+
    ‹°³j+  ã                   ó¸  € R t ^ RIHtHtHtHtHtHt ^ RIH	t	 ^ RI
Ht ^ RIHt ^ RIHtHtHtHt ^ RIHtHtHtHtHtHt ^ RIHt ^ RIHt ^ R	IHt ^ R
I H!t!H"t" ^ RI#H$t$ ^ RI%H&t&H't'H(t( ]'R'R l4       t)]'R'R l4       t*]'RR/R l4       t+]']R3R l4       t,]'R(R l4       t-R t.R t/R t0R t1R t2R t3^ RI4H5t5 R t6R t7R t8R t9R  t:R! t;R" t<R# t=R$ t>R% t?R& t@R# ))zIFunctions for generating interesting polynomials, e.g. for benchmarking. )ÚAddÚMulÚSymbolÚsympifyÚDummyÚsymbols)ÚTuple)ÚS)Ú	nextprime)Údmp_add_termÚdmp_negÚdmp_mulÚdmp_sqr)Údmp_zeroÚdmp_oneÚ
dmp_groundÚdup_from_raw_dictÚ	dmp_raiseÚ
dup_random)ÚZZ)Údup_zz_cyclotomic_poly)ÚDMP)ÚPolyÚPurePoly)Ú_analyze_gens)ÚsubsetsÚpublicÚ
filldedentNFc                ó¢  € V ^ 8:  d   \        RV ,          4      hVe   \        V4       M\        R4      pV ^8”  de   ^ RIHp ^RIHp ^pV! ^4      .p\        ^V ^,           4       F%  p\        V4      pVP                  V! V4      4       K'  	  V! \        V!  WR7      # V ^8X  d   V^,          ^,
          pMƒV ^8X  d'   V^,          ^
V^,          ,          ,
          ^,           pMVV ^8X  dP   V^,          ^(V^,          ,          ,
          RV^,          ,          ,           RV^,          ,          ,
          R,           pV'       d   \        XV4      # X# )	zöGenerates n-th Swinnerton-Dyer polynomial in `x`.

Parameters
----------
n : int
    `n` decides the order of polynomial
x : optional
polys : bool, optional
    ``polys=True`` returns an expression, otherwise
    (default) returns an expression.
z6Cannot generate Swinnerton-Dyer polynomial of order %sÚx)Úsqrt)Úminimal_polynomial)Úpolysi`  iÀ  i@  )Ú
ValueErrorr   r   Ú(sympy.functions.elementary.miscellaneousr    Únumberfieldsr!   Úranger
   Úappendr   r   )	Únr   r"   r    r!   ÚpÚaÚiÚexs	   &&&      ÚY/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/polys/specialpolys.pyÚswinnerton_dyer_polyr.      s  € ð 	ˆA„vÜØDÀqÕHóJð 	Jð 	‚}Ü��
ä�#‹Jˆàˆ1„uÝAÝ4ØˆÙ�!‹WˆIˆÜ�q˜!˜a�%–ˆAÜ˜!“ˆAØ�H‰H‘T˜!“WÖñ !ñ "¤# q¡'¨1Ô:Ð:àˆA„vØ��T�A�X‰Ø	
ˆaŒØ��T�B�q˜!•t•G�^˜aÕ‰Ø	
ˆaŒØ��T�B�q˜!•t•G�^˜c ! Q¥$�hÕ&¨¨Q°­T­Õ1°CÕ7ˆç#Œ8�B˜‹?Ð+¨Ð+ó    c                ó,  € V ^ 8:  d   \        RV ,          4      h\        \        \        V 4      \        4      \        4      pVe   \
        P                  ! W14      pM \        P                  ! V\        R4      4      pV'       d   V# VP                  4       # )zùGenerates cyclotomic polynomial of order `n` in `x`.

Parameters
----------
n : int
    `n` decides the order of polynomial
x : optional
polys : bool, optional
    ``polys=True`` returns an expression, otherwise
    (default) returns an expression.
z1Cannot generate cyclotomic polynomial of order %sr   )
r#   r   r   Úintr   r   Únewr   r   Úas_expr)r(   r   r"   Úpolys   &&& r-   Úcyclotomic_polyr5   A   sv   € ð 	ˆA„vÜØ?À!ÕCóEð 	Eô Ô%¤c¨!£f¬bÓ1´2Ó6€Dà‚}Ü�xŠx˜Ó ‰ä�|Š|˜D¤%¨£*Ó-ˆçˆ4Ð,˜dŸl™l›nÐ,r/   r"   c          	     óL  € \        V4      pV ^ 8  g   V \        V4      8”  g	   V'       g   \        RV : RV: 24      hV '       g   \        P                  pM3\        \        V\        V 4      4       Uu. uF  p\        V!  NK  	  up!  pV'       d   \        V.VO5!  # T# u upi )zË
Generates symmetric polynomial of order `n`.

Parameters
==========

polys: bool, optional (default: False)
    Returns a Poly object when ``polys=True``, otherwise
    (default) returns an expression.
z.Cannot generate symmetric polynomial of order z for )
r   Úlenr#   r	   ÚOner   r   r1   r   r   )r(   r"   Úgensr4   Úss   &$*  r-   Úsymmetric_polyr;   \   s…   € ô ˜Ó€Dàˆ1„u�”C˜“I”§TÝÓVWÒY]Ð^Ó_Ð_ßÜ�u‰u‰ä¤g¨d´C¸³FÔ&;Ó<Ñ&; ”S˜!”WÑ&;Ñ<Ñ=ˆç %Œ4�Ð�tÓÐ/¨4Ð/ùò =s   Á3B!c                ód   € \        \        WW44      WR7      pV'       d   V# VP                  4       # )a  Generates a polynomial of degree ``n`` with coefficients in
``[inf, sup]``.

