F q501 1 !V99;  !V;;= !V % % Y!^((+ +$$Q' 'r6c VPPpTpV^,'dW#,pV^,pV'gV#VP4pV^,pKGr)r>rrE)rrJr6rXs&& r4rGPolyElement._pow_genericsS IIMM 1uuCQ  AQAr6c :\\V4V4P4pVPPpVPP pVP4pVPP PpVPPpVFwrTp T p \W4F*wp wrV 'gKV!WV 4p WV ,,p K, \V 4p T pVPW4V,pV'dWV &KuW9gK}W}K V#r^) rrrMr>rrr<rrLr~r)rrJ multinomialsrrrqrr multinomialmultinomial_coeff product_monom product_coeffr rrs&& r4rFPolyElement._pow_multinomials/D 1=CCE ))33YY))  yy$$yy~~.: *K&M-M'*;'>#^e3$3M#$NM!CZ/M(? -(E!EHHU)E1E#U K#/;& r6c fVPpVPpVPp\VP 44pVP PpVP p\\V44FSpWG,pW,p \V4F1p WJ,p V!W4p V!W4WV ,,,W,&K3 KU VP^4pVPpVP4F%wrV!W4p V!W4V^,,W+&K' VP4V#)zsquare 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>rrrGrr<rrrimul_numrMr)rr>r6rrrrrk1pkjk2r rrs& r4rEPolyElement.squaresyy IIeeDIIK {{(( s4y!ABB1XW"2*S""X+5" JJqMeeJJLDAa#BMAqD(AE! r6c VPpV'g \R4hVPV4'dVPV4#\ V\ 4'd\ VP \4'd'VP PVP8XdMf\ VPP \4'd7VPP PV8XdVPV4#\#VPV4pVPV4VPV43# \d \u#i;ipolynomial division)r>ZeroDivisionErrorrrr_rr<r __rdivmod__rr quo_ground rem_groundr!rrr>s&& r4 __divmod__PolyElement.__divmod__:sww#$9: : __R 66":  K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[~~b))%% :$BMM"%r}}R'89 9 "! ! "s D>>EEc VPpVPV4pVPV4# \d \u#i;ir^)r>rrr!rras&& r4r^PolyElement.__rdivmod__PEww $B66":  "! ! "0AAc VPpV'g \R4hVPV4'dVPV4#\ V\ 4'd\ VP \4'd'VP PVP8XdMf\ VPP \4'd7VPP PV8XdVPV4#\#VPV4pVPV4# \d \u#i;ir[) r>r]rremr_rr<r__rmod__rrr`r!ras&& r4__mod__PolyElement.__mod__Ysww#$9: : __R 66":  K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[{{2&%% %$B==$ $ "! ! " D--EEc VPpVPV4pVPV4# \d \u#i;ir^)r>rrir!rras&& r4rjPolyElement.__rmod__orfrgc VPpV'g \R4hVPV4'dVPV4#\ V\ 4'd\ VP \4'd'VP PVP8XdMf\ VPP \4'd7VPP PV8XdVPV4#\#VPV4pVPV4# \d \u#i;ir[) r>r]rquor_rr<r __rtruediv__rrr_r!ras&& r4 __floordiv__PolyElement.__floordiv__xsww#$9: : __R 66":  K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[r**%% %$B==$ $ "! ! "rmc VPpVPV4pVPV4# \d \u#i;ir^)r>rrqr!rras&& r4 __rfloordiv__PolyElement.__rfloordiv__rfrgc VPpV'g \R4hVPV4'dVPV4#\ V\ 4'd\ VP \4'd'VP PVP8XdMf\ VPP \4'd7VPP PV8XdVPV4#\#VPV4pVPV4# \d \u#i;ir[) r>r]rexquor_rr<rrrrrr_r!ras&& r4 __truediv__PolyElement.__truediv__sww#$9: : __R 88B<  K ( ($++~664;;;K;Krww;VBGGNNN;;@S@SW[@[r**%% %$B==$ $ "! ! "rmc VPpVPV4pVPV4# \d \u#i;ir^)r>rryr!rras&& r4rrPolyElement.