a Z>m?Z?m@Z@ddlAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPddlQmRZRmSZSmTZTddlUmVZVmWZWmXZXmYZYmZZZddl[m\Z\dd l]m^Z^dd l_m`Z`maZambZbmcZcdd ldmeZemfZfmgZgdd lhmiZidd ljmkZlmmZnddZoddZpddZqddZrddZsddZtddZuddZvddZwdZd!d"Zxd#d$Zyd%d&Zzd'd(Z{d)d*Z|d+d,Z}d-d.Z~d/d0Zd1d2Zd3d4Zd5d6Zd7d8Zd[d:d;Zdd?Zd@dAZdBdCZdDdEZdFdGZdHdIZdJdKZdLdMZdNdOZdPdQZdRdSZdTdUZdVdWZdXdYZd9S)\z:Polynomial factorization routines in characteristic zero. ) gf_from_int_polygf_to_int_poly gf_lshift gf_add_mulgf_mulgf_divgf_remgf_gcdexgf_sqf_p gf_factor_sqf gf_factor)dup_LCdmp_LC dmp_ground_LCdup_TC dup_convert dmp_convert dup_degree dmp_degree dmp_degree_indmp_degree_list dmp_from_dict dmp_zero_pdmp_onedmp_nest dmp_raise dup_strip dmp_ground dup_inflate dmp_exclude dmp_include dmp_inject dmp_eject dup_terms_gcd dmp_terms_gcd)dup_negdmp_negdup_adddmp_adddup_subdmp_subdup_muldmp_muldup_sqrdmp_powdup_divdmp_divdup_quodmp_quo dmp_expand dmp_add_mul dup_sub_mul dmp_sub_mul dup_lshift dup_max_norm dmp_max_norm dup_l1_normdup_mul_grounddmp_mul_grounddup_quo_grounddmp_quo_ground)dup_clear_denomsdmp_clear_denoms dup_truncdmp_ground_trunc dup_content dup_monicdmp_ground_monic dup_primitivedmp_ground_primitive dmp_eval_tail dmp_eval_indmp_diff_eval_in dmp_compose dup_shift dup_mirror) dmp_primitive dup_inner_gcd dmp_inner_gcd) dup_sqf_p dup_sqf_norm dmp_sqf_norm dup_sqf_part dmp_sqf_part) _sort_factors)query)ExtraneousFactors DomainErrorCoercionFailedEvaluationFailed) nextprimeisprime factorint)subsets)ceillogcCsPg}|D]>}d}t|||\}}|s8||d}}qq8q|||fqt|S)zc Determine multiplicities of factors for a univariate polynomial using trial division. r)r/appendrV)ffactorsKresultfactorkqrrlg/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/polys/factortools.pydup_trial_divisionOsrnc CsXg}|D]F}d}t||||\}}t||r@||d}}qq@q|||fqt|S)ze Determine multiplicities of factors for a multivariate polynomial using trial division. rrb)r0rrcrV) rdreurfrgrhrirjrkrlrlrmdmp_trial_divisionfs rpc Csddlm}t|}t|d}t|d}|tdd|D}||d|}||d|d}|t||} |||| } | t||7} t| dd} | S)a The Knuth-Cohen variant of Mignotte bound for univariate polynomials in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**3 + 14*x**2 + 56*x + 64 >>> R.dup_zz_mignotte_bound(f) 152 By checking `factor(f)` we can see that max coeff is 8 Also consider a case that `f` is irreducible for example `f = 2*x**2 + 3*x + 4` To avoid a bug for these cases, we return the bound plus the max coefficient of `f` >>> f = 2*x**2 + 3*x + 4 >>> R.dup_zz_mignotte_bound(f) 6 Lastly,To see the difference between the new and the old Mignotte bound consider the irreducible polynomial:: >>> f = 87*x**7 + 4*x**6 + 80*x**5 + 17*x**4 + 9*x**3 + 12*x**2 + 49*x + 26 >>> R.dup_zz_mignotte_bound(f) 744 The new Mignotte bound is 744 whereas the old one (SymPy 1.5.1) is 1937664. References ========== ..