Parameters
----------
x
    `x` is the independent term of polynomial
n : int
    `n` decides the order of polynomial
inf
    Lower limit of range in which coefficients lie
sup
    Upper limit of range in which coefficients lie
domain : optional
     Decides what ring the coefficients are supposed
     to belong. Default is set to Integers.
polys : bool, optional
    ``polys=True`` returns an expression, otherwise
    (default) returns an expression.
)Údomain)r   r   r3   )r   r(   ÚinfÚsupr=   r"   r4   s   &&&&&& r-   Úrandom_polyr@   t   s*   € ô, ”
˜1 3Ó/°ÔB€Dçˆ4Ð,˜dŸl™l›nÐ,r/   c           	     ón  € \        VRR4      p\        V\        4      '       d   \        V: RV : 24      pM*V'       d#   V\	        V!  P
                  ,          '       d   Rp\        V\        4      '       d   \        V: RV : 24      pM*V'       d#   V\	        V!  P
                  ,          '       d   RpV'       g   \        \        R4      4      h. p\        \        V 4       Uu. uF  qaW&,          ,
          NK  	  up!  p\        V 4       Fl  pWqW&,          ,
          ,          p\        \        V 4       U	u. uF   q–V	8w  g   K  W&,          W),          ,
          NK"  	  up	!  p
VP                  WŠ,          4       Kn  	  \        \        WS4       UUu. uF  w  r¼W¼,          NK  	  upp!  # u upi u up	i u uppi )z£Construct Lagrange interpolating polynomial for ``n``
data points. If a sequence of values are given for ``X`` and ``Y``
then the first ``n`` values will be used.
Úfree_symbolsNÚ:Fz~
            Expecting symbol for x that does not appear in X or Y.
            Use `interpolate(list(zip(X, Y)), x)` instead.)ÚgetattrÚ
isinstanceÚstrr   r   rB   r#   r   r   r&   r'   r   Úzip)r(   r   ÚXÚYÚokÚcoeffsr+   ÚnumertÚnumerÚjÚdenomÚcoeffÚys   &&&&         r-   Úinterpolating_polyrR   �   sW  € ô 
��N DÓ	)€Bä�!”S×ÒÜ›q¢!Ð$Ó%‰ß	�”U˜A‘Y×+Ñ+×+Ô+Øˆä�!”S×ÒÜ›q¢!Ð$Ó%‰ß	�”U˜A‘Y×+Ñ+×+Ô+ØˆçÜœð %>ó ?ó @ð 	@ð €FÜ¤U¨1¤XÓ.¡X �q•t—8�8¡XÑ.Ñ/€Fä�1ŽXˆØ˜A�D�Õ!ˆÜ¬U°1¬XÓ@©X¨¸a¹”{�q•t˜a�d—{�{©XÑ@ÑAˆØ�‰�e•kÖ"ñ ô
 ¬¨V¬Ô8©™X˜U�—�©Ò8Ñ9Ð9ùò /ùò Aùó 9s   Ã$F'Ä5F,
ÅF,
ÆF1
c           	     ó>  € \        V ^,           4       Uu. uF  p\        R\        V4      ,           4      NK   	  ppV^ ,          V^,          rCV\        VR,          !  ,           pV^,          \        VR,           Uu. uF  qf^,          NK  	  up!  ,           pV^,           V^,           ,          P                  ! V!  pV^,           RV,          V^,          ,          V^,          ,           ^,
          ,          P                  ! V!  p	\        ^.VO5!  p
W‰V
3# u upi u upi )ú%Fateman's GCD benchmark: trivial GCD Úy_ºé   NNéýÿÿÿ)r&   r   rF   r   Úas_polyr   )r(   r+   rI   Úy_0Úy_1ÚurQ   ÚvÚFÚGÚHs   &          r-   Úfateman_poly_F_1ra   ±   så   € ä(-¨a°!­e¬Ó5© 1Œ�”s˜1“v•Ö	©€AÐ5à��t�Q�q•TˆàŒc�1�R•5‰kÕ€AØˆQ�” Q r¦UÓ+¡U ˜!—t�t¡UÑ+Ñ,Õ,€Aà
ˆa�%�!�a•%�×!Ò! 1Ñ%€AØ
ˆa�%�"�S•&˜˜a�•- # q¥&Õ(¨1Õ,Õ	-×6Ò6¸Ñ:€AäˆQˆ�‹€Aà�ˆ7€Nùò 	6ùò
 ,s   •$DÁ;D
c                ód  € V! ^4      V! ^ 4      .p\        V 4       F  p\        W14      V.pK  	  V! ^4      V! ^ 4      V! ^ 4      .p\        ^ V 4       F  p\        W14      \        V4      V.pK  	  V ^,
          p\        V\	        V! ^4      V4      ^ W4      p\        V\	        V! ^4      V4      ^ W4      pV! ^4      ) V! ^ 4      .. V! ^4      V! ^ 4      V! ^4      ) ..p\        V\	        V! ^4      V4      ^ W4      p	\        W…^V4      p