__rtruediv__sEww $B88B<  "! ! "rgc aaaVPPoVPPpVPoVPPoVP 'd VVV3RlpV#VVV3RlpV#)cX<Vwr#VwrEVS 8XdTpMS!W$4pVe VS!W543#R#r^r2 a_lm_a_lc b_lm_b_lca_lma_lcb_lmb_lcr domain_quorrs && r4term_div'PolyElement._term_div..term_divs@& & 2: E(4E$ *T"888r6ct<Vwr#VwrEVS 8XdTpMS!W$4pVeW5,'g VS!W543#R#r^r2rs && r4rrsE& & 2: E(4E  *T"888r6)r>rr<rqrr|)rr<rrrrs& @@@r4 _term_divPolyElement._term_divsZ YY ! !!!ZZ yy-- ??? 0 r6c VPpRp\V\4'dRpV.p\V4'g \ R4hV'g/V'dVP VP 3#.VP 3#VFpVPV8wgK\ R4h \V4p\V4Uu.uFqbP NK ppVP4pVP p VP4p VU u.uFqP4NK p p V'd^p^p We8dV ^8XdVP4pV !WV,3W,W,W,,34pVe@VwppWv,PVV34Wv&VPW,VV)34p^p KV^, pKV 'dKVP4pV PWV,34p WKXVP8Xd W, p V'd"V'gVP V 3#V^,V 3#Wy3#uupiuup i)aDivision 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 FTr\z"self and f must have the same ring)r>r_rrir]rrrrrrr _iadd_monom_iadd_poly_monomr)rfvr> ret_singler{rcrqvr6rrfxexpvs divoccurredrtermexpv1rXs&& r4rPolyElement.divs8yy b+ & &JB2ww#$9: :yy$))++499}$Avv~ !EFF G!&q *Aii * IIK II>>#-/0Rr"R0AK%K1,~~'w%(BE%(O1LM##HE1E--uaj9BE**2551"+>A"#KFA;(MM44/2G 4?? " FA yy!|#!uax5L=+1s 9H8?H=c &Tp\V\4'dV.p\V4'g \R4hVPpVP pVP pVPpVP pVP4pVPp VP4pVPp V'dVFp V!WP4p V fKV wrV P4F7wppV!W4pV !VV4VV,, pV'gVVK2VVV&K9 VP4pVe VVV,3p K V wppVV9dVV;;,V, uu&MVVV&VVVP4pVfKVVV,3p KV#r[)r_rrir]r>r<rrrLTrrrhr)rGr{r>r<rrrrltfrgtqrWrXmgcgm1c1ltmltcs&& r4riPolyElement.rem#sh  a % %A1vv#$9: :vv{{(( II;;=dd FFHeec44(>DA"#++-B)"0 T]QrT1! !"$&AbE #0..*C!1S6k"S!8cFcMF AcFcFnn&?qv+Cr6c 2VPV4^,#r|)r)r{rs&&r4rqPolyElement.quoPsuuQx{r6c RVPV4wr#V'gV#\W4hr^)rr#)r{rqrs&& r4ryPolyElement.exquoSs$uuQxH%a+ +r6c WPP9dVP4pMTpVwr4VPV4pVfWBV&V#WT, pV'dWRV&V#W#V#)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)rmccpselfrrrXs&& r4rPolyElement._iadd_monom[sr8 99&& &YY[FF  JJt  9 4L JA t  L r6c vTpW3PP9dVP4pVwrEVPpVPPP pVPP pVP4F2wrV!W4p V!W4W,,p V 'dWV &K0W;K4 V#)aadd 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>rrrr<rrrM) rrrrrWrXrrrrrkars &&& r4rPolyElement._iadd_poly_monoms* "" "Bffww~~""ww++ HHJDAa#BMAC'E2F  r6c aVPPV4oV'g\#S^8d^#\V3RlVP 444#)zx The leading degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (``float('-inf')``) c34<"TF qS,xK R#5ir^r2rbrrs& r4rd%PolyElement.degree..