MZd$dZddlmZddlmZddlmZmZmZddl m Z Gdde Z ddl m Z mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-dd l.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;mZ>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMdd lNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\dd l]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZgdd lhmiZimjZjmkZkmlZlmmZmmnZnmoZodd lpmqZqmrZrmsZsmtZtddlumvZvmwZwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZmZddlmZmZdZGdde eZdZGdde eZdZGdde eZy)z1OO layer for several polynomial representations. )oo) CantSympify)CoercionFailed NotReversible NotInvertible)PicklableWithSlotsc2eZdZdZdZdZdZedZy) GenericPolyz5Base class for low-level polynomial representations. cT|j|jjSzMake the ground domain a ring. ) set_domaindomget_ringfs 9/usr/lib/python3/dist-packages/sympy/polys/polyclasses.pyground_to_ringzGenericPoly.ground_to_ring s||AEENN,--cT|j|jjSz Make the ground domain a field. )r r get_fieldrs rground_to_fieldzGenericPoly.ground_to_field||AEEOO-..rcT|j|jjSzMake the ground domain exact. )r r get_exactrs rground_to_exactzGenericPoly.ground_to_exactrrcd|r|\}}Dcgc]\}}|||f}}}|r|fS|Scc}}wN)perresultincludecoefffactorsgks r_perify_factorszGenericPoly._perify_factorssD #NE7,35DAqSVQK55 '> !N 6s,N) __name__ __module__ __qualname____doc__rrr classmethodr(r rrr r s(?.//  rr )! dmp_validate dup_normal dmp_normal dup_convert dmp_convertdmp_from_sympy dup_strip dup_degree dmp_degree_indmp_degree_listdmp_negative_pdup_LC dmp_ground_LCdup_TC dmp_ground_TCdmp_ground_nthdmp_one dmp_ground dmp_zero_p dmp_one_p dmp_ground_p dup_from_dict dmp_from_dict dmp_to_dict dmp_deflate dmp_inject dmp_eject dmp_terms_gcddmp_list_terms dmp_exclude dmp_slice_in dmp_permute dmp_to_tuple)dmp_add_grounddmp_sub_grounddmp_mul_grounddmp_quo_grounddmp_exquo_grounddmp_absdup_negdmp_negdup_adddmp_adddup_subdmp_subdup_muldmp_muldmp_sqrdup_powdmp_powdmp_pdivdmp_premdmp_pquo dmp_pexquodmp_divdup_remdmp_remdmp_quo dmp_exquo dmp_add_mul dmp_sub_mul dmp_max_norm dmp_l1_normdmp_l2_norm_squared)dmp_clear_denomsdmp_integrate_in dmp_diff_in dmp_eval_in dup_revertdmp_ground_truncdmp_ground_contentdmp_ground_primitivedmp_ground_monic dmp_compose dup_decompose dup_shift dup_transformdmp_lift) dup_half_gcdex dup_gcdex dup_invertdmp_subresultants dmp_resultantdmp_discriminant dmp_inner_gcddmp_gcddmp_lcm dmp_cancel) dup_gff_listdmp_norm dmp_sqf_p dmp_sqf_norm dmp_sqf_part dmp_sqf_listdmp_sqf_list_include)dup_cyclotomic_pdmp_irreducible_pdmp_factor_listdmp_factor_list_include) dup_isolate_real_roots_sqfdup_isolate_real_rootsdup_isolate_all_roots_sqfdup_isolate_all_rootsdup_refine_real_rootdup_count_real_rootsdup_count_complex_roots