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JrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJ r J!r!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r-J.r.J/r/J0r0J1r1J2r2J3r3J4r4J5r5J6r6J7r7J8r8J9r9J:r:J;r;J<r<J=r=  SSK>J?r?J@r@JArAJBrBJCrC  SSKDJErE  SSKFJGrGJHrHJIrIJJrJJKrKJLrLJMrMJNrNJOrOJPrPJQrQJRrRJSrSJTrTJUrUJVrVJWrWJXrXJYrYJZrZJ[r[J\r\J]r]J^r^J_r_  SSK`JaraJbrbJcrcJdrdJereJfrfJgrgJhrhJiriJjrj  SSKkJlrlJmrm  SS	KnJoroJprpJqrqJrrrJsrsJtrt  SS
KuJvrvJwrwJxrxJyryJzrz  SSK{J|r|  SSK}J~r~JrJ€r€J�r�J‚r‚JƒrƒJ„r„J…r…J†r†J‡r‡JˆrˆJ‰r‰JŠrŠJ‹r‹JŒrŒJ�r�JŽrŽJ�r�J�r�J‘r‘J’r’J“r“J”r”J•r•J–r–J—r—J˜r˜J™r™JšršJ›r›JœrœJ�r�  SSKžJŸrŸ  SSK J¡r¡J¢r¢J£r£J¤r¤J¥r¥  SSK¦J§r§J¨r¨J©r©JªrªJ«r«J¬r¬J­r­  SSK®J¯r¯J°r°J±r±J²r²J³r³  SSK´JµrµJ¶r¶J·r·  SSK¸J¹r¹  SSKºJ»r»J¼r¼J½r½J¾r¾  SSK¿JÀrÀJÁrÁJÂrÂJÃrÃ  g)z:Polynomial manipulation algorithms and algebraic objects. )°ÚPolyÚPurePolyÚpoly_from_exprÚparallel_poly_from_exprÚdegreeÚtotal_degreeÚdegree_listÚLCÚLMÚLTÚpdivÚpremÚpquoÚpexquoÚdivÚremÚquoÚexquoÚ
half_gcdexÚgcdexÚinvertÚsubresultantsÚ	resultantÚdiscriminantÚ	cofactorsÚgcd_listÚgcdÚlcm_listÚlcmÚ	terms_gcdÚtruncÚmonicÚcontentÚ	primitiveÚcomposeÚ	decomposeÚsturmÚgff_listÚgffÚsqf_normÚsqf_partÚsqf_listÚsqfÚfactor_listÚfactorÚ	intervalsÚrefine_rootÚcount_rootsÚ	all_rootsÚ
real_rootsÚnrootsÚground_rootsÚnth_power_roots_polyÚcancelÚreducedÚgroebnerÚis_zero_dimensionalÚGroebnerBasisÚpolyÚ
symmetrizeÚhornerÚinterpolateÚrational_interpolateÚvieteÚtogetherÚBasePolynomialErrorÚExactQuotientFailedÚPolynomialDivisionFailedÚOperationNotSupportedÚHeuristicGCDFailedÚHomomorphismFailedÚIsomorphismFailedÚExtraneousFactorsÚEvaluationFailedÚRefinementFailedÚCoercionFailedÚNotInvertibleÚNotReversibleÚNotAlgebraicÚDomainErrorÚPolynomialErrorÚUnificationFailedÚGeneratorsErrorÚGeneratorsNeededÚComputationFailedÚUnivariatePolynomialErrorÚMultivariatePolynomialErrorÚPolificationFailedÚOptionErrorÚ	FlagErrorÚminpolyÚminimal_polynomialÚprimitive_elementÚfield_isomorphismÚto_number_fieldÚisolateÚ	round_twoÚprime_decompÚprime_valuationÚgalois_groupÚitermonomialsÚMonomialÚlexÚgrlexÚgrevlexÚilexÚigrlexÚigrevlexÚCRootOfÚrootofÚRootOfÚComplexRootOfÚRootSumÚrootsÚDomainÚFiniteFieldÚIntegerRingÚRationalFieldÚ	RealFieldÚComplexFieldÚPythonFiniteFieldÚGMPYFiniteFieldÚPythonIntegerRingÚGMPYIntegerRingÚPythonRationalÚGMPYRationalFieldÚAlgebraicFieldÚPolynomialRingÚFractionFieldÚExpressionDomainÚ	