[1] [Abbott2013]_ r)binomialcSsg|] }|dqS)rrrl.0cfrlrlrm z)dup_zz_mignotte_bound..rb) Z(sympy.functions.combinatorial.factorialsrqr_ceilsqrtsumabsr r8) rdrfrqddeltaZdelta2Z eucl_normt1t2lcboundrlrlrmdup_zz_mignotte_bound}s(   rcCsLt|||}tt|||}tt||}|||dd|||S)z7Mignotte bound for multivariate polynomials in `K[X]`. rbrr)r9r{rrzrry)rdrorfabnrlrlrmdmp_zz_mignotte_bounds rcCsJ|d}t||||}t|||}tt|||||\} } t| ||} t| ||} tt|||t| |||} tt|| |||} tt|| |||} tt|| |t|| ||} tt| |jg|||}tt|||| |\}}t|||}t|||}tt|||t|| ||} tt|||||}tt|| |||}| | ||fS)a  One step in Hensel lifting in `Z[x]`. Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s` and `t` such that:: f = g*h (mod m) s*g + t*h = 1 (mod m) lc(f) is not a zero divisor (mod m) lc(h) = 1 deg(f) = deg(g) + deg(h) deg(s) < deg(h) deg(t) < deg(g) returns polynomials `G`, `H`, `S` and `T`, such that:: f = G*H (mod m**2) S*G + T*H = 1 (mod m**2) References ========== .. [1] [Gathen99]_ rr)r5rAr/r+r'r)one)mrdghstrfMerjrkroGHrcr|STrlrlrmdup_zz_hensel_steps$     rc Csxt|}t||}|dkrHt|||||d|}t||||gS|}|d} ttt|d} t|g|} |d| D]} t | t| |||} q~t|| |} || ddD]} t | t| |||} qt | | ||\}}}t | |} t | |} t ||}t ||}t d| dD],}t ||| | ||||d\} } }}}qt|| |d| ||t|| || d||S)a Multifactor Hensel lifting in `Z[x]`. Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)` is a unit modulo `p`, monic pair-wise coprime polynomials `f_i` over `Z[x]` satisfying:: f = lc(f) f_1 ... f_r (mod p) and a positive integer `l`, returns a list of monic polynomials `F_1,\ F_2,\ \dots,\ F_r` satisfying:: f = lc(f) F_1 ... F_r (mod p**l) F_i = f_i (mod p), i = 1..r References ========== .. [1] [Gathen99]_ rbrrrN)lenr r;ZgcdexrAintrx_logrrr rrangerdup_zz_hensel_lift)prdZf_listlrfrkrFrrir|rZf_irrr_rlrlrmrs0      *rcCs(||dkr||}|sdS||dkS)NrrTrrl)fcrjplrlrlrm_test_pl4s  rcspt|}|dkr|gS|d}t||}t||}tt|||dd|||}t|dd||d|d}ttdt|d}td|t|} g} td| dD]z} t | r|| dkrq| | } t || } t | | |sqt | | |d} | | | ft| dks0t| dkrq8qt| dd d \}tttd|d}fd d |D}t||||}tt|}t|}gd}}|}d|t|krft||D]|dkrd}D]}|||d}q||}t|||srqn\|g}D]}t||||}q t|||}t||d}|d}|rr||dkrrq|g}t|}|dkr|g}D]}t||||}qt|||}|D]}t||||}qt|||}t||}t||}|||kr|}fd d |D}t||d}t||d}||t||}qq|d7}q||gS)z4Factor primitive square-free polynomials in `Z[x]`. rbrrrcSs t|dS)Nrb)r)xrlrlrm[rwz#dup_zz_zassenhaus..)keycsg|]}t|qSrl)r)rtff)rrlrmrv_rwz%dup_zz_zassenhaus..csg|]}|vr|qSrlrl)rti)rrlrmrvrw)rr8r rr{ryrxrrr]convertrr r rcrminrsetr_rr+rArFr:)rdrfrrArBCgammarrZpxrZfsqfxZfsqfrZmodularrZsorted_TrrerrrjrrrZT_SZG_normZH_normrl)rrrmdup_zz_zassenhaus;s  *$                rcCsbt||}t||}t|dd|}|r^tt|}|D]}||r>||dr>dSq>dS)z2Test irreducibility using Eisenstein's criterion. rbNrrT)r