\        WgW4      p\        WšW4      p\        W4      pW¼V3# )rT   )r&   r   r   r   r   r   r   )r(   ÚKr\   r+   r]   ÚmÚUÚVÚfÚWrI   r^   r_   r`   s   &&            r-   Údmp_fateman_poly_F_1ri   Â   s0  € á	
ˆ1‹‰q�‹tˆ€Aä�1ŽXˆÜ�Q‹]˜AÐŠñ ñ 
ˆ1‹‰q�‹t‘Q�q“TÐ€Aä�1�aŽ[ˆÜ�Q‹]œH Q›K¨Ð+Šñ ð 	
ˆA�€Aä�Qœ
¡1 Q£4¨Ó+¨Q°Ó5€AÜ�Qœ
¡1 Q£4¨Ó+¨Q°Ó5€AáˆQ‹4ˆ%‘�1“ˆ˜™Q˜q›T¡1 Q£4©!¨A«$¨Ð/Ð0€Aä�Qœ
¡1 Q£4¨Ó+¨Q°Ó5€AÜ�!˜˜1Ó€Aä��aÓ€AÜ��aÓ€Aä�‹€Aà�ˆ7€Nr/   c           	     óš  € \        V ^,           4       Uu. uF  p\        R\        V4      ,           4      NK   	  ppV^ ,          p\        VR,          !  p\	        W4,           ^,           ^,          .VO5!  p\	        W4,
          ^,
          ^,          .VO5!  p\	        W4,           ^,           ^,          .VO5!  pWV,          WW,          V3# u upi )ú7Fateman's GCD benchmark: linearly dense quartic inputs rU   rV   ©r&   r   rF   r   r   )r(   r+   rI   rZ   r\   r`   r^   r_   s   &       r-   Úfateman_poly_F_2rm   à   s¤   € ä(-¨a°!­e¬Ó5© 1Œ�”s˜1“v•Ö	©€AÐ5à
ˆA�$€CäˆQˆr�U‰€Aäˆc�g˜�k˜AÕÐ" Ó"€Aäˆc�g˜�k˜AÕÐ" Ó"€AÜˆc�g˜�k˜AÕÐ" Ó"€Aà�3�•�Qˆ;Ðùò 	6s   •$Cc                óØ  € V! ^4      V! ^ 4      .p\        V ^,
          4       F  p\        W14      V.pK  	  V ^,
          p\        V\        V! ^4      V^,
          4      ^ W4      p\	        \        WA4      \        WTV4      .W4      p\	        \        WA4      V.W4      p\        V\        V^,
          V4      ^ W4      p\	        \        WA4      V.W4      p\        WhW4      \        WxW4      V3# )rk   )r&   r   r   r   r   r   r   )	r(   rc   r\   r+   rd   r]   rg   ÚgÚhs	   &&       r-   Údmp_fateman_poly_F_2rq   ð   sÙ   € á	
ˆ1‹‰q�‹tˆ€Aä�1�q•5Ž\ˆÜ�Q‹]˜AÐŠñ ð 	
ˆA�€Aä�Qœ
¡1 Q£4¨¨Q­Ó/°°AÓ9€Aä”˜“¤¨¨aÓ 0Ð1°1Ó8€AÜ”˜“ Ð" AÓ)€Aä�Qœ  A¥ qÓ)¨1¨aÓ3€Aä”˜“ Ð" AÓ)€Aä�1˜Ó¤¨¨aÓ 3°QÐ6Ð6r/   c           	     ó2  € \        V ^,           4       Uu. uF  p\        R\        V4      ,           4      NK   	  ppV^ ,          p\        VR,           Uu. uF  qDV ^,           ,          NK  	  up!  p\	        W0^,           ,          V,           ^,           ^,          .VO5!  p\	        W0^,           ,          V,
          ^,
          ^,          .VO5!  p\	        W0^,           ,          V,           ^,           ^,          .VO5!  pWg,          Wh,          V3# u upi u upi )ú8Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) rU   rV   rl   )	r(   r+   rI   rZ   rQ   r\   r`   r^   r_   s	   &        r-   Úfateman_poly_F_3rt     sß   € ä(-¨a°!­e¬Ó5© 1Œ�”s˜1“v•Ö	©€AÐ5à
ˆA�$€Cä ! B¦%Ó(¡%˜Q�!�a•%�jˆj¡%Ñ(Ñ)€Aäˆc˜•E�l˜QÕ Õ" QÕ&Ð+¨Ó+€Aäˆc˜•E�l˜QÕ Õ" QÕ&Ð+¨Ó+€AÜˆc˜•E�l˜QÕ Õ" QÕ&Ð+¨Ó+€Aà�3�•�Qˆ;Ðùò 	6ùò )s   •$DÁDc           	     óÔ  € \        V ^,           VP                  /V4      p\        ^ V ^,
          4       F*  p\        V.\	        W14      V ^,           V^,           V4      pK,  	  \        V\        V! ^4      V ^,
          4      ^ W4      p\        \        \        W@^,
          V4      .\	        V ^,
          V4      V ^,           W4      W4      p\        \        V.\	        V ^,
          V4      V ^,           W4      W4      p\        V\	        V ^,
          V4      ^ V ^,
          V4      p\        \        V.\	        V ^,