dup_sturmdup_cauchy_upper_bounddup_cauchy_lower_bounddup_mignotte_sep_bound_squared)UnificationFailedPolynomialErrorc2tt|||||Sr)DMPr0)replevrs rinit_normal_DMPrs z#sC(#s 33rcFeZdZdZdZddZdZdZdZddZ e dd Z e dd Z e d Z e d Zdd ZddZdZdZdZe dZe ddZdZdZdZdZddZddZddZddZdZdZdZ dZ!d Z"dd!Z#dd"Z$d#Z%d$Z&d%Z'd&Z(d'Z)d(Z*d)Z+d*Z,d+Z-d,Z.d-Z/d.Z0d/Z1d0Z2d1Z3d2Z4d3Z5d4Z6d5Z7d6Z8d7Z9d8Z:d9Z;dd:Zd=Z?d>Z@d?ZAd@ZBdAZCdBZDdCZEdDZFdEZGddFZHddGZIddHZJdIZKdJZLdKZMdLZNdMZOddNZPdOZQdPZRdQZSdRZTddSZUdTZVdUZWdVZXdWZYdXZZdYZ[dZZ\d[Z]d\Z^d]Z_d^Z`d_Zad`ZbdaZcdbZddcZedddZfddeZgdfZhdgZiddhZjddiZkddjZlddkZmendlZoendmZpendnZqendoZrendpZsendqZtendrZuendsZvendtZwenduZxendvZyendwZzdxZ{dyZ|dzZ}d{Z~d|Zd}Zd~ZdZdZdZdZdZdZdZdZdZddZddZdZdZdZdZdZdZy)rz)Dense Multivariate Polynomials over `K`. )rrrringNc|Kt|turt|||}n:t|ts*t |j ||}nt|\}}||_||_ ||_ ||_ yr) typedictrD isinstancelistr?convertr.rrrr)selfrrrrs r__init__z DMP.__init__sj ?CyD #Cc2T* S!137#C(HC rc|jjd|jd|jd|jdSN(, )) __class__r)rrrrs r__repr__z DMP.__repr__s'#$;;#7#7qvvNNrct|jj|j|j|j |j fSr)hashrr)to_tuplerrrrs r__hash__z DMP.__hash__s4Q[[))1::<qvvNOOrct|tr|j|jk7rtd|d||j|jk(rR|j |j k(r9|j|j|j |j|jfS|j|jj|j}}|j |j *j|j n |j t|j||j|}t|j||j|}||dffd }|||||fS)z7Unify representations of two multivariate polynomials. Cannot unify  with Fc6|r |s|S|dz}t|||SNr)rrrkillrs rr!zDMP.unify..pers(" q3S$//r) rrrrrrr!runifyr2)rr&rrFGr!rs @rrz DMP.unifys!S!QUUaee^#A$FG G 55AEE>aff.55!%%quu4 4uuaeekk!%%0C66Dvv!#::aff-D66DAEE3s3AAEE3s3A c 0S!Q& &rc|j}|r |s|S|dz}| |j}| |j}t||||Sz.Create a DMP out of the given representation. r)rrrr)rrrrrrs rr!zDMP.persLee  q ;%%C <66D3S$''rctd|||SNrrclsrrrs rzerozDMP.zero1c3%%rctd|||Srrrs ronezDMP.onerrc.|t||d|||S)zCCreate an instance of ``cls`` given a list of native coefficients. N)r2rrrrs r from_listz DMP.from_lists;sCs3S#>>rc,|t|||||S)zBCreate an instance of ``cls`` given a list of SymPy coefficients. )r3rs rfrom_sympy_listzDMP.from_sympy_lists>#sC0#s;;rc\t|j|j|j|S)AConvert ``f`` to a dict representation with native coefficients. r)rErrr)rrs rto_dictz DMP.to_dicts155!%%T::rct|j|j|j|}|j D]#\}}|jj |||<%|S)@Convert ``f`` to a dict representation with SymPy coefficients. r)rErrritemsto_sympy)rrrr'vs r to_sympy_dictzDMP.to_sympy_dictsT!