B dd-@sdZddlmZddlmZddlmZmZmZddl m Z Gddde 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~mZmZddlmZmZddZGddde eZddZGddde eZddZGddde eZdS)z1OO layer for several polynomial representations. )oo) CantSympify)CoercionFailed NotReversible NotInvertible)PicklableWithSlotsc@s4eZdZdZddZddZddZedd Zd S) GenericPolyz5Base class for low-level polynomial representations. cCs||jS)zMake the ground domain a ring. ) set_domaindomget_ring)fr d/work/yifan.wang/ringdown/master-ringdown-env/lib/python3.7/site-packages/sympy/polys/polyclasses.pyground_to_ring szGenericPoly.ground_to_ringcCs||jS)z Make the ground domain a field. )r r get_field)r r r rground_to_fieldszGenericPoly.ground_to_fieldcCs||jS)zMake the ground domain exact. )r r get_exact)r r r rground_to_exactszGenericPoly.ground_to_exactcs2|r |\}}fdd|D}|r*||fS|SdS)Ncsg|]\}}||fqSr r ).0gk)perr r sz/GenericPoly._perify_factors..r )rresultincludecoefffactorsr )rr_perify_factorss zGenericPoly._perify_factorsN) __name__ __module__ __qualname____doc__rrr classmethodrr r r rr s r)! 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)UnificationFailedPolynomialErrorcCstt|||||S)N)DMPr%)replevr r r rinit_normal_DMPsrc@sheZdZdZdZd ddZddZdd Zd d Zd!d dZ e d"ddZ e d#ddZ e ddZ e ddZd$ddZd%ddZddZddZdd Ze d!d"Ze d&d#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd'd.d/Zd(d0d1Zd)d2d3Zd*d4d5Zd6d7Zd8d9Zd:d;Z dd?Z"d+d@dAZ#d,dBdCZ$dDdEZ%dFdGZ&dHdIZ'dJdKZ(dLdMZ)dNdOZ*dPdQZ+dRdSZ,dTdUZ-dVdWZ.dXdYZ/dZd[Z0d\d]Z1d^d_Z2d`daZ3dbdcZ4dddeZ5dfdgZ6dhdiZ7djdkZ8dldmZ9dndoZ:dpdqZ;d-drdsZdxdyZ?dzd{Z@d|d}ZAd~dZBddZCddZDddZEddZFddZGd.ddZHd/ddZId0ddZJddZKddZLddZMddZNddZOd1ddZPddZQddZRddZSddZTd2ddZUddZVddZWddZXddZYddZZddZ[ddZ\ddZ]ddZ^ddZ_ddZ`ddZaddZbddÄZcddńZdddDŽZed3ddɄZfd4dd˄Zgdd̈́ZhddτZid5ddфZjd6ddӄZkd7ddՄZld8ddׄZmenddلZoenddۄZpendd݄Zqendd߄ZrenddZsenddZtenddZuenddZvenddZwenddZxenddZyenddZzddZ{ddZ|ddZ}ddZ~ddZddZddZddZddZddZddZddZdd Zd d Zd d ZddZd9ddZd:ddZddZddZddZddZddZddZdS(;rz)Dense Multivariate Polynomials over `K`. )rrr ringNcCsf|dk r>t|tkr"t|||}qJt|tsJt|||}n t|\}}||_||_ ||_ ||_ dS)N) typedictr9 isinstancelistr4convertr#rrr r)selfrr rrr r r__init__s   z DMP.__init__cCsd|jj|j|j|jfS)Nz%s(%s, %s, %s)) __class__rrr r)r r r r__repr__sz DMP.__repr__cCs t|jj||j|j|jfS)N)hashrrto_tuplerr r)r r r r__hash__sz DMP.