          V4      V ^,           W4      W4      p\        WWW4      \        WgW4      V3# )rs   )	r   Úoner&   r   r   r   r   r   r   )r(   rc   r\   r+   r]   rg   ro   rp   s   &&      r-   Údmp_fateman_poly_F_3rw     s3  € ä˜1˜q�5 !§%¡%˜.¨!Ó,€Aä�1�a˜!•eŽ_ˆÜ˜!˜œg a›m¨Q°­U°A¸µE¸1Ó=Šñ ô 	�Qœ
¡1 Q£4¨¨Q­Ó/°°AÓ9€AäÜ”g˜a Q¥¨Ó*Ð+¬W°Q¸µU¸AÓ->ÀÀAÅÀqÓLÈaó	T€Aä”˜a˜S¤'¨!¨a­%°Ó"3°Q¸µU¸AÓAÀ1ÓH€Aä�Qœ  A¥ qÓ)¨1¨a°!­e°QÓ7€Aä”˜a˜S¤'¨!¨a­%°Ó"3°Q¸µU¸AÓAÀ1ÓH€Aä�1˜Ó¤¨¨aÓ 3°QÐ6Ð6r/   )Úringc                  ó>  € \        R \        4      w  rr#V^,          V,          V^,          ,          ^V^,          ,          V,          V,          ,           ^V^,          ,          V,          ,           ^V^,          ,          ,           ^V,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          V,          ,           ^V^,          ,          ,           W#^,          ,          ,           ^V,          V,          ,           V,           ^,           # ©úx,y,z©rx   r   ©ÚRr   rQ   Úzs       r-   Ú_f_0r€   ,  sÍ   € Ü�gœrÓ"�J€Aˆ!Øˆa�4��6�!�Q•$�;˜˜1˜a�4� � !�Õ# a¨¨1­¥f¨Q¥hÕ.°°1°aµ4µÕ7¸!¸A½#Õ=ÀÀ!ÀQÅ$ÅÀqÈ!ÅtÅÕKÈaÐPQÐSTÕPTÍfÐUVÍhÕVÐYZÐ[\Ð^_Õ[_ÕY_Õ_ÐbcÐghÕdhÕbhÕhÐklÐmnÕknÐopÕkpÕpÐstÕtÐwxÕxÐxr/   c                  óè  € \        R \        4      w  rr#V^,          V,          V,          V^,          V^,          ,          V^,          ,          ,           V^,          V^,          ,          ,           ^V^,          ,          V,          V,          ,           ^V^,          ,          V,          ,           V^,          V^,          ,          ,           ^
V^,          ,          V,          ,           W^,          ,          V,          ,           ^V,          V^,          ,          V,          ,           ^V,          V^,          ,          ,           W,          V^,          ,          ,           ^
V,          V,          V^,          ,          ,           W,          V,          ,           RV,          V,          ,           ^V,          V^,          ,          ,           ^æV,          V,          ,           RV,          ,           V^,          V^,          ,          ,           ^
V^,          ,          V,          ,           ^V,          V^,          ,          ,           RV,          V,          ,           ^ÈV,          ,           RV,          ,           R,           # )r{   ib  i,  i@  iX  ip  r|   r}   s       r-   Ú_f_1r‚   0  sZ  € Ü�gœrÓ"�J€Aˆ!Øˆa�4��6�!�8�a˜•d˜1˜a�4•i  1¥•nÕ$ q¨!¥t¨A¨q­D¥yÕ0°2°a¸µdµ7¸1µ9¸Qµ;Õ>ÀÀAÀqÅDÅÈÅÕJÈQÐPQÍTÐRSÐUVÕRVÍYÕVÐY[Ð\]Ð_`Õ\`ÕY`ÐabÕYbÕbÐefÐjkÕgkÕekÐlmÕemÕmÐprÐstÕptÐuvÐxyÕuyÕpyÐz{Õp{Õ{ð  Að  BCõ  Cð  DEð  GHõ  DHõ  Hõ  Hð  KLõ  KNð  OPð  RSõ  OSõ  KSõ  Sð  VXð  YZõ  VZð  [\õ  V\ð  ]^ð  `aõ  ]aõ  Vaõ  að  deõ  dgð  hiõ  diõ  ið  loð  pqõ  lqð  rsõ  lsõ  sð  vxð  yzõ  vzð  {|ð  ~õ  {õ  võ  ð  BEð  FGõ  BGð  HIõ  BIõ  Ið  LOð  PQõ  LQõ  Qð  TUð  WXõ  TXð  YZð  \]õ  Y]õ  T]õ  ]ð  `bð  cdð  fgõ  cgõ  `gð  hiõ  `iõ  ið  lnð  opõ  lpð  qrð  tuõ  quõ  luõ  uð  x{ð  |}õ  x}ð  ~õ  xõ  ð  BEð  FGõ  BGõ  Gð  JMð  NOõ  JOõ  Oð  RVõ  Vð  Vr/   c                  ód  € \        R \        4      w  rr#V^,          V^,          ,          V^,          V^,          ,          V,          ,           V^,          V,          V^,          ,          ,           V^,          V^,          ,          ,           V^,          V^,          ,          ,           V^,          V,          V,          ,           ^ZV^,          ,          V,          ,           ^ZV^,          ,          V,          ,           V^,          V^,          ,          V,          ,           ^V^,          ,          V^,          ,          ,
