+
    i                        R t . RNRNRNRNRNRNRNRNR	NR
NRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNR NR!NR"NR#NR$NR%NR&NR'NR(NR)NR*NR+NR,NR-NR.NR/NR0NR1NR2NR3NR4NR5NR6NR7NR8NR9NR:NR;NR<NR=NR>NR?NR@NRANRBNRCNRDNRENRFNRGNRHNRINRJNRKNRLNRMNRNNRONRPNRQNRRNRSNRTNRUNRVNRWNRXNRYNRZNR[NR\NR]NR^NR_NR`NRaNRbNRcNRdNReNRfNRgNRhNRiNRjNRkNRlNRmNRnNRoNRpNRqNRrNRsNRtNRuNRvNRwNRxNRyNRzNR{NR|NR}NR~NRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNt^RIHtHtHtHtHtHtH	t	H
t
HtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtH t H!t!H"t"H#t#H$t$H%t%H&t&H't'H(t(H)t)H*t*H+t+H,t,H-t-H.t.H/t/H0t0H1t1H2t2H3t3H4t4H5t5H6t6H7t7H8t8H9t9H:t:H;t;H<t<H=t= ^RI>H?t?H@t@HAtAHBtBHCtC ^RIDHEtE ^RIFHGtGHHtHHItIHJtJHKtKHLtLHMtMHNtNHOtOHPtPHQtQHRtRHStSHTtTHUtUHVtVHWtWHXtXHYtYHZtZH[t[H\t\H]t]H^t^H_t_ ^RI`HataHbtbHctcHdtdHeteHftfHgtgHhthHitiHjtj ^RIkHltlHmtm ^RInHotoHptpHqtqHrtrHstsHttt ^RIuHvtvHwtwHxtxHytyHztz ^RI{H|t| ^RI}H~t~HtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHt ^RIHt ^RIHtHtHtHtHt ^RIHtHtHtHtHtHtHt ^RIHtHtHtHtHt ^RIHtHtHt ^RIHt ^RIHtHtHtHt ^RIHtHtHtHt R# )z:Polynomial manipulation algorithms and algebraic objects. PolyPurePolypoly_from_exprparallel_poly_from_exprdegreetotal_degreedegree_listLCLMLTpdivprempquopexquodivremquoexquo
half_gcdexgcdexinvertsubresultants	resultantdiscriminant	cofactorsgcd_listgcdlcm_listlcm	terms_gcdtruncmoniccontent	primitivecompose	decomposesturmgff_listgffsqf_normsqf_partsqf_listsqffactor_listfactor	intervalsrefine_rootcount_roots	all_roots
real_rootsnrootsground_rootsnth_power_roots_polycancelreducedgroebneris_zero_dimensionalGroebnerBasispoly
symmetrizehornerinterpolaterational_interpolatevietetogetherBasePolynomialErrorExactQuotientFailedPolynomialDivisionFailedOperationNotSupportedHeuristicGCDFailedHomomorphismFailedIsomorphismFailedExtraneousFactorsEvaluationFailedRefinementFailedCoercionFailedNotInvertibleNotReversibleNotAlgebraicDomainErrorPolynomialErrorUnificationFailedGeneratorsErrorGeneratorsNeededComputationFailedUnivariatePolynomialErrorMultivariatePolynomialErrorPolificationFailedOptionError	FlagErrorminpolyminimal_polynomialprimitive_elementfield_isomorphismto_number_fieldisolate	round_twoprime_decompprime_valuationgalois_groupitermonomialsMonomiallexgrlexgrevlexilexigrlexigrevlexCRootOfrootofRootOfComplexRootOfRootSumrootsDomainFiniteFieldIntegerRingRationalField	RealFieldComplexFieldPythonFiniteFieldGMPYFiniteFieldPythonIntegerRingGMPYIntegerRingPythonRationalGMPYRationalFieldAlgebraicFieldPolynomialRingFractionFieldExpressionDomain	FF_pythonFF_gmpy	ZZ_pythonZZ_gmpy	QQ_pythonQQ_gmpyGFFFZZQQZZ_IQQ_IRRCCEXEXRAWconstruct_domainswinnerton_dyer_polycyclotomic_polysymmetric_polyrandom_polyinterpolating_polyjacobi_polychebyshevt_polychebyshevu_polyhermite_polyhermite_prob_polylegendre_polylaguerre_polybernoulli_polybernoulli_c_polygenocchi_poly