__hash__cst|tr|j|jkr&td||f|j|jkrV|j|jkrV|j|j|j|j|jfS|j|j|j}}|j|jdk rdk r|jn|jt |j||j|}t |j||j|}||dffdd }|||||fSdS)z7Unify representations of two multivariate polynomials. zCannot unify %s with %sNFcs"|r|s |S|d8}t|||S)N)r)rr rkill)rr rrs zDMP.unify..per) rrrrr rrrunifyr')r rrr FGrr )rrrs  z DMP.unifyFcCsD|j}|r|s|S|d8}|dkr(|j}|dkr6|j}t||||S)z.Create a DMP out of the given representation. rN)rr rr)r rr rrrr r rrszDMP.percCstd|||S)Nr)r)clsrr rr r rzeroszDMP.zerocCstd|||S)Nr)r)rrr rr r roneszDMP.onecCs|t||d|||S)zCCreate an instance of ``cls`` given a list of native coefficients. N)r')rrrr r r r from_listsz DMP.from_listcCs|t|||||S)zBCreate an instance of ``cls`` given a list of SymPy coefficients. )r()rrrr r r rfrom_sympy_listszDMP.from_sympy_listcCst|j|j|j|dS)zAConvert ``f`` to a dict representation with native coefficients. )r)r:rrr )r rr r rto_dictsz DMP.to_dictcCs@t|j|j|j|d}x$|D]\}}|j|||<q W|S)z@Convert ``f`` to a dict representation with SymPy coefficients. )r)r:rrr itemsto_sympy)r rrrvr r r to_sympy_dictszDMP.to_sympy_dictcCs|jS)zAConvert ``f`` to a list representation with native coefficients. )r)r r r rto_listsz DMP.to_listcsfddjS)z@Convert ``f`` to a list representation with SymPy coefficients. csBg}x8|D]0}t|tr(||q |j|q W|S)N)rrappendr r)routval)r sympify_nested_listr rrs   z.DMP.to_sympy_list..sympify_nested_list)r)r r )r rr to_sympy_lists zDMP.to_sympy_listcCst|j|jS)zx Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. )rCrr)r r r rr sz DMP.to_tuplecCs|t|||||S)zBConstruct and instance of ``cls`` from a ``dict`` representation. )r9)rrrr r r r from_dictsz DMP.from_dictcCstttt|||||S)N)rrrzip)rmonomscoeffsrr rr r rfrom_monoms_coeffsszDMP.from_monoms_coeffscCs||jS)zMake the ground domain a ring. )rr r )r r r rto_ringsz DMP.to_ringcCs||jS)z Make the ground domain a field. )rr r)r r r rto_field!sz DMP.to_fieldcCs||jS)zMake the ground domain exact. )rr r)r r r rto_exact%sz DMP.to_exactcCs0|j|kr|Stt|j|j|j|||jSdS)z$Convert the ground domain of ``f``. N)r rr'rr)r r r r rr)s z DMP.convertrc Cs|t|j||||j|jS)z1Take a continuous subsequence of terms of ``f``. )rrArrr )r mnjr r rslice0sz DMP.slicecCs ddt|j|j|j|dDS)z;Returns all non-zero coefficients from ``f`` in lex order. cSsg|] \}}|qSr r )r_cr r rr6szDMP.coeffs..)order)r?rrr )r rr r rr4sz DMP.coeffscCs ddt|j|j|j|dDS)z8Returns all non-zero monomials from ``f`` in lex order. cSsg|] \}}|qSr r )rrrr r rr:szDMP.monoms..)r)r?rrr )r rr r rr8sz DMP.monomscCst|j|j|j|dS)z4Returns all non-zero terms from ``f`` in lex order. )r)r?rrr )r rr r rterms<sz DMP.termscCs2|js&|s|jjgSdd|jDSntddS)z%Returns all coefficients from ``f``. cSsg|]}|qSr r )rrr r rrFsz"DMP.all_coeffs..z&multivariate polynomials not supportedN)rr rrr)r r r r all_coeffs@s  zDMP.all_coeffscsD|js8t|jdkrdgSfddt|jDSntddS)z"Returns all monomials from ``f``. r)rcsg|]\}}|fqSr r )rir)rr rrRsz"DMP.all_monoms..z&multivariate polynomials not supportedN)rr*r enumerater)r r )rr all_monomsJs  zDMP.all_monomscsL|js@t|jdkr&d|jjfgSfddt|jDSntddS)z