          V^,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,
          W#,          ,           ^V,          ,
          ^ZV,          ,           R,
          # )r{   iÞ  r|   r}   s       r-   Ú_f_2r„   4  sè  € Ü�gœrÓ"�J€Aˆ!Øˆa�4��1•�9�q˜!•t˜A˜q�D•y •{Õ" Q¨¥T¨!¥V¨A¨q­D¥[Õ0°1°aµ4¸¸1½µ9Õ<¸qÀ!½tÀAÀqÅD½yÕHÈ1ÈaÍ4ÐPQÍ6ÐRSÍ8ÕSÐVXÐYZÐ\]ÕY]ÕV]Ð^_ÕV_Õ_ÐbdÐefÐhiÕeiÕbiÐjkÕbkÕkÐnoÐqrÕnrÐstÐvwÕswÕnwÐxyÕnyÕyÐ|~ð  @Að  CDõ  @Dõ  }Dð  EFð  HIõ  EIõ  }Iõ  Ið  LMð  OPõ  LPð  QRð  TUõ  QUõ  LUõ  Uð  XZð  [\ð  ^_õ  [_õ  X_ð  `að  cdõ  `dõ  Xdõ  dð  ghõ  gjõ  jð  moð  pqõ  mqõ  qð  tvð  wxõ  txõ  xð  {~õ  ~ð  ~r/   c                  ó„  € \        R \        4      w  rr#V^,          V^,          ,          V^,          V^,          ,          ,           V^,          ,           V^,          V^,          ,          V,          ,           V^,          V,          ,           V^,          V^,          ,          ,           V^,          V^,          ,          V^,          ,          ,           V^,          V,          V^,          ,          ,           V^,          V,          V,          ,           W^,          ,          V^,          ,          ,           W^,          ,          ,           W,          V^,          ,          ,           W,          V^,          ,          ,           W,          V^,          ,          ,           V^,          V,          ,           W#^,          ,          ,           # rz   r|   r}   s       r-   Ú_f_3r†   8  sñ  € Ü�gœrÓ"�J€Aˆ!Øˆa�4��1•�9�q˜!•t˜A˜q�D•yÕ  1 a¥4Õ'¨!¨Q­$¨q°!­t­)°A­+Õ5¸¸1½¸Q½Õ>ÀÀAÅÀaÈÅdÅÕJÈQÐPQÍTÐRSÐUVÕRVÍYÐWXÐZ[ÕW[Í^Õ[Ð^_ÐabÕ^bÐcdÕ^dÐefÐhiÕeiÕ^iÕiÐlmÐopÕlpÐqrÕlrÐstÕltÕtÐwxÐ|}Õy}Õw}Ð~ð  BCõ  Cõ  xCõ  Cð  FGð  KLõ  HLõ  FLõ  Lð  OPõ  ORð  STð  VWõ  SWõ  OWõ  Wð  Z[õ  Z]ð  ^_ð  abõ  ^bõ  Zbõ  bð  efõ  ehð  ijð  lmõ  imõ  emõ  mð  pqð  stõ  ptð  uvõ  pvõ  vð  yzð  ~õ  {õ  yõ  ð  r/   c                  ó~  € \        R \        4      w  rr#V^	,          ) V^,          ,          V,          V^,          V^,          ,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,
          V^,          V^	,          ,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,           V^,          V,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,           V^,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,           V^,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,           V^,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V,          V^,          ,          ,
          V^,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           W^,          ,          V^,          ,          ,           ^V,          V^,          ,          V^,          ,          ,           ^V,          V^,          ,          V^,          ,          ,           ^V,          V^,          ,          V^,          ,          ,           V^,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          ,           ^V^,          ,          ,           # rz   r|   r}   s       r-   Ú_f_4rˆ   <  sã	  € Ü�gœrÓ"�J€Aˆ!Øˆq�Dˆ5��A•�:�a�<˜!˜Q�$˜q !�t�) A q¥D�.Õ(¨1¨a­4°°2µ­:°a¸µd­?