\hSrSSKJr SSKJr SSKJrJrJrJ r J r J r J r J r JrJrJr SSKJrJrJrJrJrJrJrJrJrJrJrJrJrJrJ r J!r!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r) SSK*J+r+J,r,J-r-J.r.J/r/J0r0J1r1J2r2J3r3J4r4J5r5J6r6J7r7J8r8J9r9J:r:J;r;Jr>J?r?J@r@JArAJBrBJCrCJDrD SSKEJFrFJGrGJHrHJIrIJJrJJKrKJLrLJMrMJNrNJOrOJPrPJQrQJRrRJSrSJTrT SSKUJVrVJWrWJXrX SS KYJZrZJ[r[J\r\J]r]J^r^J_r_J`r` SS KaJbrb SS KcJdrd SS KeJfrfJgrgJhrhJiri SS KjJkrk SSKlJmrnJorpJqrr \S:XaSSKsJtrt OSrtSruSrvSrwSrxSrySrzSr{Sr|Sr}S7Sjr~SrSrSrSrS rS!rS"rS#rS$rS%rS&rS8S'jrS(rS)rS*rS+rS,rS-rS.rS/rS0rS1rS2rS3rS4rS5rS6rg)9z:Polynomial factorization routines in characteristic zero. ) GROUND_TYPES)_randint) gf_from_int_polygf_to_int_poly gf_lshift gf_add_mulgf_mulgf_divgf_remgf_gcdexgf_sqf_p gf_factor_sqf gf_factor)dup_LCdmp_LC dmp_ground_LCdup_TC dup_convert dmp_convert dup_degree dmp_degree dmp_degree_indmp_degree_list dmp_from_dict dmp_zero_pdmp_onedmp_nest dmp_raise dup_strip dmp_ground dup_inflate dmp_exclude dmp_include dmp_inject dmp_eject dup_terms_gcd dmp_terms_gcd)dup_negdmp_negdup_adddmp_adddup_subdmp_subdup_muldmp_muldup_sqrdmp_powdup_divdmp_divdup_quodmp_quo dmp_expand dmp_add_mul dup_sub_mul dmp_sub_mul dup_lshift dup_max_norm dmp_max_norm dup_l1_normdup_mul_grounddmp_mul_grounddup_quo_grounddmp_quo_ground)dup_clear_denomsdmp_clear_denoms dup_truncdmp_ground_trunc dup_content dup_monicdmp_ground_monic dup_primitivedmp_ground_primitive dmp_eval_tail dmp_eval_indmp_diff_eval_in dup_shift dmp_shift dup_mirror) dmp_primitive dup_inner_gcd dmp_inner_gcd) dup_sqf_p dup_sqf_norm dmp_sqf_norm dup_sqf_part dmp_sqf_part_dup_check_degrees_dmp_check_degrees) _sort_factors)query)ExtraneousFactors DomainErrorCoercionFailedEvaluationFailed)subsets)ceilloglog2flint) fmpz_polyNc/nUHGnSn[XU5upgU(dXeS-pPOOMUS:Xa [S5eURXE45 MI [U5$)z Determine multiplicities of factors for a univariate polynomial using trial division. An error will be raised if any factor does not divide ``f``. rtrial division failed)r2 RuntimeErrorappendr[)ffactorsKresultfactorkqrs O/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/factortools.pydup_trial_divisionruXsmF 1a(DAa%1  667 7 vk"   c/nUHPnSn[XX#5upx[X5(aXvS-p`OOM'US:Xa [S5eURXV45 MR [ U5$)z Determine multiplicities of factors for a multivariate polynomial using trial division. An error will be raised if any factor does