Returns all terms from a ``f``. r)rcsg|]\}}|f|fqSr r )rrr)rr rr^sz!DMP.all_terms..z&multivariate polynomials not supportedN)rr*rr rrr)r r )rr all_termsVs  z DMP.all_termscCs |jt|j|j|j|jjdS)z-Convert algebraic coefficients to rationals. )r )rrprrr )r r r rliftbszDMP.liftcCs$t|j|j|j\}}|||fS)z2Reduce degree of `f` by mapping `x_i^m` to `y_i`. )r;rrr r)r Jrr r rdeflatefsz DMP.deflatecCs,t|j|j|j|d\}}|||jj|S)z,Inject ground domain generators into ``f``. )front)r<rrr r)r rrrr r rinjectksz DMP.injectcCs.t|j|j||d}||||jt|jS)z2Eject selected generators into the ground domain. )r)r=rrrlensymbols)r r rrr r rejectpsz DMP.ejectcCs,t|j|j|j\}}}||||j|fS)ao Remove useless generators from ``f``. Returns the removed generators and the new excluded ``f``. Examples ======== >>> 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)) )r@rrr r)r rrur r rexcludeusz DMP.excludecCs|t|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) )rrBrrr )r Pr r rpermutesz DMP.permutecCs$t|j|j|j\}}|||fS)z/Remove GCD of terms from the polynomial ``f``. )r>rrr r)r rrr r r terms_gcdsz DMP.terms_gcdcCs"|t|j|j||j|jS)z.Add an element of the ground domain to ``f``. )rrDrr rr)r rr r r add_groundszDMP.add_groundcCs"|t|j|j||j|jS)z5Subtract an element of the ground domain from ``f``. )rrErr rr)r rr r r sub_groundszDMP.sub_groundcCs"|t|j|j||j|jS)z5Multiply ``f`` by a an element of the ground domain. )rrFrr rr)r rr r r mul_groundszDMP.mul_groundcCs"|t|j|j||j|jS)z8Quotient of ``f`` by a an element of the ground domain. )rrGrr rr)r rr r r quo_groundszDMP.quo_groundcCs"|t|j|j||j|jS)z>Exact quotient of ``f`` by a an element of the ground domain. )rrHrr rr)r rr r r exquo_groundszDMP.exquo_groundcCs|t|j|j|jS)z)Make all coefficients in ``f`` positive. )rrIrrr )r r r rabsszDMP.abscCs|t|j|j|jS)z"Negate all coefficients in ``f``. )rrKrrr )r r r rnegszDMP.negcCs&||\}}}}}|t||||S)z2Add two multivariate polynomials ``f`` and ``g``. )rrM)r rrr rrrr r raddszDMP.addcCs&||\}}}}}|t||||S)z7Subtract two multivariate polynomials ``f`` and ``g``. )rrO)r rrr rrrr r rsubszDMP.subcCs&||\}}}}}|t||||S)z7Multiply two multivariate polynomials ``f`` and ``g``. )rrQ)r rrr rrrr r rmulszDMP.mulcCs|t|j|j|jS)z(Square a multivariate polynomial ``f``. )rrRrrr )r r r rsqrszDMP.sqrcCs8t|tr$|t|j||j|jStdt|dS)z+Raise ``f`` to a non-negative power ``n``. z``int`` expected, got %sN) rintrrTrrr TypeErrorr)r rr r rpows zDMP.powc Cs6||\}}}}}t||||\}}||||fS)z/Polynomial pseudo-division of ``f`` and ``g``. )rrU) r rrr rrrqrr r rpdivszDMP.pdivcCs&||\}}}}}|t||||S)z0Polynomial pseudo-remainder of ``f`` and ``g``. )rrV)r rrr rrrr r