<^EQxx^)r>rrrzrr{xrs&&@r4degreePolyElement.degree? FFLLOK Urnr~rHrzrGrLrrs&r4degreesPolyElement.degrees>7166<<' 'S$sALLN';"<=> >r6c aVPPV4oV'g\#S^8d^#\V3RlVP 444#)zu The tail degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (``float('-inf')``) c34<"TF qS,xK R#5ir^r2rs& r4rd*PolyElement.tail_degree..rr)r>rrminrrs&&@r4 tail_degreePolyElement.tail_degreerr6c  V'g#\3VPP,#\\ \ \ \VP4!444#)zx A tuple containing tail degrees in all variables. Note that the degree of 0 is negative infinity (``float('-inf')``) ) rr>rnr~rHrrGrLrrs&r4 tail_degreesPolyElement.tail_degreesrr6c LV'dVPPV4#R#)a Leading 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&r4rPolyElement.leading_expvs 99))$/ /r6c `VPWPPP4#r^)rr>r<rrrs&&r4 _get_coeffPolyElement._get_coeffs!xxii..3344r6c V^8Xd&VPVPP4#VPPV4'dj\ VP 44p\ V4^8XdAV^,wr4W@PPP8XdVPV4#\RV,4h)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 zexpected a monomial, got %s) rr>rrrGrhrr<rr)rrrqrrs&& r4rPolyElement.coeffs4 a<??499#7#78 8 YY ! !' * ***,-E5zQ$Qx II,,000??5116@AAr6c LVPVPP4#)z"Returns the constant coefficient. )rr>rrs&r4rPolyElement.const styy3344r6c @VPVP44#r^)rrrs&r4rPolyElement.LC$st00233r6c ZVP4pVfVPP#V#r^)rr>rrs& r4LMPolyElement.LM(s*  " <99'' 'Kr6c VPPpVP4pV'd#VPPPW&V#)z 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>rrr<rrr6rs& r4 leading_monomPolyElement.leading_monom0s> IINN  " ii&&**AGr6c VP4pVf7VPPVPPP3#WP V43#r^)rr>rr<rrrs& r4rPolyElement.LTEsL  " <II(($))*:*:*?*?@ @//$/0 0r6c nVPPpVP4pVe W,W&V#)zLeading 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>rrrs& r4 leading_termPolyElement.leading_termMs3 IINN  "  jAGr6c aSfVPPoM\P!S4oS\Jd\ VRRR7#\ VV3RlRR7#)NcV^,#r|r2)rs&r4rt%PolyElement._sorted..hsqr6T)ryreversec"<S!V^,4#r|r2)rr1s&r4rtrjs uQxr6)r>r1rrrsorted)rr\r1s&&fr4_sortedPolyElement._sortedasL =IIOOE''.E C<##94H H##@$O Or6c XVPV4UUu.uFwr#VNK upp#uuppi)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] rq)rr1_rs&& r4rTPolyElement.coeffsls)0(,zz%'8:'881'8::: &c XVPV4UUu.uFwr#VNK upp#uuppi)aOrdered 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)] r)rr1rrs&& r4monomsPolyElement.monomss)0(,zz%'8:'885'8:::rc TVP\VP44V4#)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)] )rrGrM)rr1s&&r4rqPolyElement.termss 0||D.66r6c 4\VP44#)z,Iterator over coefficients of a polynomial. )iterrJrs&r4 