%%D9IIK 'DAqUU^^A&CF ' rc|jSzAConvert ``f`` to a list representation with native coefficients. rrs rto_listz DMP.to_list uu rc6fdjS)@Convert ``f`` to a list representation with SymPy coefficients. cg}|D]T}t|tr|j|+|jjj |V|Sr)rrappendrr)routvalrsympify_nested_lists rrz.DMP.to_sympy_list..sympify_nested_listsSC 4c4(JJ2378JJquu~~c23  4 Jrr)rrs`@r to_sympy_listzDMP.to_sympy_lists #155))rcBt|j|jS)x Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. )rNrrrs rrz DMP.to_tuple s AEE155))rc,|t|||||S)zBConstruct and instance of ``cls`` from a ``dict`` representation. )rDrs r from_dictz DMP.from_dicts=c3/c::rc Vtttt|||||Sr)rrrzip)rmonomscoeffsrrrs rfrom_monoms_coeffszDMP.from_monoms_coeffss$4S012CdCCrcT|j|jjSr )rrrrs rto_ringz DMP.to_ringsyy)**rcT|j|jjSr)rrrrs rto_fieldz DMP.to_field!yy*++rcT|j|jjSr)rrrrs rto_exactz DMP.to_exact%rrc|j|k(r|Stt|j|j|j|||jS)z$Convert the ground domain of ``f``. )rrr2rr)rrs rrz DMP.convert)s< 55C<H{155!%%@da!a%@@!"JK KAsA!c|jsat|j}|dkrd|jjfgSt |jDcgc] \}}||z f|fc}}St dcc}}w)z Returns all terms from a ``f``. rrr )rr5rrrrrrs r all_termsz DMP.all_termsVsnuu155!A1uquuzz*++3>> from sympy.polys.polyclasses import DMP >>> from sympy.polys.domains import ZZ >>> DMP([[[ZZ(1)]], [[ZZ(1)], [ZZ(2)]]], ZZ).exclude() ([2], DMP([[1], [1, 2]], ZZ, None)) )rKrrrr)rrrus rexcludez DMP.excludeus@ aeeQUUAEE21a!++a***rcx|jt|j||j|jS)a Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`. Examples ======== >>> from sympy.polys.polyclasses import DMP >>> from sympy.polys.domains import ZZ >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 0, 2]) DMP([[[2], []], [[1, 0], []]], ZZ, None) >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 2, 0]) DMP([[[1], []], [[2, 0], []]], ZZ, None) )r!rMrrr)rPs rpermutez DMP.permutes*"uu[155!%%899rct|j|j|j\}}||j |fS)z/Remove GCD of terms from the polynomial ``f``. )rIrrrr!rs r terms_gcdz DMP.terms_gcds2QUUAEE15511!%%({rc|jt|j|jj ||j |jS)z.Add an element of the ground domain to ``f``. )r!rOrrrrrrs r add_groundzDMP.add_ground5uu^AEE155==+;QUUAEEJKKrc|jt|j|jj ||j |jS)z5Subtract an element of the ground domain from ``f``. )r!rPrrrrr-s r sub_groundzDMP.sub_groundr/rc|jt|j|jj ||j |jS)z5Multiply ``f`` by a an element of the ground domain. )r!rQrrrrr-s r mul_groundzDMP.mul_groundr/rc|jt|j|jj ||j |jS)z8Quotient of ``f`` by a an element of the ground domain. )r!rRrrrrr-s r quo_groundzDMP.quo_groundr/rc|jt|j|jj ||j |jS)z>Exact quotient of ``f`` by a an element of the ground domain. )r!rSrrrrr-s r exquo_groundzDMP.exquo_ground6uu%aeeQUU]]1-=quuaeeLMMrcv|jt|j|j|jS)z)Make all coefficients in ``f`` positive. )r!rTrrrrs rabszDMP.abs&uuWQUUAEE155122rcv|jt|j|j|jS)"Negate all coefficients in ``f``. )r!rVrrrrs rnegzDMP.negr;rcX|j|\}}}}}|t||||S)z2Add two multivariate polynomials ``f`` and ``g``. )rrXrr&rrr!rrs raddzDMP.add0ggajS#q!71ac*++rcX|j|\}}}}}|t||||S)z7Subtract two multivariate polynomials ``f`` and ``g``. )rrZr@s rsubzDMP.subrBrcX|j|\}}}}}|t||||S)z7Multiply two multivariate polynomials ``f`` and ``g``. )rr\r@s rmulzDMP.mulrBrcv|jt|j|j|jS)z(Square a multivariate polynomial ``f``. )r!r]rrrrs rsqrzDMP.sqrr;rct|tr;|jt|j||j |j Stdt|z)+Raise ``f`` to a non-negative power ``n``. ``int`` expected, got %s) rintr!r_rrr TypeErrorrrrs rpowzDMP.powsG a 55155!