not divide ``f``. rrhri)r3rrjrkr[) rlrmurnrorprqrrrss rtdmp_trial_divisionrytsuF 1a+DA!a%1  667 7 vk"   rvc\SSKJn [U5n[US- 5n[US- 5nUR [ SU555nU"US- U5nU"US- US- 5nUR [X55n Xv-X--n U [X5- n [U S- 5S-n U $)ae 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] [Abbott13]_ r)binomialc3*# UH oS-v M g7f)r|N.0cfs rt (dup_zz_mignotte_bound..s0QrUQrh) (sympy.functions.combinatorial.factorialsr{r_ceilsqrtsumabsrr;) rlrnr{ddeltadelta2 eucl_normt1t2lcbounds rtdup_zz_mignotte_boundrsRB1 A !a%LE 519 F0Q002I %!)V $B %!)VaZ (B va| B NRW $E \! E %!) q E Lrvc[XU5n[[XU55n[[ X55nUR U"US-55SU--U-U-$)z7Mignotte bound for multivariate polynomials in `K[X]`. rhr|)r<rrrrr)rlrxrnabns rtdmp_zz_mignotte_boundrsXQ1A M! "#A OA !"A 66!AE( AqD  "1 $$rvcUS-n[XX65n[XU5n[[XHU5X65up[XU5n [XU5n [ [XXU5[XU5U5n [[ X+U5Xv5n [[ X:U5Xv5n [ [XLU5[X]U5U5n [[ XR /U5Xv5n[[XNU5X5unn[XU5n[UXv5n[ [X^U5[XU5U5n [[ UUU5Xv5n[[ X[U5Xv5nXUU4$)a One step in Hensel lifting in `Z[x]`. Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s` and `t` such that:: f = g*h (mod m) s*g + t*h = 1 (mod m) lc(f) is not a zero divisor (mod m) lc(h) = 1 deg(f) = deg(g) + deg(h) deg(s) < deg(h) deg(t) < deg(g) returns polynomials `G`, `H`, `S` and `T`, such that:: f = G*H (mod m**2) S*G + T*H = 1 (mod m**2) References ========== .. [1] [Gathen99]_ r|)r8rDr2r.r*r,one)mrlghstrnMerrrsrxGHrcrSTs rtdup_zz_hensel_steprsH8 1AA!A!A 71#Q *DA!A!Aa '!"2A6A'!"A)A'!"A)Aa '!"2A6A'!eeWa(!/A 71#Q *DAq!A!QAa '!"2A6A'!Q"A)A'!"A)A A:rvc [U5n[X5nUS:Xa1[XRX`U-5SU5n[ XpU-U5/$UnUS-n [ [ [U555n [U/U5n USU Hn [U [X5X5n M [X)U5n X)S-SHn [U [X5X5n M [XX5upn[X5n [X5n [X5n[X5n[SU S-5Hn[XXXU5US-suppnM [X USU X45[X X)SX45-$)a Multifactor Hensel lifting in `Z[x]`. Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)` is a unit modulo `p`, monic pair-wise coprime polynomials `f_i` over `Z[x]` satisfying:: f = lc(f) f_1 ... f_r (mod p) and a positive integer `l`, returns a list of monic polynomials `F_1,\ F_2,\ \dots,\ F_r` satisfying:: f = lc(f) F_1 ... F_r (mod p**l) F_i = f_i (mod p), i = 1..r References ========== .. [1] [Gathen99]_ rhrr|N)lenrr>gcdexrDintr_log2rr r rrangerdup_zz_hensel_lift)prlf_listlrnrsrFrrqrrf_irrr_s rtrrsq. F A BAv 1ggbQ$/2A 61dA&(( A QA E%(OA"q!Abqz 1&s. 5 A&A!ef~ 1&s. 5qQ"GA!qAqAqAqA 1a!e_,Q1qA1a4 qa aF2AJ 5 Q6":q 4 55rvc8XS-:aX- nU(dgX-S:H$)Nr|Trr~)fcrrpls