itercoeffsPolyElement.itercoeffsDKKM""r6c 4\VP44#)z)Iterator over monomials of a polynomial. )rrrs&r4rPolyElement.itermonomsDIIK  r6c 4\VP44#)z%Iterator over terms of a polynomial. )rrMrs&r4rhPolyElement.itertermsDJJL!!r6c 4\VP44#)z+Unordered list of polynomial coefficients. )rGrJrs&r4 listcoeffsPolyElement.listcoeffsrr6c 4\VP44#)z(Unordered list of polynomial monomials. )rGrrs&r4 listmonomsPolyElement.listmonomsrr6c 4\VP44#)z$Unordered list of polynomial terms. rrs&r4 listtermsPolyElement.listtermsrr6c WPP9d W,#V'gVP4R#VFpW;;,V,uu&K V#)amultiply 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)r6rXr s&& r4rTPolyElement.imul_numsD4   3J GGI C FaKFr6c VPPpVPpVPpVP 4F pV!W$4pK V#)z*Returns GCD of polynomial's coefficients. )r>r<rrr)r{r<contrrs& r4rPolyElement.contentsB{{jj\\^Et#D$ r6c VP4pWPPP8XdW3#WP V43#)z,Returns content and a primitive polynomial. )rr>r<rr_)r{r s& r4 primitivePolyElement.primitives;yy{ 66==%% %9 \\$'''r6c LV'gV#VPVP4#)z5Divides all coefficients by the leading coefficient. )r_rrs&r4monicPolyElement.monicsH<<% %r6c V'gVPP#VP4UUu.uFwr#W#V,3NK pppVPV4#uuppir^)r>rrhr)r{rrrrqs&& r4rPolyElement.mul_groundsJ66;; 78{{}F}|u5'"}FuuU|GsAc VPPpVP4UUu.uFwr4V!W14V3NK pppVPV4#uuppir^)r>rrMr)r{rrf_monomf_coeffrqs&& r4 mul_monomPolyElement.mul_monomsQvv** RSRYRYR[]R[>Ng</9R[]uuU|^sAc hVwr#V'd V'gVPP#W PP8XdVPV4#VPPpVP 4UUu.uFwrVV!WR4Wc,3NK pppVP V4#uuppir^)r>rrrrrMr)r{rrrrrrrqs&& r4mul_termPolyElement.mul_terms 66;;  ff'' '<<& &vv** XYX_X_XacXaDTG</?XacuuU|ds?B.c  VPPpV'g \R4hV'dWP8XdV#VP'd8VP pVP 4UUu.uFwrEWC!WQ43NK pppM;VP 4UUu.uFwrEWQ,'dKWEV,3NK! pppVPV4#uuppiuuppir[)r>r<r]rr|rqrhr)r{rr<rqrrrqs&& r4r_PolyElement.quo_ground&s#$9: :AOH ???**CABPuc%m,EPE>?kkm`mleTYT]T])uqj)mE`uuU| Q`s5C C7Cc Vwr#V'g \R4hV'gVPP#W PP8XdVP V4#VP 4pVP 4Uu.uF qT!WQ4NK ppTPVUu.uF qUfKVNK up4#uupiuupir[)r]r>rrr_rrhr)r{rrrrtrqs&& r4quo_termPolyElement.quo_term6s #$9: :66;;  ff'' '<<& &;;=-.[[]<](1#]<uu%:%Qqq%:;;=:sB;$C.Cc VPPP'dL.pVP4F4wr4WA,pWA^,8d WA, pVP W434K6 M+VP4UUu.uFwr4W4V,3NK pppVP V4pVP 4V#uuppi)r0)r>r<is_ZZrhrrr)r{r6rqrrrs&& r4 trunc_groundPolyElement.trunc_groundEs 66==   E !  6>!IE e^, !.>?