%%89 96a@A Arcr|j|\}}}}}t||||\}}||||fS)z/Polynomial pseudo-division of ``f`` and ``g``. )rr` rr&rrr!rrqrs rpdivzDMP.pdivs@ggajS#q!1c3'11vs1v~rcX|j|\}}}}}|t||||S)z0Polynomial pseudo-remainder of ``f`` and ``g``. )rrar@s rpremzDMP.prem0ggajS#q!8Aq#s+,,rcX|j|\}}}}}|t||||S)z/Polynomial pseudo-quotient of ``f`` and ``g``. )rrbr@s rpquozDMP.pquorWrcX|j|\}}}}}|t||||S)z5Polynomial exact pseudo-quotient of ``f`` and ``g``. )rrcr@s rpexquoz DMP.pexquos0ggajS#q!:aC-..rcr|j|\}}}}}t||||\}}||||fS)z7Polynomial division with remainder of ``f`` and ``g``. )rrdrQs rdivzDMP.divs@ggajS#q!q!S#&11vs1v~rcX|j|\}}}}}|t||||S)z2Computes polynomial remainder of ``f`` and ``g``. )rrfr@s rremzDMP.remrBrcX|j|\}}}}}|t||||S)z1Computes polynomial quotient of ``f`` and ``g``. )rrgr@s rquozDMP.quorBrc|j|\}}}}}|t||||}|jr(||jvrddlm}||||j|S)z7Computes polynomial exact quotient of ``f`` and ``g``. rExactQuotientFailed)rrhrsympy.polys.polyerrorsrd) rr&rrr!rrresrds rexquoz DMP.exquos\ggajS#q!)Aq#s+, 66c' B%aAFF3 3 rct|tr!t|j||jSt dt |z)z0Returns the leading degree of ``f`` in ``x_j``. rK)rrLr6rrrMr)rrs rdegreez DMP.degrees8 a  1551 16a@A ArcBt|j|jS)z$Returns a list of degrees of ``f``. )r7rrrs r degree_listzDMP.degree_list squuaee,,rcBtd|jDS)z#Returns the total degree of ``f``. c32K|]}t|ywrsum).0rs r z#DMP.total_degree..s.a3q6.s)maxrrs r total_degreezDMP.total_degrees.188:...rc|j}i}|t|jddk(}|jD][}t|d}||kr||z }nd}|r|d||d|fz<0t |d}||xx|z cc<|d|t |<]t ||j|jt|z|jS)z&Return homogeneous polynomial of ``f``rr) rsr!r rortuplerrrrLr) rstdr" new_symboltermdrls r homogenizezDMP.homogenizes ^^ 3qwwy|A// GGI +DDG A2vF)-atAw!~&aM! #'7uQx  +6155!%%#j/"9166BBrc|jrt S|j}t|d}|D]}t|}||k7sy|S)z(Returns the homogeneous order of ``f``. rN)is_zerorrro)rrtdegmonom_tdegs rhomogeneous_orderzDMP.homogeneous_order&sQ 993J6!9~ EJE}    rcXt|j|j|jSz*Returns the leading coefficient of ``f``. )r:rrrrs rLCzDMP.LC6QUUAEE15511rcXt|j|j|jSz+Returns the trailing coefficient of ``f``. )r<rrrrs rTCzDMP.TC:rrctd|Dr,t|j||j|jSt d)z+Returns the ``n``-th coefficient of ``f``. c3<K|]}t|tywr)rrL)rprs rrqzDMP.nth..@s-az!S!-sza sequence of integers expected)allr=rrrrM)rNs rnthzDMP.nth>s9 -1- -!!