rt_test_plrGs$7{ F  6Q;rvc F[U5nUS:XaU/$SSKJn USn[X5n[ X5n[ [ URU"US-55SU--U-U-55n[ US-SU--USU-S- --5n[ [S[U5-55n [ SU -[U 5-5n /n [SU S-5Hn U"U 5(aXl-S:XaMURU 5n [X 5n [XU5(dMI[XU5SnU R!X45 [#U5S:d[#U 5S:dM O [%U S S 9unn[ [[SU-S-U555nUVs/sHn['UU5PM nn[)XUUU5n[[#U55n[+U5n/SnnUU-nSU-[#U5::Ga[-UU5GHjnUS:Xa0SnUHnUUUS-nM UU-n[/UUU5(dM9OOU/nUHn[1UUUU5nM [3UUU5n[5UU5SnUSnU(a UU-S:waMU/n[+U5nUU- nUS:Xa)U/nUHn[1UUUU5nM [3UUU5nUHn[1UUUU5nM [3UUU5n[7WU5n [7UU5n!U U!-U::dGMUnUVs/sH nUU;dM UPM nn[5UU5Sn[5UU5SnUR!U5 [ X5n O US- nSU-[#U5::aGMUU/-$s snfs snf) z4Factor primitive square-free polynomials in `Z[x]`. rhr)isprimer|c[US5$)Nrh)r)xs rt#dup_zz_zassenhaus..ps 3qt9rv)key)r sympy.ntheoryrr;rrrrrr_logrconvertrr rrkrminrrsetrarr.rDrIr=)"rlrnrrrArBCgammarrpxrfsqfxrfsqfrffmodularrsorted_TrrmrrrrrirrT_SG_normH_norms" rtdup_zz_zassenhausrNs1 AAvs % 2BQAq A CqQx A%a') *+A QUacN1qsQw< '(A aaj! "E %U # $E AAuqy!r{{afk  YYr] Q #q!! aQ'* " u:?c!fqj "!,-GAt E$qsQw" #$A/34t~b!$tG41!Q/ASV}H H AQQG AB A#Q-1%A AvA!A$r( AFAr**+CA1Q4+AaQ'!!Q'*bEa1AAAa%CAvCA1Q4+AaQ'AqtQ'!R#A A&F A&Ff}!'/>x!1A:Ax>!!Q'*!!Q'*q!1Le&h FAk A#Q-n aS=A5h?sN% N3Nc[X5n[X5n[USSU5nU(aHSSKJn U"[ U55nUR 5HnX'-(dMX7S--(dM g gg)z2Test irreducibility using Eisenstein's criterion. rhNr factorintr|T)rrrFrrrkeys)rlrnrtce_fcre_ffrs rtdup_zz_irreducible_prsc B B qua D +T#ARQ$YY  rvcUR(aXR5p[XU5nOUR(dg[ X5n[ X5nUS:wd US:waUS:wagU(d%[X5upgXaR:wdXpS4/:wag[U5n//p[USS5Hn U RSX 5 M [US- SS5Hn U RSX 5 M [[U 5U5n [[U 5U5n [U [U SU5U5n UR![ X55(a [#X5n X:Xag[%X5n UR![ X55(a [#X5n X:Xa['X5(ag[)X5n [X5U :Xa['X5(agg![a gf=f)a  Efficiently test if ``f`` is a cyclotomic polynomial. Examples ======== >>> 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 References ========== Bradford, Russell J., and James H. Davenport. "Effective tests for cyclotomic polynomials." In International Symposium on Symbolic and Algebraic Computation, pp. 244-251. Springer, Berlin, Heidelberg, 1988. FrhrrT)is_QQget_ringrr_is_ZZrrdup_factor_listrrrinsertr0rr,r: is_negativer(rPdup_cyclotomic_prW)rlrn irreducibleK0rrcoeffrmrrrrrrs rtrrs4 ww zz|A1%AWW B B Qw28a (. EE>WQ01 A rq 1b"  AD1q5"b ! AD"  ! a A ! a A:aA&*A}}VA\"" AMv1A}}VA\"" AMv"1((QAq}.q44 e  sG!! G.