[[]L]\Uuai(]ELuuU|  MsCc TpVP4pVP4pVPPPW44pVP V4pVP V4pWRV3#r^)rr>r<rr_)rrr{fcgcrs&& r4extract_groundPolyElement.extract_groundYs[  YY[ YY[ffmm' LL  LL qyr6c V'g!VPPP#VPPPpT!VP 4Uu.uF q2!V4NK up4#uupir^)r>r<rabsr)r{ norm_func ground_absrs&& r4_normPolyElement._normesT66==%% %**JallnNnUz%0nNO ONsA4c ,VP\4#r^)r2rzrs&r4max_normPolyElement.max_normlwws|r6c ,VP\4#r^)r2rIrs&r4l1_normPolyElement.l1_normor7r6c VPpV.\V4,p^.VP,pVFBpVP4F+p\ V4Fwrx\ WG,V4WG&K K- KD \ V4FwryV 'dK^WG&K \ V4p\;QJdRV4F 'dK RM RM !RV44'dWC3#.p VFjpVPp VP4F6wr\W4UUu.uF wr~W~,NK pppW\ V4&K8 V PV 4Kl WJ3#uuppi)r}c3*"TF q^8HxK R#5irr2)rbrss& r4rd&PolyElement.deflate..s!q!AvqsFT) r>rGrnrrr r~rirrhrLr)r{rr>rYJr6rrrWrsHhIrrWNs&* r4deflatePolyElement.deflaters"vvd1g  C NA%e,DAa=AD-( aLDA1! !H 3!q!333!q! ! !8O A AKKM),Q4aff4#%( * HHQKt 5sE c VPPpVP4F6wr4\W14UUu.uF wrVWV,NK pppWB\ V4&K8 V#uuppir^)r>rrhrLr~)r{r>rrArrrWrBs&& r4inflatePolyElement.inflatesVvv{{ HA"%a)-)$!!##)A-"qN& .sA#c rTpVPPpVP'g6VP4wrBVP4wrQVP WE4pW!,P VP V44pVP'gVPX4#VP4#r^) r>r<r|rrrqrrr)rrr{r<r*r+rXr@s&& r4rPolyElement.lcms KKMEBKKMEB 2"A SIIaeeAh <<? "779 r6c 2VPV4^,#r|) cofactorsr{rs&&r4rPolyElement.gcds{{1~a  r6c .V'g#V'gVPPpW"V3#V'gVPV4wr4pW4V3#V'gVPV4wr5pW4V3#\V4^8XdVP V4wr4pW4V3#\V4^8XdVP V4wr5pW4V3#VP V4wpwrVP V4wr4pVPV4VPV4VPV43#r)r>r _gcd_zeror _gcd_monomrC_gcdrF)r{rrr@cffcfgr>s&& r4rKPolyElement.cofactorss66;;Dt# #++a.KAC3; ++a.KAC3;  Vq[,,q/KAC3;  Vq[,,q/KAC3; IIaL 6AffQi  ! ckk!nckk!n==r6c VPPVPPr2VP'dWV3#V)W2)3#r^)r>rrr)r{rrrs&& r4rOPolyElement._gcd_zeros=FFJJ T   C< 2tT> !r6c  8VPpVPPpVPPpVPpVP p\ VP44^,wrxYxrVP4FwrV!W4p V!W4p K VPW3.4p VPV!Wy4V!W43.4pTPVP4U U u.uFwrV!W4V!W43NK up p 4pWV3#uup p ir|) r>r<rrqrrrGrhr)r{rr> ground_gcd ground_quorrmfcf_mgcd_cgcdrrr@rRrSs&& r4rPPolyElement._gcd_monomsvv[[__ [[__ (( ** akkm$Q'ukkmFB +Eu)E$ EEE>" #eemB. 20EFGHeeUVU`U`UbcUb62mB. 20EFUbcds{ds/D c VPpVPP'dVPV4#VPP'dVP V4#VP W4#r^)r>r<is_QQ_gcd_QQr&_gcd_ZZ dmp_inner_gcd)r{rr>s&& r4rQPolyElement._gcdsXvv ;;   99Q<  [[   99Q< %%a+ +r6c \W4#r^rrLs&&r4rbPolyElement._gcd_ZZs a|r6c jTpVPpVPVPP4R7pVP 4wrRVP 4wraVP V4pVP V4pVP V4wrxp VP V4pVPVP4rzVP V4PVPPW44pV P V4PVPPW44p WxV 3#)r) r>rr<r~rrrrbrrrrq) rrr{r>rpr[rr@rRrSrXs && r4raPolyElement._gcd_QQs vv::T[[%9%9%;:<   JJx  JJx iil  JJt ttQWWY1ll4 ++DKKOOA,BCll4 ++DKKOOA,BCs{r6c NTpVPpV'gW#P3#VPpVP'dVP'gVP V4wrVpMVP VP4R7pVP4wrVP4wrVPV4pVPV4pVP