%%AEE1559 9=> >rcXt|j|j|jS)zReturns maximum norm of ``f``. )rkrrrrs rmax_normz DMP.max_normEsAEE155!%%00rcXt|j|j|jS)zReturns l1 norm of ``f``. )rlrrrrs rl1_normz DMP.l1_normIs155!%%//rcXt|j|j|jS)z!Return squared l2 norm of ``f``. )rmrrrrs rl2_norm_squaredzDMP.l2_norm_squaredMs"155!%%77rct|j|j|j\}}||j |fS)z0Clear denominators, but keep the ground domain. )rnrrrr!)rr$rs r clear_denomszDMP.clear_denomsQs2#AEE155!%%8qaeeAhrc t|tstdt|zt|tstdt|z|j t |j |||j|jS)zEComputes the ``m``-th order indefinite integral of ``f`` in ``x_j``. rK) rrLrMrr!rorrrrrrs r integratez DMP.integrateVsi!S!6a@A A!S!6a@A Auu%aeeQ155!%%@AArc t|tstdt|zt|tstdt|z|j t |j |||j|jS)z !=> >rc|j|\}}}}}|s(t|||\}}} |||||| fStd)z-Extended Euclidean algorithm, if univariate. r)rr}r) rr&rrr!rrrvtrs rgcdexz DMP.gcdex|sVggajS#q!1c*GAq!q63q63q6) )=> >rcp|j|\}}}}}|s|t|||Std)z(Invert ``f`` modulo ``g``, if possible. r)rr~rr@s rinvertz DMP.inverts=ggajS#q!z!Q,- -=> >rc|js0|jt|j||jSt d)z"Compute ``f**(-1)`` mod ``x**n``. r)rr!rrrrrrNs rrevertz DMP.reverts5uu55AEE1aee45 5=> >rcv|j|\}}}}}t||||}tt||S)z7Computes subresultant PRS sequence of ``f`` and ``g``. )rrrmap)rr&rrr!rrRs r subresultantszDMP.subresultantss;ggajS#q! aC -CQK  rc|j|\}}}}}|r2t|||||\}} ||dtt|| fS|t||||dS)z/Computes resultant of ``f`` and ``g`` via PRS. ) includePRSTr)rrrr) rr&rrrr!rrrfrs r resultantz DMP.resultantshggajS#q! "1acjIFCs&Sa[(99 9=AsC0t<?@ @=> >rc|jsI|jt|j|jj ||jSt d)z/Efficiently compute Taylor shift ``f(x + a)``. r)rr!ryrrrr)rrs rshiftz DMP.shiftsBuu55155!%%--*:AEEBC C=> >rc,|jr td|j|\}}}}}|j||||\}}}}}||||j||||\}}}}}|s|t||||Std)z5Evaluate functional transformation ``q**n * f(p/q)``.r)rrrrz) rrrRrrr!r(Qrs r transformz DMP.transforms 55=> >ggajS#q!ggc!S#&67S#q!!!S#.44SC5EFS#q!}Q1c23 3=> >rc |js=tt|jt |j |j Std)z&Computes the Sturm sequence of ``f``. r)rrrr!rrrrrs rsturmz DMP.sturms:uuAEE9QUUAEE#:;< <=> >rcp|js t|j|jSt d)z7Computes the Cauchy upper bound on the roots of ``f``. r)rrrrrrs rcauchy_upper_boundzDMP.cauchy_upper_bound*uu)!%%7 7=> >rcp|js t|j|jSt d)z?Computes the Cauchy lower bound on the nonzero roots of ``f``. r)rrrrrrs rcauchy_lower_boundzDMP.cauchy_lower_boundrrcp|js t|j|jSt d)zBComputes the squared Mignotte bound on root separations of ``f``. r)rrrrrrs rmignotte_sep_bound_squaredzDMP.mignotte_sep_bound_squareds*uu1!%%? ?=> >rc|jsDt|j|jDcgc]\}}|j ||fc}}St dcc}}w)z4Computes greatest factorial factorization of ``f``. r)rrrrr!r)rr&r's rgff_listz DMP.gff_listsKuu/;AEE155/IKtq!aeeAh]K K=> >LsAct|j|j|j}|j ||jjS)zComputes ``Norm(f)``.r)rrrrr!)rrSs rnormzDMP.norms7 QUUAEE155 )uuQAEEIIu&&rct|j|j|j\}}}||j ||j ||jjfS)z$Computes square-free norm of ``f``. r)rrrrr!)rrvr&rSs rsqf_normz DMP.sqf_norm"sLquuaeeQUU31a!