euler_poly
andre_polyapart
apart_listassemble_partfrac_listOptionsringxringvringsringfieldxfieldvfieldsfield);r   r   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   )r=   r>   r?   r@   rA   )rB   )rC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   rO   rP   rQ   rR   rS   rT   rU   rV   rW   rX   rY   rZ   r[   )
r\   r]   r^   r_   r`   ra   rb   rc   rd   re   )rf   rg   )rh   ri   rj   rk   rl   rm   )rn   ro   rp   rq   rr   )rs   ) rt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   )r   )r   r   r   r   r   )r   r   r   r   r   r   r   )r   r   r   r   r   )r   r   r   )r   )r   r   r   r   )r   r   r   r   N)__doc____all__	polytoolsr   r   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   	polyfuncsr=   r>   r?   r@   rA   rationaltoolsrB   
polyerrorsrC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   rO   rP   rQ   rR   rS   rT   rU   rV   rW   rX   rY   rZ   r[   numberfieldsr\   r]   r^   r_   r`   ra   rb   rc   rd   re   	monomialsrf   rg   	orderingsrh   ri   rj   rk   rl   rm   rootoftoolsrn   ro   rp   rq   rr   	polyrootsrs   domainsrt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   constructorr   specialpolysr   r   r   r   r   
orthopolysr   r   r   r   r   r   r   
appellseqsr   r   r   r   r   partfracr   r   r   polyoptionsr   ringsr   r   r   r   fieldsr   r   r   r        R/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/polys/__init__.py<module>r      s   @?
??(?*C?EM??!?#'?)-?/3?5;?=C?EK? ? ? ? "? $+? -9? ;B? DL? 	? !	? #1	? 3>	? @J	?
 
?
 ?
 ?
 *?
 ,3?
 5<?
 >G? ? ? (? *1? 3=? ?D? ? ? '? ).? 0=? ?G? ? ? !.? 0;? =I? ? ? 5? 7?? AJ? ? &? (7? 9?? ? ? *? ,B? DK? ?  !?  1!?  3M!?" #?" 2#?" 4H#?$ %?$ -%?$ /A%?& '?& )'?& +:'?& <K'?( )?( ")?( $5)?( 7J)?* +?* *+?* ,?+?,  -?, "?-?. /?. (/?. *5/?2 3?2 $3?2 &93?2 ;N3?4 5?4 !5?4 #.5?4 0>5?6 7?6 &7?: ;?:  ;?> 
??> ??> ??>  &??> (0??> 2<??B C?B C?B "C?B $3C?B 5>C?F G?J K?J K?J +K?J -<K?J >IK?L M?L (M?L *;M?N O?N +O?N -=O?P Q?P *Q?P ,<Q?P >MQ?R S?R $S?R &/S?R 1<S?R >GS?T U?T U?T !U?T #'U?T )-U?T /3U?T 5;U?T =CU?T EIU?V 	W?V W?V W?Z [?^ _?^ ._?^ 0@_?` a?` (a?d e?d %e?d '8e?d :He?f g?f )g?f +:g?j k?j )k?j +:k?j <Hk?l m?p q?p q?p 4q?t u?x y?x y?x y?x &y?| }?| }?|  }?| "*}?B	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	% % $      ' ' ' / B B H H / / / / / / / / / *9 9G G G    @ ?   , , 1 1r   