V4wrVpVPP W4wrZp VPV4pVPV4pVPV 4pVPV 4pVP4p WP8XdWg3#WP)8Xd V)V)rvWg3#VPV 4pVPV 4pWg3#)z 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>rr<r|r}rKrr~rrrrcanonical_unit) rrr{r>r<rr6rrpcqcpus && r4cancelPolyElement.cancelsi vvhh; F$9$9$9kk!nGA!zz):z;HNN$EBNN$EB 8$A 8$Akk!nGA! 11"9IA2 4 A 4 A R A R A     ? t ::+ 2rq t  QA QAt r6c dVPPpVPVP4#r^)r>r<rjr)r{r<s& r4rjPolyElement.canonical_unit6 s$$$QTT**r6c .VPpVPV4pVPV4pVPpVP 4FFwrgWc,'gKVP Wd4pVP WvV,,4WX&KH V#)zComputes 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>rrrrhrr) r{rr>rrWrrres && r4diffPolyElement.diff: syvv JJqM    " II;;=KDww&&t/u!W}5)r6c 0^\V4u;8dVPP8:d<MM8VP\ \ VPP V444#\RVPP: R\V4: 24h)r}z expected at least 1 and at most z values, got )rr>rnevaluaterGrLr:r)r{rJs&*r4rSPolyElement.__call__S sb s6{ *affll *::d3qvv{{F#;<= =TUTZTZT`T`beflbmno or6c bTp\V\4'dlVfhV^,VR,uwrBpVPWB4pV'gV#VUUu.uFwrRVPV4V3NK pppVPV4#VPpVP V4pVP PV4pVP^8XdIVP PpVP4FwwrWW),,, pK V#VPV4Pp VP4FewrW,V RVW^,R,rWV ,,p W9d"WV ,,p V 'dWV &KSWKWV 'gKaWV &Kg V #uuppi)Nr) r_rGrwrr>rr<rrnrrh) rrrrr{XYr>rresultrJrrrs &&& r4rwPolyElement.evaluateY se  a  19!aeIFQA 1 A3461!qvvay!n16zz!}$vv JJqM KK   " ::?[[%%F {{} *$ -M99Q<$$D !  8U2AYst%<5d =!K/E&+U  Ku&+U !.KA7sF+c Tp\V\4'd#VfVFwrBVPWB4pK V#VPpVP V4pVP P V4pVP^8XdXVP PpVP4FwwrWyW(,,, pK VPV4#VPp VP4FlwrW,V RVR,W^,R,rWV,,p W9d"WV ,,p V 'dWV &KZWK^V 'gKhWV &Kn V #)Nr|) r_rGsubsr>rr<rrnrrhr) rrrrr{rzr>rr|rJrrrs &&& r4rPolyElement.subs s'  a  19FF1LHvv JJqM KK   " ::?[[%%F {{} *$ -??6* *99D !  8U2AY%5cd %C5d =!K/E&+U  Ku&+U !.Kr6c 8aaaVP4pVPpVPpV'gWP.3#\ V4Uu.uFqBP V^,4NK upo/oVV3Rlp\ \ V^, 44p\ \ V^R44pVPpV'Ed4RRRrp \VP44Ftwpwop \;QJdV3RlV4F 'dK RM RM!V3RlV44'gKK\R\VS444p W8gKoT ST rp Kv V R8wdYuop MM.p\SSR,R ,4FwppVPVV, 4K WP\V4V 4, pT p\V4FwrCVV!WC4,pK VV,pEK<\ \VPS44pWV3#uupi) a 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 cP<W3S9dSV,V,SW3&SW3,#r^r2)rrJ poly_powersrYs&&r4get_poly_power.PolyElement.symmetrize..get_poly_power s.v[(&+Ahk QF#v& &r6Nc3X<"TFpSV,SV^,,8xK! R#5irr2)rbrrs& r4rd)PolyElement.symmetrize.. s"AAuQx5Q</s'*FTc36"TFwrW,xK R#5ir^r2)rbrJrWs& r4rdr s E1D1Dsrrr|)rr>rnrrrKrGrrqrirzrLrrr~r:)rr{r>rJrrrweights symmetric_height_monom_coeffrheight exponentsrm2productr rrrYs& @@@r4 symmetrizePolyElement.symmetrize sv IIKvv JJii# #388<8a$$QqS)8<  ' uQU|$uQ2'II a&($VG%.qwwy%9!