rrCr^rkeys)rdrfrtcZe_fcZe_ffrrlrlrmdup_zz_irreducible_ps    rFcCs|jr>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1 >>> R.dup_cyclotomic_p(f) False >>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1 >>> R.dup_cyclotomic_p(g) True FrbrrT)Zis_QQget_ringrrZis_ZZr rdup_factor_listrrrinsertr-rr)r7 is_negativer%rMdup_cyclotomic_prT)rdrfZ irreducibleK0rrcoeffrerrrrrrrlrlrmrsL          rcCsP|j|j g}t|D]0\}}tt|||||}t|||d|}q|S)z1Efficiently generate n-th cyclotomic polynomial. rb)rr^itemsr1r)rrfrrrirlrlrmdup_zz_cyclotomic_polys rcsvjj gg}t|D]T\}fdd|D}||td|D]"}fdd|D}||qLq|S)Ncs g|]}tt||qSrl)r1r)rtrrfrrlrmrv rwz-_dup_cyclotomic_decompose..rbcsg|]}t|qSrl)r)rtrjrrlrmrvrw)rr^rextendr)rrfrriQrrlrrm_dup_cyclotomic_decomposes rcCst||t||}}t|dkr&dS|dks6|dvr:dStdd|ddDrXdSt|}t||}||sx|Sg}td||D]}||vr||q|SdS) a Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` returns a list of factors of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for `n >= 1`. Otherwise returns None. Factorization is performed using cyclotomic decomposition of `f`, which makes this method much faster that any other direct factorization approach (e.g. Zassenhaus's). References ========== .. [1] [Weisstein09]_ rNrb)rrbcss|]}t|VqdS)N)boolrsrlrlrm 0rwz+dup_zz_cyclotomic_factor..rrr)r rranyris_onerc)rdrfZlc_fZtc_frrrrrlrlrmdup_zz_cyclotomic_factors     rcCst||\}}t|}t||dkr6| t||}}|dkrF|gfS|dkrX||gfStdrtt||rt||gfSd}tdrt||}|durt||}|t|ddfS)z:Factor square-free (non-primitive) polynomials in `Z[x]`. rrbUSE_IRREDUCIBLE_IN_FACTORNUSE_CYCLOTOMIC_FACTORF)multiple) rFrr r%rWrrrrV)rdrfcontrrrerlrlrmdup_zz_factor_sqfBs"     rcCst||\}}t|}t||dkr6| t||}}|dkrF|gfS|dkr\||dfgfStdr|t||r|||dfgfSt||}d}tdrt||}|durt||}t |||}||fS)a Factor (non square-free) polynomials in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, ..., f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Zassenhaus algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Examples ======== Consider the polynomial `f = 2*x**4 - 2`:: >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_zz_factor(2*x**4 - 2) (2, [(x - 1, 1), (x + 1, 1), (x**2 + 1, 1)]) In result we got the following factorization:: f = 2 (x - 1) (x + 1) (x**2 + 1) Note that this is a complete factorization over integers, however over Gaussian integers we can factor the last term. By default, polynomials `x**n - 1` and `x**n + 1` are factored using cyclotomic decomposition to speedup computations. To disable this behaviour set cyclotomic=False. References ========== .. [1] [Gathen99]_ rrbrNr) rFrr r%rWrrTrrrn)rdrfrrrrrerlrlrm dup_zz_factor_s&.     