rpremszDMP.premcCs&||\}}}}}|t||||S)z/Polynomial pseudo-quotient of ``f`` and ``g``. )rrW)r rrr rrrr r rpquoszDMP.pquocCs&||\}}}}}|t||||S)z5Polynomial exact pseudo-quotient of ``f`` and ``g``. )rrX)r rrr rrrr r rpexquosz DMP.pexquoc Cs6||\}}}}}t||||\}}||||fS)z7Polynomial division with remainder of ``f`` and ``g``. )rrY) r rrr rrrrrr r rdivszDMP.divcCs&||\}}}}}|t||||S)z2Computes polynomial remainder of ``f`` and ``g``. )rr[)r rrr rrrr r rremszDMP.remcCs&||\}}}}}|t||||S)z1Computes polynomial quotient of ``f`` and ``g``. )rr\)r rrr rrrr r rquoszDMP.quoc CsT||\}}}}}|t||||}|jrP||jkrPddlm}||||j|S)z7Computes polynomial exact quotient of ``f`` and ``g``. r)ExactQuotientFailed)rr]rsympy.polys.polyerrorsr) r rrr rrrresrr r rexquos  z DMP.exquocCs.t|trt|j||jStdt|dS)z0Returns the leading degree of ``f`` in ``x_j``. z``int`` expected, got %sN)rrr+rrrr)r rr r rdegrees z DMP.degreecCst|j|jS)z$Returns a list of degrees of ``f``. )r,rr)r r r r degree_list szDMP.degree_listcCstdd|DS)z#Returns the total degree of ``f``. css|]}t|VqdS)N)sum)rrr r r sz#DMP.total_degree..)maxr)r r r r total_degreeszDMP.total_degreec Cs|}i}|t|ddk}xz|D]n}t|d}||krP||}nd}|rp|d||d|f<q.t|d}|||7<|d|t|<q.Wt||j|jt ||j S)z&Return homogeneous polynomial of ``f``rr) rrrrrtuplerr rrr) r stdrZ new_symboltermdrlr r r homogenizes   zDMP.homogenizecCsF|jr t S|}t|d}x |D]}t|}||kr&dSq&W|S)z(Returns the homogeneous order of ``f``. rN)is_zerorrr)r rZtdegmonomZ_tdegr r rhomogeneous_order&s  zDMP.homogeneous_ordercCst|j|j|jS)z*Returns the leading coefficient of ``f``. )r/rrr )r r r rLC6szDMP.LCcCst|j|j|jS)z+Returns the trailing coefficient of ``f``. )r1rrr )r r r rTC:szDMP.TCcGs2tdd|Dr&t|j||j|jStddS)z+Returns the ``n``-th coefficient of ``f``. css|]}t|tVqdS)N)rr)rrr r rr@szDMP.nth..za sequence of integers expectedN)allr2rrr r)r Nr r rnth>szDMP.nthcCst|j|j|jS)zReturns maximum norm of ``f``. )r`rrr )r r r rmax_normEsz DMP.max_normcCst|j|j|jS)zReturns l1 norm of ``f``. )rarrr )r r r rl1_normIsz DMP.l1_normcCst|j|j|jS)z!Return squared l2 norm of ``f``. )rbrrr )r r r rl2_norm_squaredMszDMP.l2_norm_squaredcCs$t|j|j|j\}}|||fS)z0Clear denominators, but keep the ground domain. )rcrrr r)r rrr r r clear_denomsQszDMP.clear_denomsrcCsPt|tstdt|t|ts4tdt||t|j|||j|jS)zEComputes the ``m``-th order indefinite integral of ``f`` in ``x_j``. z``int`` expected, got %s) rrrrrrdrrr )r rrr r r integrateVs   z DMP.integratecCsPt|tstdt|t|ts4tdt||t|j|||j|jS)z||\}}}}}t||||\}}} |||||| fS)z4Returns GCD of ``f`` and ``g`` and their cofactors. )rrw) r rrr rrrrZcffcfgr r r cofactorssz DMP.cofactorscCs&||\}}}}}|t||||S)z+Returns polynomial GCD of ``f`` and ``g``. )rrx)r rrr rrrr r