-G.cSSKJn URUR*/nU"U5R5H-upE[ [ X4U5X15n[ X4US- -U5nM/ U$)z1Efficiently generate n-th cyclotomic polynomial. rrrh)rrritemsr4r!)rrnrrrrqs rtdup_zz_cyclotomic_polyrs_' A! ""$ Ka(! / q1u:q )% Hrvc SSKJn URUR*//nU"U5R5H|upEUVs/sHn[ [ XdU5Xa5PM nnUR U5 [SU5H0nUV s/sHn [ XU5PM nn UR U5 M2 M~ U$s snfs sn f)Nrrrh)rrrrr4r!extendr) rrnrrrrqrQrrrs rt_dup_cyclotomic_decomposer&s' %%!%%A! ""$;< >1agk!*A11 >  q!A0131+aA&A3 HHQK % H ?4s B9B>c\[X5[X5p2[U5S::agUS:wdUS;ag[SUSS55(ag[U5n[ XA5nUR U5(dU$/n[ SU-U5HnXu;dM UR U5 M U$)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]_ rNrh)rrhc38# UHn[U5v M g7f)N)boolrs rtr+dup_zz_cyclotomic_factor..Ps &g488gsrr|)rrranyris_onerk)rlrnlc_ftc_frrrrs rtdup_zz_cyclotomic_factorr6s$va|$!} qyD' &a"g &&&1 A!!'A 88D>> *1Q32Az 3rvcH[X5up#[U5n[X15S:a U*[X15p2US::aU/4$US:XaX#/4$[ S5(a[ X15(aX#/4$Sn[ S5(a [ X15nUc [X15nU[USS94$)z:Factor square-free (non-primitive) polynomials in `Z[x]`. rrhUSE_IRREDUCIBLE_IN_FACTORNUSE_CYCLOTOMIC_FACTORF)multiple) rIrrr(r\rrrr[)rlrncontrrrms rtdup_zz_factor_sqfrbsA!GD1 A a|a%aAvRx aSy ())  % %9 G $%%*10#A) w7 77rvcZ[S:Xa\[USSS25nUR5up4UVVs/sHupVUR5SSS2U4PM nnnU[ U54$[ X5up7[ U5n[Xq5S:a U*[Xq5psUS::aU/4$US:XaX7S4/4$[S5(a[Xq5(aX7S4/4$[Xq5nSn [S5(a [Xq5n U c [Xq5n [X U5n[X5 X44$s snnf)a Factor (non square-free) polynomials in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, ..., f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Zassenhaus algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Examples ======== Consider the polynomial `f = 2*x**4 - 2`:: >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_zz_factor(2*x**4 - 2) (2, [(x - 1, 1), (x + 1, 1), (x**2 + 1, 1)]) In result we got the following factorization:: f = 2 (x - 1) (x + 1) (x**2 + 1) Note that this is a complete factorization over integers, however over Gaussian integers we can factor the last term. By default, polynomials `x**n - 1` and `x**n + 1` are factored using cyclotomic decomposition to speedup computations. To disable this behaviour set cyclotomic=False. References ========== .. [1] [Gathen99]_ reNrrrhrr)rrfrpcoeffsr[rIrrr(r\rrWrrrurY) rlrnf_flintrrmfacexprrrs rt dup_zz_factorrs6\wAddG$( =DEWCJJL2&,WE]7+++A!GD1 A a|a%aAvRx a!fX~ ())  % %a&> !QA A $%% $Q *y a # q)Gq" =AFs$D'cX-/nUHmn[U5n[U5H?nUS:waURXe5nXV-nUS:waMURU5(dM> g UR U5 Mo USS$)z,Wang/EEZ: Compute a set of valid divisors. rhN)rreversedgcdrrk)Ecsctrnrorrrss rtdmp_zz_wang_non_divisorsr suYF  F&!Aq&EE!KFq&xx{{ "  a !":rvc [[X5X4S- U5(d [S5e[XXE5n[Xe5(d [S5e[ Xe5upxUR [ X55(a U*[X5pUS- n UV V s/sHup[XX5PM n n n [XX%5n U bXxU 4$[S5es sn n f)z2Wang/EEZ: Test evaluation points for suitability. rhzno luck) rKrr`rTrIrrr(r )rlrr rrxrnrrrvrrr Ds rtdmp_zz_wang_test_pointsrs qa% 3 3y))aA!A Q??y))  DA}}VA\""r71=1 AA013-a #A3 r-A}Qwy)) 4sC c  /S/[U5-US- pnUHn [X5n [X5U-n [[ [U555HUnSX>XsnnunnU U-(dU U-US-pU U-(dMUS:wdM8[ U [ UXU5X5SsoU'MW URU 5 M [U 5(d[e//nn[X5Hup[XX5n [X5nURU5(aUU -nO+URUU 5nU U-UU-nn [XU5X--p+[U UX5n URU 5 URU 5 M URU5(aUUU4$//nn[UU5H<upUR[XX55 UR[XSU55 M> [X[U5S- -Xg5nUUU4$)z0Wang/EEZ: Compute correct leading coefficients. rrh)rrrrrr/r1rkallr]ziprKrr r>r?)rlrr r rrrxrnrJrrrrrrqrrrCCHHrccrCCCHHHs rtdmp_zz_wang_lead_coeffsrs1#c!f*a!e!A  AM 1LO%A-(AadADLAq&1a1u!tQU11uuAv!!WQa%8!?Q4)   q66 BA  ! % A\ 88B<<QBb! AqD"a%rA"1+RUr 1b! ' !  !   xx||"by2CB  >!./ >!A./ qs1vz*A1A c3;rvc [U5S:XawUupE[XB5n[XR5n[XvX#5upn [XU5n[XU5n [ XX#5up[ XXrU5n [ X5n[ X5n X/n U $US/n [USS5H"nU RS[XjSU55 M$ /S//p[X 5H>upg[Xv/US/SUSU5upURU 5 U RU5 M@ /XS/-p[X5HNup[X5n[Xb5n[[XU5XbU5n[ X5nU RU5 MP U $)z2Wang/EEZ: Solve univariate Diophantine equations. r|rrhr)rrr rr rrrrr.rdmp_zz_diophantinerkr )rrrrnrrrlrrrrrrrorrrss rtdup_zz_diophantiner7s 1v{ Q " Q "1&a aA  aA aA! qQ1 % 1  1 2 M/rUG!Ab'"A HHQQ4+ ,#QC51IDA%qfaeRAq!DDA HHQK HHQK rUG IDA &A &Ayq)13Aq$A MM!  Mrvc U(dUVs/sHn/PM nn[U5n [U5HaupU (dM[X U - XF5n [[X55H,un up[ XU5n[ [ XU5XF5X'M. Mc U$[U5n [XU5nUSUSSnn//nnUH<nUR[UUXV55 UR[UUXU55 M> [UUXU5nUS- n[UUX#UUU5nUVs/sHn[USUU5PM nn[UU5Hunn[XUXV5nM [XXV5n[!UR"U*/X5n[%X5n['SU5GH n[)X5(a O[+UUXV5n[-UUS-UXU5n[)UU5(aMH[/UUR1U"U5S-5UU5n[UUX#UUU5n [U 5H up[+[USUU5UXV5X'M" [[X55Hun up[3XXV5X'M [U U5Hunn[XUXV5nM [XXV5nGM UVs/sHn[XXV5PM nnU$s snfs snfs snf)z4Wang/EEZ: Solve multivariate Diophantine equations. rNrhr)r enumeraterrr>rDr*rr6rkr5rLrrr9rErrrrrr/rMrA factorialr+)rrrrrrxrnrrrrrrjrrrrrrrlrrrrrrqs rtrrgs  !Qb!  qM!! HA"1!eQ2A&s1y1 6A"1Q/ q!1182 %v Hc F qQ uaf121A HHWQ1( ) HH[AqQ/ 0 1aA & E q!Q1a 3-. 