Õ:¸Q¸qÀ!½t½VÀAÀqÅD½[ÕHÈ1ÈaÍ4ÐPQÐSTÕPTÍ9ÐUVÐXYÕUYÍ>ÕYÐ\]Ð_`Õ\`ÐabÐdeÕaeÕ\eÐfgÐijÕfjÕ\jÕjÐmnÐopÐrsÕosÕmsÐtuÐwxÕtxÕmxÐyzÕmzÕzÐ}~ð  @Að  CDõ  @Dõ  ~Dð  EFð  HIõ  EIõ  ~Ið  JKð  MNõ  JNõ  ~Nõ  Nð  QRð  TUõ  QUð  VWð  YZõ  VZõ  QZð  [\ð  ^_õ  [_õ  Q_õ  _ð  bcð  efõ  bfð  ghð  jkõ  gkõ  bkð  lmð  opõ  lpõ  bpõ  pð  stð  uvð  xyõ  uyõ  syð  z{ð  }~õ  z~õ  s~ð  @ð  BCõ  Cõ  sCõ  Cð  FGð  IJõ  FJð  KLõ  FLð  MNð  PQõ  MQõ  FQõ  Qð  TUð  WXõ  TXð  YZð  \^õ  Y^õ  T^ð  _`ð  bcõ  _cõ  Tcõ  cð  fgð  hið  klõ  hlõ  flð  mnð  prõ  mrõ  frð  stð  vwõ  swõ  fwõ  wð  z{ð  }~õ  z~ð  @ð  BCõ  Cõ  zCð  DEð  GHõ  DHõ  zHõ  Hð  KLð  MNð  PQõ  MQõ  KQð  RSð  UVõ  RVõ  KVð  WXð  Z[õ  W[õ  K[õ  [ð  ^`ð  abð  deõ  aeõ  ^eð  fgð  ijõ  fjõ  ^jõ  jð  mnð  opð  rsõ  osõ  msð  tuð  wxõ  txõ  mxð  yzð  |}õ  y}õ  m}õ  }ð  @Að  CDõ  @Dð  EFð  HIõ  EIõ  @Ið  JKð  MNõ  JNõ  @Nõ  Nð  QRð  STð  VWõ  SWõ  QWð  XYð  [\õ  X\õ  Q\ð  ]^ð  `aõ  ]aõ  Qaõ  að  deð  ghõ  dhð  ijð  lmõ  imõ  dmð  noð  qrõ  nrõ  drõ  rð  uvð  wxð  z{õ  w{õ  u{ð  |}ð  @õ  |@õ  u@ð  ABð  DEõ  AEõ  uEõ  Eð  HJð  KLð  NOõ  KOõ  HOð  PQð  STõ  PTõ  HTð  UVð  XYõ  UYõ  HYõ  Yð  \]ð  _`õ  \`ð  abð  deõ  aeõ  \eð  fgð  ijõ  fjõ  \jõ  jð  mnð  opð  rsõ  osõ  msð  tuð  wxõ  txõ  mxð  yzð  |}õ  y}õ  m}õ  }ð  @Að  BCð  EFõ  BFõ  @Fð  GHõ  @Hð  IJð  LMõ  IMõ  @Mõ  Mð  PQð  STõ  PTð  UVð  XYõ  UYõ  PYõ  Yð  \]ð  ^_ð  abõ  ^bõ  \bð  cdð  fgõ  cgõ  \gõ  gð  jkð  opõ  lpõ  jpð  qrð  tuõ  quõ  juõ  uð  xyð  z{õ  x{ð  |}ð  @õ  |@õ  x@ð  ABð  DEõ  AEõ  xEõ  Eð  HIð  JKõ  HKð  LMð  OPõ  LPõ  HPð  QRð  TUõ  QUõ  HUõ  Uð  XZð  [\õ  X\ð  ]^ð  `aõ  ]aõ  Xað  bcð  efõ  bfõ  Xfõ  fð  ijð  lmõ  imð  noð  qrõ  nrõ  irõ  rð  uvð  wxð  z{õ  w{õ  u{ð  |}ð  @	õ  |@	õ  u@	õ  @	ð  C	D	ð  E	F	ð  H	I	õ  E	I	õ  C	I	õ  I	ð  L	N	ð  O	P	ð  R	S	õ  O	S	õ  L	S	õ  S	ð  S	r/   c                  óü  € \        R \        4      w  rr#V^,          ) ^V^,          ,          V,          ,
          ^V^,          ,          V,          ,           ^V,          V^,          ,          ,
          ^V,          V,          V,          ,           ^V,          V^,          ,          ,
          V^,          ,
          ^V^,          ,          V,          ,           ^V,          V^,          ,          ,
          V^,          ,           # rz   r|   r}   s       r-   Ú_f_5rŠ   @  s¸   € Ü�gœrÓ"�J€Aˆ!Øˆq�Dˆ5�1�Q˜•T•6˜!•8Õ˜a  1¥�f Q�hÕ&¨¨1­¨Q°­T­Õ1°A°aµC¸µE¸!µGÕ;¸aÀ½cÀ!ÀQÅ$½hÕFÈÈAÍÕMÐPQÐRSÐUVÕRVÕPVÐWXÕPXÕXÐ[\Ð]^Õ[^Ð_`ÐbcÕ_cÕ[cÕcÐfgÐijÕfjÕjÐjr/   c                  ó:  € \        R \        4      w  rr#pRV^,          ,          V,          ^-V^,          ,          V^,          ,          V^,          ,          ,           ^-V^,          ,          V^,          ,          ,
          RV,          V^,          ,          ,
          ^/V,          V^,          ,          ,
          ^�V,          V,          V^,          ,          ,           ^^V,          V,          V,          V,          ,           ^	V^,          ,          V^,          ,          V^,          ,          ,
          ^	V^,          ,          V^,          ,          ,           V^,          V^,          ,          V^,          ,          ,
          V^,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,
          ^V,          V^,          ,          ,