rcCsp||g}|D]T}t|}t|D]4}|dkrD|||}||}q&||r"dSq"||q|ddS)z,Wang/EEZ: Compute a set of valid divisors. rbN)r{reversedgcdrrc)Ecsctrfrgrjrkrlrlrmdmp_zz_wang_non_divisorss       rc stt||ds tdt||}t|s@tdt|\}}t|rp| t|}}|dfdd|D} t| ||} | dur||| fStddS)z2Wang/EEZ: Test evaluation points for suitability. rbzno luckcsg|]\}}t|qSrl)rH)rtrrrrfvrlrmrvrwz+dmp_zz_wang_test_points..N) rHrr[rQrFrr r%r) rdrrrrorfrrrrDrlrrmdmp_zz_wang_test_pointss  rc Csgdgt||d}} } |D]} t| |} t| ||} ttt|D]f}d||||}}\}}| |s| ||d} }qn|dkrNt| t||| || |d} | |<qN|| q"t| st gg}}t ||D]\} } t | || |} t| |}| |r|| }n4| || }| |||} }t| | ||| } }t| || |} || || q| |r|||fSgg}}t ||D]2\} } |t| || ||t| |d|qt||t|d||}|||fS)z0Wang/EEZ: Compute correct leading coefficients. rrb)rrr rrr,r.rcallrXziprHrrr;r<)rdrrrrrrorfrJrrrr|rrirrrCCZHHrccrZCCCZHHHrlrlrmdmp_zz_wang_lead_coeffssB $           rc Cst|dkr|\}}t||}t||}t||||\}} } t|||}t| ||} t||||\} }t| | |||} t||}t| |} || g} n|dg} t|ddD]}| dt || d|qgdgg} }t || D]<\}}t ||g|dgd|d|\} }| | | |qg| |dg} } t | |D]H\}}t||}t||}t t||||||}t||}| |q@| S)z2Wang/EEZ: Solve univariate Diophantine equations. rrrrbr)rrr rrrrrrr+rdmp_zz_diophantinercr)rrrrfrrrdrrrrrjrgrrrkrlrlrmdup_zz_diophantines8               rc s|sdd|D}t|}t|D]`\} } | s0q"t||| } tt|| D]0\} \} }t|| }tt| ||| <qPq"n,t|}t|}|d|dd}}gg}}|D].}| t ||| t |||qt |||}dt ||||}fdd|D}t||D]\} }t || |}q:t|}tj| g|}t|}td|D]}t|rqt||}t||d||}t|st||d}t ||||} t| D]&\} }tt|d|| | <qtt|| D] \} \} }t| ||| <qDt| |D]\}}t |||}qpt|}qfdd|D}|S) z4Wang/EEZ: Solve multivariate Diophantine equations. cSsg|]}gqSrlrlrtrrlrlrmrvArwz&dmp_zz_diophantine..rNrbcsg|]}t|dqSrb)rrtr)rfrrlrmrv]rwrcsg|]}t|qSrl)rBr)rfrrorlrmrv}rw)r enumeraterrr;rAr'rr3rcr2rIrr6rBrrrmaprrr,rJr> factorialrr()rrrr|rrorfrrrrrjrrrrrrrdrrrrrirl)rfrrorrmr>sV        rc Cs|gt||d}}} t|}tt|ddD]>\} } t|d| || || |} |dt| || | |q6tt||dd} t t d|d||D]\}} } t||d}}|d|d||dd}}tt ||D]L\} \}}tt ||| |||d|}|gt |ddd|d||| <qt |j| g||}t||}t| t|||||}t| ||}|t d|D]}t||rqt||||}t||d| |||}t||dst|||d|d|}t|||| ||d|}tt ||D]>\} \}}t|t |d|d||||}t|||||| <qt| t|||||}t||||}qqt||||krtn|SdS)z-Wang/EEZ: Parallel Hensel lifting algorithm. rbNrrr)rlistrrrIrrBmaxrrrrHrrrrr*r3rrrr,rJr>rrr4rX)rdrLCrrrorfrrrrrrr|rrwIrrrrrrZdjrirrrrlrlrmdmp_zz_wang_hensel_liftings@""&   rNc sHddlm}||tt||d\}}t||}t|} dur`|dkr\dndtgjg|df\} } } } zRt|||| |\}}}t |\}}t |} | dkr|gWS||||| fg} Wnt yYn0t d}t d}t d}t | |krt |D]}fd d t |D} t| | vr| t| nqzt|||| |\}}}Wnt yYqYn0t |\}}t |}| dur|| kr|| krg|} } nqn|} | dkr|gS| ||||| ft | |krqq|7qd \}}}| D]D\}}}}}t|}|durb||krf|}|}n|}|d7}q,| |\}}}}} |}z4t|||||| |\}}}t|||| | |}Wn<tyt d rt||dYStd Yn0g}|D]@}t||\}}t||r6t||}||q|S)a` Factor primitive square-free polynomials in `Z[X]`. Given a multivariate polynomial `f` in `Z[x_1,...,x_n]`, which is primitive and square-free in `x_1`, computes factorization of `f` into irreducibles over integers. The procedure is based on Wang's Enhanced Extended