%%(AEE!E333rcv|jt|j|j|jS)z$Computes square-free part of ``f``. )r!rrrrrs rsqf_partz DMP.sqf_part's&uu\!%%677rct|j|j|j|\}}||Dcgc]\}}|j ||fc}}fScc}}wz0Returns a list of square-free factors of ``f``. )rrrrr!)rrr$r%r&r's rsqf_listz DMP.sqf_list+sK%aeeQUUAEE3?w';$!Qq1 ;;;;sAct|j|j|j|}|Dcgc]\}}|j ||fc}}Scc}}wr)rrrrr!)rrr%r&r's rsqf_list_includezDMP.sqf_list_include0sB&quuaeeQUUC@+2441a!%%(A444sAct|j|j|j\}}||Dcgc]\}}|j ||fc}}fScc}}wz0Returns a list of irreducible factors of ``f``. )rrrrr!)rr$r%r&r's r factor_listzDMP.factor_list5sI(quu=w';$!Qq1 ;;;;sAct|j|j|j}|Dcgc]\}}|j ||fc}}Scc}}wr)rrrrr!)rr%r&r's rfactor_list_includezDMP.factor_list_include:s@)!%%>+2441a!%%(A444sAcd|js|sL|s%t|j|j||||St |j|j||||S|s%t |j|j||||St |j|j||||Std)z0Compute isolating intervals for roots of ``f``. )epsinfsupfastz1Cannot isolate roots of a multivariate polynomial)rrrrrrrr)rrrrrrsqfs r intervalsz DMP.intervals?suu1!%%CSVY`dee5aeeQUUQTZ]dhii03CUX_cdd4QUUAEEsPSY\cghh!CE Erc ||js&t|j|||j|||St d)zu Refine an isolating interval to the given precision. ``eps`` should be a rational number. )rstepsrz1Cannot refine a root of a multivariate polynomial)rrrrr)rrvrrrrs r refine_rootzDMP.refine_rootPs=uu'q!QUU5W[\ \!CE ErcHt|j|j||S)zReturns ``True`` if ``f`` is an element of the ground domain. N)rBrrrs r is_groundz DMP.is_groundosAEE4//rcXt|j|j|jS)z7Returns ``True`` if ``f`` is a square-free polynomial. )rrrrrs ris_sqfz DMP.is_sqftrrc|jjt|j|j|jS)z=Returns ``True`` if the leading coefficient of ``f`` is one. )rrr:rrrs ris_monicz DMP.is_monicys,uu||M!%%>??rc|jjt|j|j|jS)zAReturns ``True`` if the GCD of the coefficients of ``f`` is one. )rrrtrrrs r is_primitivezDMP.is_primitive~s-uu||.quuaeeQUUCDDrctdt|j|j|jj DS)z:Returns ``True`` if ``f`` is linear in all its variables. c38K|]}t|dkyw)rNrnrprs rrqz DMP.is_linear..Xu3u:?XrrErrrkeysrs r is_linearz DMP.is_linear3X AEE155!%%0P0U0U0WXXXrctdt|j|j|jj DS)z=Returns ``True`` if ``f`` is quadratic in all its variables. c38K|]}t|dkyw)Nrnrs rrqz#DMP.is_quadratic..rrrrs r is_quadraticzDMP.is_quadraticrrc:t|jdkS)z8Returns ``True`` if ``f`` is zero or has only one term. r)r!rrs r is_monomialzDMP.is_monomials199;1$$rc&|jduS)z7Returns ``True`` if ``f`` is a homogeneous polynomial. N)rrs ris_homogeneouszDMP.is_homogeneouss""$D00rcXt|j|j|jS)z:Returns ``True`` if ``f`` has no factors over its domain. )rrrrrs ris_irreduciblezDMP.is_irreducibles!quu55rc\|js t|j|jSy)z6Returns ``True`` if ``f`` is a cyclotomic polynomial. 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