>E53AA333AAAA EWe1D EEF'28% &:"}% uIeU2Y%56B  b)7 uY'7? ?IG!),>!//- LAs499e,-W$$S=s Hc a VPpVPp\\VP\ VP 444o VeW3.pMc\V\4'd \V4pMA\V\4'd!\VP4V 3RlR7pM \R4h\V4F$wpwrS V,VPV43WV&K& VP4Fqwr\V4pVPp VF+wrW,^uqV &V 'gKWV ,,p K- V P!\#V4V 34p WJ, pKs V#)Nc$<SV^,,#r|r2)rgens_maps&r4rt%PolyElement.compose..$ s x!~r6rxz9expected a generator, value pair a sequence of such pairs)r>rrKrLr:rrnr_rGrrMrrrrhrrr~)r{rrrr>r replacementsrrrrsubpolyrrJrs&&& @r4rBPolyElement.compose s+vvyyDIIuTZZ'89: =F8L!T""#Aw At$$%aggi5MN  !\]]"<0IAv'{DMM!,<=LO1KKMLEKEhhG$#h 81!tOG% &&e e'<=G OD* r6c TpVPPV4pVP4UUu.uFwrVWT,V8XgKWV3NK pppV'gVPP#\ V!wrVUu.uFqURVR,WT^,R,NK ppVPP \ \ W444#uuppiuupi)am Coefficient of ``self`` with respect to ``x**deg``. Treating ``self`` as a univariate polynomial in ``x`` this finds the coefficient of ``x**deg`` as a polynomial in the other generators. Parameters ========== x : generator or generator index The generator or generator index to compute the expression for. deg : int The degree of the monomial to compute the expression for. Returns ======= :py:class:`~.PolyElement` The coefficient of ``x**deg`` as a polynomial in the same ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x, y, z = ring("x, y, z", ZZ) >>> p = 2*x**4 + 3*y**4 + 10*z**2 + 10*x*z**2 >>> deg = 2 >>> p.coeff_wrt(2, deg) # Using the generator index 10*x + 10 >>> p.coeff_wrt(z, deg) # Using the generator 10*x + 10 >>> p.coeff(z**2) # shows the difference between coeff and coeff_wrt 10 See Also ======== coeff, coeffs Nr|)r>rrhrrLrNrK) rrdegr6rrWrXrqrrTs &&& r4 coeff_wrtPolyElement.coeff_wrt9 sT  FFLLO$%KKMAMDAQTS[!MA66;; e4:;FqBQ%$,q56**F;vvS%8 9::B $Cc TpVPPV4pVPV4pVPV4pV^8d \R4hY4rvWE8dV#WE, ^,pVP W%4p VPP V,p VP W'4p Wu, V^, rWi,p W,W,,pW, pVPV4pWu8gK`Y,pYo,#)ag 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 r\r>rrr]rr:)rrrr{dfdgrdrrBlc_gxplc_rrWRrrXs&&& r4premPolyElement.premn sl  FFLLO XXa[ XXa[ 6#$9: :2 7H GaK{{1! VV[[^;;q%D7AEqA25 AA!Bw Iu r6c pTpVPPV4pVPV4pVPV4pV^8d \R4hY#TrpWE8dWg3#WE, ^,p VP W%4p VPP V,p VP W(4p W, V ^, rWj,pWW,,,pWz,pW,W,,pVV, pVPV4pW8gK~Y,pTT,pTT,pYg3#)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. r\r)rrrr{rrrrrrBrrrrWQrrrXs&&& r4pdivPolyElement.pdiv