Zassenhaus algorithm. The algorithm works by viewing `f` as a univariate polynomial in `Z[x_2,...,x_n][x_1]`, for which an evaluation mapping is computed:: x_2 -> a_2, ..., x_n -> a_n where `a_i`, for `i = 2, \dots, n`, are carefully chosen integers. The mapping is used to transform `f` into a univariate polynomial in `Z[x_1]`, which can be factored efficiently using Zassenhaus algorithm. The last step is to lift univariate factors to obtain true multivariate factors. For this purpose a parallel Hensel lifting procedure is used. The parameter ``seed`` is passed to _randint and can be used to seed randint (when an integer) or (for testing purposes) can be a sequence of numbers. References ========== .. [1] [Wang78]_ .. [2] [Geddes92]_ r)_randintrbNrrZEEZ_NUMBER_OF_CONFIGSZEEZ_NUMBER_OF_TRIESZEEZ_MODULUS_STEPcsg|]} qSrlrlrrfmodrandintrlrmrvrwzdmp_zz_wang..)NrrZEEZ_RESTART_IF_NEEDEDz3we need to restart algorithm with better parameters)Zsympy.core.randomr dmp_zz_factorrrr\rzerorrrr[rWrtupleaddrcr8rrrX dmp_zz_wangrGrrr&)rdrorfrseedrrrrrhistoryZconfigsrrkrrrrrZeez_num_configsZ eez_num_triesZ eez_mod_steprrZs_normZs_argrZ_s_normorig_frrergrlrrmrs                    rc Cs|st||St||r"|jgfSt|||\}}t|||dkrV| t|||}}tddt||Drv|gfSt|||\}}g}t ||dkrt |||}t |||}t ||||}t ||d|dD]\}}|d|g|fq|t|fS)a Factor (non square-free) polynomials in `Z[X]`. Given a multivariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, \dots, f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Consider polynomial `f = 2*(x**2 - y**2)`:: >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_zz_factor(2*x**2 - 2*y**2) (2, [(x - y, 1), (x + y, 1)]) In result we got the following factorization:: f = 2 (x - y) (x + y) References ========== .. [1] [Gathen99]_ rcss|]}|dkVqdSrNrlrtr|rlrlrmrsrwz dmp_zz_factor..rb)rrrrGrr&rrrNrrUrrprrrV) rdrorfrrrrerrirlrlrmrDs$$     rcsJt|}t|\}}fdd|D}|}||fS)z>Factor univariate polynomials into irreducibles in `QQ_I[x]`. cs g|]\}}t||fqSrl)rrtfacrrK1rlrmrvrwz#dup_qq_i_factor..)as_AlgebraicFieldrrr)rdrrrerlrrmdup_qq_i_factors   rc Cs|}t|||}t||\}}g}|D]T\}}t||\}} t| ||} t| d|\} } || |||}|| |fq*|}|||}||fS)z>Factor univariate polynomials into irreducibles in `ZZ_I[x]`. r) get_fieldrrr?rGrcr) rdrrrre new_factorsrr fac_denomfac_num fac_num_ZZ_Icontentfac_primrlrlrmdup_zz_i_factors    r csPt|}t|\}}fdd|D}|}||fS)z@Factor multivariate polynomials into irreducibles in `QQ_I[X]`. cs"g|]\}}t||fqSrl)rrrrrorlrmrvrwz#dmp_qq_i_factor..)rrdmp_factor_listr)rdrorrrerlr rmdmp_qq_i_factors  r cCs|}t||||}t|||\}}g}|D]X\}}t|||\} } t| |||} t| ||\} } || || |}|| |fq.|}|||}||fS)z@Factor multivariate polynomials into irreducibles in `ZZ_I[X]`. )rrr r@rGrcr)rdrorrrrerrrrrrrr rlrlrmdmp_zz_i_factors  rcCst|t||}}t||}|dkr.