0Qi1a#Q 01IDAqA!Q*A Q1 ( aeeaR[! ' AMq!A!1a#A AE1aA6Aa##"1akk!A$(&;QB&q!Q1a;%aLDA"9Q1a#8!QBAD)"+3q9!5IAv"1.AD"6 1IDAq#A!Q2A&%Q10),56 7AqqQ*A 7 H} 8 1@ 8s K %K1Kc 8U/[U5US- pn[U5n[[USS55H:up[ USXU - XZ- U5n UR S[ XX- U55 M< [[X5SS5n [[SUS-5Xs5GHupn [U5US- nnUSUS- X>S- Snn[[X55H?un unn[ [UUX5UUS- U5nU/[USSSUS- U5-X'MA [URU */UU5n[UU5n[!U [#UUU5UU5n[%U UU5n[SU5GHn['UU5(a M[)UUUU5n[+UUS-U UUU5n['UUS- 5(aMO[-UUR/U"U5S-5US- U5n[1UUUXUS- U5n[[UU55H7un unn[3U[USUS- U5UUU5n[ UUUU5X'M9 [!U [#UUU5UU5n[ UUUU5nGM GM [#XU5U:wa[4eU$)z-Wang/EEZ: Parallel Hensel lifting algorithm. rhNrr|)rlistr rrLrrEmaxrrrrKrrrrr-r6rrr/rMrAr!rr7r])rlrLCrrrxrnrrrrrrrr"rwIrrrrrrdjrqrrrs rtdmp_zz_wang_hensel_liftingr*sc3q61q5!A QA(1QR5/* !aQq 1 $Q15!45+ OA !!" %&AuQA-aAwA1!a%y!EF)1#CJ/JAw2!-Aq" 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) nextprimerhNr|EEZ_NUMBER_OF_CONFIGSEEZ_NUMBER_OF_TRIESEEZ_MODULUS_STEP)NrrEEZ_RESTART_IF_NEEDEDz3we need to restart algorithm with better parameters)rr,r dmp_zz_factorrrrzerorrrr`r\rtupleaddrkr;rr*r] dmp_zz_wangrJrrr)) rlrxrnmodseedr,randintr rrrhistoryconfigsrrsr rr rreez_num_configs eez_num_tries eez_mod_steprrs_norms_argr_s_normorig_fr&rmros rtr5r5sc<(tnG &,Aq 1EBaA&A )A,A { 6CC UB D8Ga  *1=Aq A&1 F 63Jr1a#$34O/0M+,L g, (}%A16q;A!GSD#&'A;Qxw& E!H% 21EAq%Q*DAqQB}7Av%'  Avs NNAr1a+ ,7|.A&D < CG g, (J"FE1 1aAq!$   F Q!U^NAr1a FG*1Q1C1b,Q2qQBF #A!,1 ==qQ/ 0 0a A a  Mc    <$  \ G ( ) )vqS1W5 5#EG G GsB53J)J<J%J*#J< J"!J"* J98J9<*K4( K4cU(d [X5$[X5(aUR/4$[XU5up4[ XAU5S:aU*[ XAU5pC[ S[XA555(aU/4$[XAU5upT/n[XA5S:a$[XAU5n[XAU5n[XX5n[XQS- U5SHupHURSU/U45 M [XU5 U[!U54$)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]_ rc3*# UH oS:*v M g7frNr~rrs rtr dmp_zz_factor.. 10a60rrh)rrr2rJrr)rrrQrrXr5ryr1rrZr[) rlrxrnrrrrmrrqs rtr1r1msH Q""!vvrz"1+GDQ1!%q)a 1?10 111Rx q !DAG!! q ! a $Q10aQ*1-qA3(#.qW% w' ''rvc UR5n[XU5n[X5up4UVVs/sHupV[XRU5U4PM nnnURX25nX44$s snnf)z>Factor univariate polynomials into irreducibles in `QQ_I[x]`. )as_AlgebraicFieldrrr)rlrK1rrmrrs rtdup_qq_i_factorrLsh   BA2A$Q+NE;BC7 CR(!,7GC JJu !E >DsA c&UR5n[XU5n[X5up4/nUHKupg[Xb5up[XU5n [ U SU5upX;U--X--nUR X45 MM UnUR X25nX44$)z>Factor univariate polynomials into irreducibles in `ZZ_I[x]`. r) get_fieldrrLrBrJrkr) rlrrKrrm new_factorsrr fac_denomfac_num fac_num_ZZ_Icontentfac_prims rtdup_zz_i_factorrUs BA2A$Q+NEK-c6 "73 0q"EA%).8H=)G JJu !E >rvc UR5n[XX#5n[XU5upEUVVs/sHupg[XaX25U4PM nnnURXC5nXE4$s snnf)z@Factor multivariate polynomials into irreducibles in `QQ_I[X]`. )rJrdmp_factor_listr)rlrxrrKrrmrrs rtdmp_qq_i_factorrXsj   BA"!A$Q2.NE>EFgFC CB+Q/gGF JJu !E >GsA!c(UR5n[XX#5n[XU5upE/nUHKupx[XqU5up[XX25n [ XU5upXLU--X--nUR