s `  FFLLO XXa[ XXa[ 6#$9: :b 74K GaK{{1! VV[[^;;q%D7AEqA25L AA25 AAA!Bw G E Et r6c 6TpVPW4^,#)a 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)rrrr{s&&& r4pquoPolyElement.pquoG s6 vva|Ar6c jTpVPW4wrEVP'dV#\W14h)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#)rrrr{rrs&&& r4pexquoPolyElement.pexquoe s1: vva| 999H%a+ +r6c TpVPPV4pVPV4pVPV4pWE8dYrYTrTV^8Xd^^.#V^8XdV^.#W1.pWE, pRV^,,pVPW4p W,p VP W%4p W,p ^V .p V )p V 'dV PV4p VP V 4WWV , 3wr1rWV )W,,pVPW4p V P V4p VP W-4p V^8d,V )V,pW^, ,pVP V4p MV )p V P V )4KV#)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] r)r>rrrrrry)rrrr{rJrWrdrsr@lcrXSrr6rs&&& r4 subresultantsPolyElement.subresultants s`D  FFLLO HHQK HHQK 5qq 6q6M 6q6M F E QUO FF1L E[[  G F B A HHQKqa%JA!af Aq A AQ"B1uSQJa%LGGAJC HHaRLr6c 8VPPW4#r^)r>dmp_half_gcdexrLs&&r4 half_gcdexPolyElement.half_gcdex svv$$Q**r6c 8VPPW4#r^)r> dmp_gcdexrLs&&r4gcdexPolyElement.gcdex vv%%r6c 8VPPW4#r^)r> dmp_resultantrLs&&r4 resultantPolyElement.resultant svv##A))r6c 8VPPV4#r^)r>dmp_discriminantrs&r4 discriminantPolyElement.discriminant svv&&q))r6c VPP'dVPPV4#\R4h)zpolynomial decomposition)r>r, dup_decomposer$rs&r4 decomposePolyElement.decompose s3 66   66''* *-.HI Ir6c VPP'dVPPW4#\R4h)zshift: use shift_list instead)r>r, dup_shiftr$r{rrs&&r4shiftPolyElement.shift s3 66   66##A) )-.MN Nr6c 8VPPW4#r^)r> dmp_shiftrs&&r4 shift_listPolyElement.shift_list rr6c VPP'dVPPV4#\R4h)zsturm sequence)r>r, dup_sturmr$rs&r4sturmPolyElement.sturm s3 66   66##A& &-.>? ?r6c 8VPPV4#r^)r> dmp_gff_listrs&r4gff_listPolyElement.gff_list vv""1%%r6c 8VPPV4#r^)r>dmp_normrs&r4normPolyElement.norm svvq!!r6c 8VPPV4#r^)r> dmp_sqf_normrs&r4sqf_normPolyElement.sqf_norm rr6c 8VPPV4#r^)r> dmp_sqf_partrs&r4sqf_partPolyElement.sqf_part rr6c :VPPWR7#))ri)r> dmp_sqf_list)r{ris&&r4sqf_listPolyElement.sqf_list svv""1"..r6c 8VPPV4#r^)r>dmp_factor_listrs&r4 factor_listPolyElement.factor_list svv%%a((r6)rr>r^)F)rrMrNrOrPrXr`rrerrrrrrrvrurrrrrrrrrrrrrrr<rrr`rRrrrrrrrrrrrrrrrrr#r&r)r.r,r4r3r?r9rHrGrFrErbr^rkrjrsrvrzrrrrrirqryrrrrrrrrrrrrrrrrrTrrqrrrhrrrrTrrrrrrr_r#r'r`r,r2r5r9rCrFrrrKrOrPrQrbrarnrjrtrSrwrrrBrrrrrrrrrrrrrrrrrrrrrT __classcell__)rs@r4rrJsU? 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