FF_pythonÚFF_gmpyÚ	ZZ_pythonÚZZ_gmpyÚ	QQ_pythonÚQQ_gmpyÚGFÚFFÚZZÚQQÚZZ_IÚQQ_IÚRRÚCCÚEXÚEXRAWÚconstruct_domainÚswinnerton_dyer_polyÚcyclotomic_polyÚsymmetric_polyÚrandom_polyÚinterpolating_polyÚjacobi_polyÚchebyshevt_polyÚchebyshevu_polyÚhermite_polyÚhermite_prob_polyÚlegendre_polyÚlaguerre_polyÚbernoulli_polyÚbernoulli_c_polyÚgenocchi_polyÚ
euler_polyÚ
andre_polyÚapartÚ
apart_listÚassemble_partfrac_listÚOptionsÚringÚxringÚvringÚsringÚfieldÚxfieldÚvfieldÚsfieldé   );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$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   )r=   r>   r?   r@   rA   )rB   )rC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   rO   rP   rQ   rR   rS   rT   rU   rV   rW   rX   rY   rZ   r[   )
r\   r]   r^   r_   r`   ra   rb   rc   rd   re   )rf   rg   )rh   ri   rj   rk   rl   rm   )rn   ro   rp   rq   rr   )rs   ) rt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r€   r�   r‚   rƒ   r„   r…   r†   r‡   rˆ   r‰   rŠ   r‹   rŒ   r�   rŽ   r�   r�   r‘   r’   r“   )r”   )r•   r–   r—   r˜   r™   )rš   r›   rœ   r�   rž   rŸ   r    )r¡   r¢   r£   r¤   r¥   )r¦   r§   r¨   )r©   )rª   r«   r¬   r­   )r®   r¯   r°   r±   N)ÄÚ__doc__Ú__all__Ú	polytoolsr   r   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   Ú	polyfuncsr=   r>   r?   r@   rA   ÚrationaltoolsrB   Ú
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[   Únumberfieldsr\   r]   r^   r_   r`   ra   rb   rc   rd   re   Ú	monomialsrf   rg   Ú	orderingsrh   ri   rj   rk   rl   rm   Úrootoftoolsrn   ro   rp   rq   rr   Ú	polyrootsrs   Údomainsrt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r€   r�   r‚   rƒ   r„   r…   r†   r‡   rˆ   r‰   rŠ   r‹   rŒ   r�   rŽ   r�   r�   r‘   r’   r“   Úconstructorr”   Úspecialpolysr•   r–   r—   r˜   r™   Ú
orthopolysrš   r›   rœ   r�   rž   rŸ   r    Ú
appellseqsr¡   r¢   r£   r¤   r¥   Úpartfracr¦   r§   r¨   Úpolyoptionsr©   Úringsrª   r«   r¬   r­   Úfieldsr®   r¯   r°   r±   © ó    ÚQ/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/polys/__init__.pyÚ<module>rÊ      s%  ðÙ @ò?€÷B	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	÷ 	ñ 	÷%õ %õ $÷÷ ÷ ÷ ÷ ÷ õ ÷'÷ '÷ '÷ /ç B× Bç HÕ Hå ÷/÷ /÷ /÷ /÷ /÷ /÷ /÷ /ó /õ *÷9õ 9÷G÷ Gñ G÷ õ  ÷ @Ñ ?å  ç ,Ó ,ç 1Ô 1rÈ   