|gfS|dkrD||dfgfSt|||}}t||\}}}t||j}t|dkr|||t|fgfS||j} t |D]@\} \} } t | |j|} t | ||\} } }t | | |} | || <qt |||}||fS)zFactor multivariate polynomials over algebraic number fields. css|]}|dkVqdSrrlrrlrlrmrrwz!dmp_ext_factor..rbr)rrrErrrUrSdmp_factor_list_includerrrrrrrrPrKrp)rdrorfrrrrrkrerrrhrrrlrlrmdmp_ext_factors$     rcCs`t|||j}t||j|j\}}t|D]"\}\}}t||j||f||<q*|||j|fS)z2Factor univariate polynomials over finite fields. )rrr rrr)rdrfrrerrirlrlrm dup_gf_factors rcCs tddS)z4Factor multivariate polynomials over finite fields. z+multivariate polynomials over finite fieldsN)NotImplementedError)rdrorfrlrlrm dmp_gf_factorsrc CsDt||\}}t||\}}|jr4t||\}}n|jrLt||\}}n|jrdt||\}}n|jr|t ||\}}n|j s|| }}t |||}nd}|j r|}t|||\}}t |||}n|}|jrt||\}}nr|jrNt|d|\}} t|| |j\}}t|D]"\} \}} t|| || f|| <q|||j}n td||j rt|D]"\} \}} t |||| f|| <qj|||}|||}|rt|D]P\} \}} t||} t|| |}t |||}|| f|| <|||| | }q|||}|}|r4|d|j |j!g|f||t"|fS);Factor univariate polynomials into irreducibles in `K[x]`. Nr#factorization not supported over %s)#r#rFis_FiniteFieldr is_Algebraicris_GaussianRingr is_GaussianFieldris_Exact get_exactris_Fieldrr?rris_Polyr!r rrr"rrYquor8r=mulpowrrrrV) rdrrrrre K0_inexactrfdenomrorrimax_normrlrlrmrsZ        rcCsXt||\}}|s"t|gdfgSt|dd||}||ddfg|ddSdS)rrbrN)rrr;)rdrfrrerrlrlrmrWs rcCs|st||St|||\}}t|||\}}|jrHt|||\}}n:|jrbt|||\}}n |jr|t|||\}}n|j rt |||\}}n|j s|| }}t ||||}nd}|jr|}t||||\} }t ||||}n|}|jrLt|||\} }} t|| |\}}t|D]$\} \}} t|| | || f|| <q$nr|jrt|||\}} t|| |j\}}t|D]"\} \}} t|| || f|| <q~|||j}n td||jrt|D]$\} \}} t ||||| f|| <q|||}||| }|rt|D]V\} \}} t|||}t||||}t ||||}|| f|| <| ||!|| }q|||}|}tt"|D]J\} }|sqd|| dd| |j#i}|$dt%||||fq||t&|fS)=Factor multivariate polynomials into irreducibles in `K[X]`. Nr)rrr)'rr$rGrrrrrrrr rrrr rr@rrrrr r!r!r rr"rrYr"r9r>r#r$rrrrrV)rdrorrrrrer%rfr&Zlevelsrrrir'rZtermrlrlrmr bsj       r cCsj|st||St|||\}}|s2t||dfgSt|dd|||}||ddfg|ddSdS)r(rbrN)rr rr<)rdrorfrrerrlrlrmrs rcCs t|d|S)z_ Returns ``True`` if a univariate polynomial ``f`` has no factors over its domain. r)dmp_irreducible_p)rdrfrlrlrmdup_irreducible_psr*cCs@t|||\}}|sdSt|dkr(dS|d\}}|dkSdS)za Returns ``True`` if a multivariate polynomial ``f`` has no factors over its domain. TrbFrN)r r)rdrorfrrerirlrlrmr)s  r))F)NN)__doc__Zsympy.polys.galoistoolsrrrrrrrr r r r Zsympy.polys.densebasicr rrrrrrrrrrrrrrrrrrr r!r"r#r$Zsympy.polys.densearithr%r&r'r(r)r*r+r,r-r.r/r0r1r2r3r4r5r6r7r8r9r:r;r<r=r>Zsympy.polys.densetoolsr?r@rArBrCrDrErFrGrHrIrJrKrLrMZsympy.polys.euclidtoolsrNrOrPZsympy.polys.sqfreetoolsrQrRrSrTrUZsympy.polys.polyutilsrVZsympy.polys.polyconfigrWZsympy.polys.polyerrorsrXrYrZr[Z sympy.ntheoryr\r]r^Zsympy.utilitiesr_mathr`rxrarrnrprrrrrrrrrrrrrrrrrrrrrrr r rrrrrrrr rr*r)rlrlrlrmsd4hpD    < 99g L ,K60D4 A   B M