
    wg                     `   d Z g dZddlmZmZmZmZmZmZm	Z	m
Z
mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z= ddl>m?Z?m@Z@mAZAmBZBmCZC ddlDmEZE ddlFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\m]Z]m^Z^m_Z_ ddl`maZambZbmcZcmdZdmeZemfZfmgZgmhZhmiZimjZj ddlkmlZlmmZm dd	lnmoZompZpmqZqmrZrmsZsmtZt dd
lumvZvmwZwmxZxmyZymzZz ddl{m|Z| ddl}m~Z~mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZ ddlmZ ddlmZmZmZmZmZ ddlmZmZmZmZmZmZmZ ddlmZmZmZmZmZ ddlmZmZmZ ddlmZ ddlmZmZmZmZ ddlmZmZmZmZ y)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        Y/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/polys/__init__.py<module>r      s%   @?B	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	% % $      ' ' ' / B B H H / / / / / / / / / *9 9G G G    @ ?   , , 1 1r   