          # )zx,y,z,tiC  i§  r|   )r~   r   rQ   r   Úts        r-   Ú_f_6r�   D  sÉ  € Ü˜¤BÓ'�M€Aˆ!�Ø��1•�9�Q�;˜˜A˜q�D�  A¥� a¨¥dÕ*Õ*¨R°°1µ­W°Q¸µT­\Õ9¸CÀ½EÀ!ÀQÅ$½JÕFÈÈAÍÈaÐQRÍdÍÕRÐUXÐYZÕUZÐ[\ÕU\Ð]^Ð`aÕ]aÕUaÕaÐdfÐghÕdhÐijÕdjÐklÕdlÐmnÕdnÕnÐqrÐstÐvwÕswÕqwÐxyÐ{|Õx|Õq|Ð}~ð  ABõ  ~Bõ  rBõ  Bð  EFð  GHð  JKõ  GKõ  EKð  LMð  OPõ  LPõ  EPõ  Pð  STð  VWõ  SWð  XYð  [\õ  X\õ  S\ð  ]^ð  `aõ  ]aõ  Saõ  að  deð  ghõ  dhð  ijð  lmõ  imõ  dmõ  mð  pqð  rsð  uvõ  rvõ  pvð  wxð  z{õ  w{õ  p{õ  {ð  ~ð  @Að  CDõ  @Dõ  ~Dð  EFð  HIõ  EIõ  ~Iõ  Ið  LMð  NOð  QRõ  NRõ  LRð  STð  VWõ  SWõ  LWõ  Wð  Z[ð  \]õ  Z]ð  ^_ð  abõ  ^bõ  Zbõ  bð  br/   c                  óø  € \        R \        4      w  rr#^V^,          ,          V^,          ,          V^,          ,          ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,
          V^,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V,          ,           V^,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V,          ,           V^,          V^,          ,          V^,          ,          ,           V^,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V^,          ,          ,
          ^	V^,          ,          V^,          ,          V,          ,           ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          V,          ,           ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,
          ^V^,          ,          V,          V^,          ,          ,           ^V,          V^,          ,          V,          ,           ^V,          V^,          ,          V^,          ,          ,
          ^V,          V,          V,          ,
          ^V,          V^,          ,          ,           ^V^,          ,          ,
          ^V,          V^,          ,          ,           # rz   r|   r}   s       r-   Ú_w_1r�   H  sL  € Ü�gœrÓ"�J€Aˆ!ØˆQ��T�6�!�Q•$�;�q˜!•tÕ˜a  1¥�f Q¨¥T�k¨!¨Q­$Õ.Õ.°°1°aµ4µ¸¸1½µ¸QÀ½TÕ1AÕAÀAÀaÈÅdÅFÈ1ÅHÈQÐPQÍTÅMÕQÐTUÐWXÕTXÐYZÐ\]ÕY]ÕT]Ð^_ÐabÕ^bÕTbÕbÐegÐhiÐklÕhlÕelÐmnÐpqÕmqÕeqÐrsÕesÕsÐvwÐyzÕvzÐ{|Ð~Õ{Õvð  ABð  DEõ  AEõ  wEõ  Eð  HJð  KLð  NOõ  KOõ  HOð  PQð  STõ  PTõ  HTð  UVð  XYõ  UYõ  HYõ  Yð  \^ð  _`ð  bcõ  _cõ  \cð  deõ  \eð  fgð  ijõ  fjõ  \jõ  jð  moð  pqð  stõ  ptõ  mtð  uvð  xyõ  uyõ  myõ  yð  |}ð  ~ð  ABõ  ~Bõ  |Bð  CDð  FGõ  CGõ  |Gõ  Gð  JKð  LMð  OPõ  LPõ  JPð  QRð  TUõ  QUõ  JUð  VWð  YZõ  VZõ  JZõ  Zð  ]^ð  _`ð  bcõ  _cõ  ]cð  deð  ghõ  dhõ  ]hð  ijõ  ]jõ  jð  mnð  opð  rsõ  osõ  msð  tuð  wxõ  txõ  mxð  yzð  |}õ  y}õ  m}õ  }ð  @Að  BCð  EFõ  BFõ  @Fð  GHð  JKõ  GKõ  @Kð  LMð  OPõ  LPõ  @Põ  Pð  STð  UVð  XYõ  UYõ  SYð  Z[ð  ]^õ  Z^õ  S^ð  _`ð  bcõ  _cõ  Scõ  cð  fgð  hið  klõ  hlõ  flð  mnõ  fnð  opð  rsõ  osõ  fsõ  sð  vwð  xyð  {|õ  x|õ  v|ð  }~õ  v~ð  @ð  BCõ  Cõ  vCõ  Cð  FGð  HIð  KLõ  HLõ  FLð  MNõ  FNð  OPð  RSõ  OSõ  FSõ  Sð  VWð  XYð  [\õ  X\õ  V\ð  ]^ð  `aõ  ]aõ  Vað  bcõ  Vcõ  cð  fgð  ijõ  fjð  klð  noõ  koõ  foð  pqð  stõ  ptõ  ftõ  tð  wxð  z{õ  w{ð  |}ð  @õ  |@õ  w@ð  ABð  DEõ  AEõ  wEõ  Eð  HIð  JKð  MNõ  JNõ  HNð  OPð  RSõ  OSõ  HSð  TUð  WXõ  TXõ  HXõ  Xð  [\ð  ]^ð  `aõ  ]aõ  [að  bcð  efõ  bfõ  [fð  ghõ  [hõ  hð  kmð  noð  qrõ  nrõ  krð  stõ  ktð  uvð  xyõ  uyõ  kyõ  yð  |~ð  @ð  BCõ  Cõ  |Cð  DEõ  |Eð  FGð  IJõ  FJõ  |Jõ  Jð  MOð  PQð  STõ  PTõ  MTð  UVð  XYõ  UYõ  MYõ  Yð  \]ð  ^_ð  abõ  ^bõ  \bð  cdð  fgõ  cgõ  \gõ  gð  jkð  lmð  opõ  lpõ  jpð  qrð  tuõ  quõ  juõ  uð  xyð  z{ð  }~õ  z~õ  x~ð  @ð  BCõ  Cõ  xCð  DEð  GHõ  