rgcdszDMP.gcdcCs&||\}}}}}|t||||S)z+Returns polynomial LCM of ``f`` and ``g``. )rry)r rrr rrrr r rlcmszDMP.lcmTc Csx||\}}}}}|r0t||||dd\}}nt||||dd\}} }}||||}}|rh||fS|| ||fSdS)z6Cancel common factors in a rational function ``f/g``. T)rFN)rrz) r rrrr rrrZcFZcGr r rcancelsz DMP.cancelcCs"|t|j|j||j|jS)z&Reduce ``f`` modulo a constant ``p``. )rrhrr rr)r pr r rtruncsz DMP.trunccCs|t|j|j|jS)z'Divides all coefficients by ``LC(f)``. )rrkrrr )r r r rmonicsz DMP.moniccCst|j|j|jS)z(Returns GCD of polynomial coefficients. )rirrr )r r r rcontentsz DMP.contentcCs$t|j|j|j\}}|||fS)z/Returns content and a primitive form of ``f``. )rjrrr r)r contrr r r primitivesz DMP.primitivecCs&||\}}}}}|t||||S)z4Computes functional composition of ``f`` and ``g``. )rrl)r rrr rrrr r rcomposesz DMP.composecCs,|js tt|jt|j|jStddS)z,Computes functional decomposition of ``f``. zunivariate polynomial expectedN)rrrrrmrr r)r r r r decomposesz DMP.decomposecCs0|js$|t|j|j||jStddS)z/Efficiently compute Taylor shift ``f(x + a)``. zunivariate polynomial expectedN)rrrnrr rr)r rr r rshiftsz DMP.shiftc Cs|jrtd||\}}}}}|||||\}}}}}||||||||\}}}}}|sx|t||||StddS)z5Evaluate functional transformation ``q**n * f(p/q)``.zunivariate polynomial expectedN)rrrro) r r%rrr rrQrr r r transforms$z DMP.transformcCs,|js tt|jt|j|jStddS)z&Computes the Sturm sequence of ``f``. zunivariate polynomial expectedN)rrrrrrr r)r r r rsturmsz DMP.sturmcCs |jst|j|jStddS)z7Computes the Cauchy upper bound on the roots of ``f``. zunivariate polynomial expectedN)rrrr r)r r r rcauchy_upper_boundszDMP.cauchy_upper_boundcCs |jst|j|jStddS)z?Computes the Cauchy lower bound on the nonzero roots of ``f``. zunivariate polynomial expectedN)rrrr r)r r r rcauchy_lower_boundszDMP.cauchy_lower_boundcCs |jst|j|jStddS)zBComputes the squared Mignotte bound on root separations of ``f``. zunivariate polynomial expectedN)rrrr r)r r r rmignotte_sep_bound_squaredszDMP.mignotte_sep_bound_squaredcs.js"fddtjjDStddS)z4Computes greatest factorial factorization of ``f``. csg|]\}}||fqSr )r)rrr)r r rrsz DMP.gff_list..zunivariate polynomial expectedN)rr{rr r)r r )r rgff_listsz DMP.gff_listcCs$t|j|j|j}|j||jjdS)zComputes ``Norm(f)``.)r )r|rrr r)r rr r rnormszDMP.normcCs6t|j|j|j\}}}||||j||jjdfS)z$Computes square-free norm of ``f``. )r )r~rrr r)r rrrr r rsqf_norm"sz DMP.sqf_normcCs|t|j|j|jS)z$Computes square-free part of ``f``. )rrrrr )r r r rsqf_part'sz DMP.sqf_partcs.tjjj|\}}|fdd|DfS)z0Returns a list of square-free factors of ``f``. csg|]\}}||fqSr )r)rrr)r r rr.sz DMP.sqf_list..)rrrr )r rrrr )r rsqf_list+sz DMP.sqf_listcs&tjjj|}fdd|DS)z0Returns a list of square-free factors of ``f``. csg|]\}}||fqSr )r)rrr)r r rr3sz(DMP.sqf_list_include..)rrrr )r rrr )r rsqf_list_include0szDMP.sqf_list_includecs,tjjj\}}|fdd|DfS)z0Returns a list of irreducible factors of ``f``. csg|]\}}||fqSr )r)rrr)r r rr8sz#DMP.factor_list..)rrrr )r rrr )r r factor_list5szDMP.factor_listcs$tjjj}fdd|DS)z0Returns a list of irreducible factors of ``f``. csg|]\}}||fqSr )r)rrr)r r rr=sz+DMP.factor_list_include..)rrrr )r rr )r rfactor_list_include:szDMP.factor_list_includecCs|jsv|s@|s&t|j|j||||dSt|j|j||||dSq~|s\t|j|j||||dSt|j|j||||dSntddS)z0Compute isolating intervals for roots of ``f``. )epsinfsupfastz1Cannot isolate roots of a multivariate polynomialN)rrrr rrrr)r rr<r=r>r?Zsqfr r r intervals?sz DMP.intervalsc Cs,|js t|j|||j|||dStddS)zu Refine an isolating interval to the given precision. ``eps`` should be a rational number. )r<stepsr?z1Cannot refine a root of a multivariate polynomialN)rrrr r)r rrr<rAr?r r r refine_rootPszDMP.refine_rootcCst|j|j||dS)z)rrr )r r=r>r r rcount_real_roots]szDMP.count_real_rootscCst|j|j||dS)z?Return the number of complex roots of ``f`` in ``[inf, sup]``. )r=r>)rrr )r r=r>r r rcount_complex_rootsaszDMP.count_complex_rootscCst|j|jS)z0Returns ``True`` if ``f`` is a zero polynomial. )r5rr)r r r rresz DMP.is_zerocCst|j|j|jS)z0Returns ``True`` if ``f`` is a unit polynomial. )r6rrr )r r r ris_onejsz DMP.is_onecCst|jd|jS)z>Returns ``True`` if ``f`` is an element of the ground domain. N)r7rr)r r r r is_groundosz DMP.is_groundcCst|j|j|jS)z7Returns ``True`` if ``f`` is a square-free polynomial. )r}rrr )r r r ris_sqftsz DMP.is_sqfcCs|jt|j|j|jS)z=Returns ``True`` if the leading coefficient of ``f`` is one. )r rEr/rr)r r r ris_monicysz DMP.is_moniccCs|jt|j|j|jS)zAReturns ``True`` if the GCD of the coefficients of ``f`` is one. )r rErirr)r r r r is_primitive~szDMP.is_primitivecCs$tddt|j|j|jDS)z:Returns ``True`` if ``f`` is linear in all its variables. css|]}t|dkVqdS)rN)r)rrr r rrsz DMP.is_linear..)rr:rrr keys)r r r r is_linearsz DMP.is_linearcCs$tddt|j|j|jDS)z=Returns ``True`` if ``f`` is quadratic in all its variables. css|]}t|dkVqdS)N)r)rrr r rrsz#DMP.is_quadratic..)rr:rrr rJ)r r r r is_quadraticszDMP.is_quadraticcCst|dkS)z8Returns ``True`` if ``f`` is zero or has only one term. r)rr)r r r r is_monomialszDMP.is_monomialcCs |dk S)z7Returns ``True`` if ``f`` is a homogeneous polynomial. N)r)r r r ris_homogeneousszDMP.is_homogeneouscCst|j|j|jS)z:Returns ``True`` if ``f`` has no factors over its domain. )rrrr )r r r ris_irreducibleszDMP.is_irreduciblecCs|jst|j|jSdSdS)z6Returns ``True`` if ``f`` is a cyclotomic polynomial. 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