DHõ  xHõ  Hð  KLð  MNð  PQõ  MQõ  KQð  RSð  UVõ  RVõ  KVð  WXõ  KXõ  Xð  [\ð  ]^ð  `aõ  ]aõ  [að  bcõ  [cð  deð  ghõ  dhõ  [hõ  hð  klð  mnð  pqõ  mqõ  kqð  rsõ  ksð  tuð  wxõ  txõ  kxõ  xð  {|ð  }~ð  @	A	õ  }A	õ  {A	ð  B	C	õ  {C	ð  D	E	ð  G	H	õ  D	H	õ  {H	õ  H	ð  K	L	ð  M	N	õ  K	N	ð  O	P	ð  R	S	õ  O	S	õ  K	S	ð  T	U	õ  K	U	õ  U	ð  X	Y	ð  Z	[	õ  X	[	ð  \	]	ð  _	`	õ  \	`	õ  X	`	ð  a	b	ð  d	e	õ  a	e	õ  X	e	õ  e	ð  h	i	ð  j	k	õ  h	k	ð  l	m	õ  h	m	ð  n	o	õ  h	o	õ  o	ð  r	s	ð  t	u	õ  r	u	ð  v	w	ð  y	z	õ  v	z	õ  r	z	õ  z	ð  }	~	ð  	@
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r/   c                  ó
  € \        R \        4      w  rp^V^,          ,          V^,          ,          ^0V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^HV^,          ,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           ^V^,          ,          V^,          ,          ,           ^V^,          ,          V,          ,           ^V^,          ,          ,           V^,          V^,          ,          ,           V^,          V^,          ,          ,           ^V^,          ,          ,
          V^,          V^,          ,          ,           V^,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,
          RV^,          ,          V^,          ,          ,           V^,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           V^,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           ^0V^,          ,          ,           ^V^,          ,          ,
          # )zx,yi$  r|   )r~   r   rQ   s      r-   Ú_w_2r‘   L  s�  € Ü�5œ"‹o�G€Aˆ!Øˆa��d�7�1�a•4�<˜"˜Q �T�' ! Q¥$�,Õ&¨¨A¨q­D­°°Aµ­Õ5¸¸1¸a½4½ÀÀ1Å½ÕDÀrÈ!ÈQÍ$ÅwÈqÐRSÍtÅ|ÕSÐVWÐXYÐ[\ÕX\ÕV\Ð]^Ð`aÕ]aÕVaÕaÐdeÐfgÐijÕfjÕdjÐklÕdlÕlÐopÐqrÐtuÕquÕouÕuÐxyÐ{|Õx|Ð}~ð  ABõ  ~Bõ  yBõ  Bð  EFð  HIõ  EIð  JKð  MNõ  JNõ  ENõ  Nð  QSð  TUð  WXõ  TXõ  QXõ  Xð  [\ð  ^_õ  [_ð  `að  cdõ  `dõ  [dõ  dð  ghð  jkõ  gkð  lmð  opõ  lpõ  gpõ  pð  stð  uvð  xyõ  uyõ  syð  z{ð  }~õ  z~õ  s~õ  ~ð  ADð  EFð  HIõ  EIõ  AIð  JKð  MNõ  JNõ  ANõ  Nð  QRð  TUõ  QUð  VWð  YZõ  VZõ  QZõ  Zð  ]^ð  _`ð  bcõ  _cõ  ]cð  deð  ghõ  dhõ  ]hõ  hð  klð  noõ  koð  pqð  stõ  ptõ  ktõ  tð  wyð  z{ð  }~õ  z~õ  w~ð  @ð  BCõ  Cõ  wCõ  Cð  FHð  IJð  LMõ  IMõ  FMõ  Mð  PRð  STð  VWõ  SWõ  PWõ  Wð  Wr/   c                  ó„   € \        4       \        4       \        4       \        4       \	        4       \        4       \        4       3# ©N)r€   r‚   r„   r†   rˆ   rŠ   r�   © r/   r-   Úf_polysr•   P  s%   € Ü‹6”4“6œ4›6¤4£6¬4«6´4³6¼4»6ÐAÐAr/   c                  ó*   € \        4       \        4       3# r“   )r�   r‘   r”   r/   r-   Úw_polysr—   S  s   € Ü‹6”4“6ˆ>Ðr/   )NF)r   rQ   )AÚ__doc__Ú
sympy.corer   r   r   r   r   r   Úsympy.core.containersr   Úsympy.core.singletonr	   Úsympy.ntheoryr
   Úsympy.polys.densearithr   r   r   r   Úsympy.polys.densebasicr   r   r   r   r   r   Úsympy.polys.domainsr   Úsympy.polys.factortoolsr   Úsympy.polys.polyclassesr   Úsympy.polys.polytoolsr   r   Úsympy.polys.polyutilsr   Úsympy.utilitiesr   r   r   r.   r5   r;   r@   rR   ra   ri   rm   rq   rt   rw   Úsympy.polys.ringsrx   r€   r‚   r„   r†   rˆ   rŠ   r�   r�   r‘   r•   r—   r”   r/   r-   Ú<module>r¦      s  ðÙ O÷ A× @Ý 'Ý "Ý #÷ó ÷÷ õ #Ý :Ý 'ß 0Ý /ß 7Ñ 7ð ó&,ó ð&,ðR ó-ó ð-ð4 ð0 5ó 0ó ð0ð. Ø')°ó -ó ð-ð4 ó:ó ð:òBò"ò<ò 7ò*ò 7õ* #òyòVò~òòS	òkòbòN
òWòBôr/   