X45 MM UnUR XC5nXE4$)z@Factor multivariate polynomials into irreducibles in `ZZ_I[X]`. )rNrrXrCrJrkr)rlrxrrKrrmrOrrrPrQrRrSrTs rtdmp_zz_i_factorrZs BA"!A$Q2.NEK-cb9 "7r6 0"EA%).8H=)G JJu !E >rvc[U5[X5p2[X5nUS::aU/4$US:XaX0S4/4$[X5Up@[ X5upVn[ XqR 5n[U5S:XaX0U[U5-4/4$XQR-n [U5H=un up[XR U5n [XU5upn[XU5n XU 'M? [XHU5n[XH5 X84$)aFactor univariate polynomials over algebraic number fields. The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). Examples ======== First define the algebraic number field `K = \mathbb{Q}(\sqrt{2})`: >>> from sympy import QQ, sqrt >>> from sympy.polys.factortools import dup_ext_factor >>> K = QQ.algebraic_field(sqrt(2)) We can now factorise the polynomial `x^2 - 2` over `K`: >>> p = [K(1), K(0), K(-2)] # x^2 - 2 >>> p1 = [K(1), -K.unit] # x - sqrt(2) >>> p2 = [K(1), +K.unit] # x + sqrt(2) >>> dup_ext_factor(p, K) == (K.one, [(p1, 1), (p2, 1)]) True Usually this would be done at a higher level: >>> from sympy import factor >>> from sympy.abc import x >>> factor(x**2 - 2, extension=sqrt(2)) (x - sqrt(2))*(x + sqrt(2)) Explanation =========== Uses Trager's algorithm. In particular this function is algorithm ``alg_factor`` from [Trager76]_. If `f` is a polynomial in `k(a)[x]` then its norm `g(x)` is a polynomial in `k[x]`. If `g(x)` is square-free and has irreducible factors `g_1(x)`, `g_2(x)`, `\cdots` then the irreducible factors of `f` in `k(a)[x]` are given by `f_i(x) = \gcd(f(x), g_i(x))` where the GCD is computed in `k(a)[x]`. The first step in Trager's algorithm is to find an integer shift `s` so that `f(x-sa)` has square-free norm. Then the norm is factorized in `k[x]` and the GCD of (shifted) `f` with each factor gives the shifted factors of `f`. At the end the shift is undone to recover the unshifted factors of `f` in `k(a)[x]`. The algorithm reduces the problem of factorization in `k(a)[x]` to factorization in `k[x]` with the main additional steps being to compute the norm (a resultant calculation in `k[x,y]`) and some polynomial GCDs in `k(a)[x]`. In practice in SymPy the base field `k` will be the rationals :ref:`QQ` and this function factorizes a polynomial with coefficients in an algebraic number field like `\mathbb{Q}(\sqrt{2})`. See Also ======== dmp_ext_factor: Analogous function for multivariate polynomials over ``k(a)``. dup_sqf_norm: Subroutine ``sqfr_norm`` also from [Trager76]_. sympy.polys.polytools.factor: The high-level function that ultimately uses this function as needed. rrh)rrrGrWrUdup_factor_list_includedomrunitr rrRrNrurY)rlrnrrrrrrsrmrrrprrs rtdup_ext_factorr_sD qM6!U(d [X5$[XU5n[XU5n[S[ X555(aU/4$[ XU5Up@[ XU5upVn[XqUR5n[U5S:XaU/nOk[U5H\un up[XURU5n [XX5upnUV s/sHoUR-PM nn [XX5n XU 'M^ [XHX5n[!XAU5 X?4$s sn f)a.Factor multivariate polynomials over algebraic number fields. The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). Examples ======== First define the algebraic number field `K = \mathbb{Q}(\sqrt{2})`: >>> from sympy import QQ, sqrt >>> from sympy.polys.factortools import dmp_ext_factor >>> K = QQ.algebraic_field(sqrt(2)) We can now factorise the polynomial `x^2 y^2 - 2` over `K`: >>> p = [[K(1),K(0),K(0)], [], [K(-2)]] # x**2*y**2 - 2 >>> p1 = [[K(1),K(0)], [-K.unit]] # x*y - sqrt(2) >>> p2 = [[K(1),K(0)], [+K.unit]] # x*y + sqrt(2) >>> dmp_ext_factor(p, 1, K) == (K.one, [(p1, 1), (p2, 1)]) True Usually this would be done at a higher level: >>> from sympy import factor >>> from sympy.abc import x, y >>> factor(x**2*y**2 - 2, extension=sqrt(2)) (x*y - sqrt(2))*(x*y + sqrt(2)) Explanation =========== This is Trager's algorithm for multivariate polynomials. In particular this function is algorithm ``alg_factor`` from [Trager76]_. See :func:`dup_ext_factor` for explanation. See Also ======== dup_ext_factor: Analogous function for univariate polynomials over ``k(a)``. dmp_sqf_norm: Multivariate version of subroutine ``sqfr_norm`` also from [Trager76]_. sympy.polys.polytools.factor: The high-level function that ultimately uses this function as needed. c3*# UH oS:*v M g7frEr~rFs rtr!dmp_ext_factor..rHrrh)r_rrHrrrXrVdmp_factor_list_includer]rr rrSr^rOryrZ)rlrxrnrrrrrsrmrrprrsirros rtdmp_ext_factorreTs^ a## qQ Bq!A 1?10 1112v a !q1#GA!%aAEE2G 7|q#'0NA{Fquua0A#A!/GA!%&'QrAFFQA'!%AAJ 1 A 1FqV$ :(sDc[XUR5n[XRUR5up#[ U5H"unup[XRU5U4X4'M$ UR X!R5U4$)z2Factor univariate polynomials over finite fields. )rr]rr6r r)rlrnrrmrrqs rt dup_gf_factorrgsnA!%% Aq%%/NEw' 6A!!UUA.2 ( 99UEE "G ++rvc[S5e)z4Factor multivariate polynomials over finite fields. z+multivariate polynomials over finite fields)NotImplementedError)rlrxrns rt dmp_gf_factorrjs K LLrvc[X5up [X5up0UR(a[X5upEGOJUR(a[ X5upEGO*UR (a[X5upEGO UR(a[X5upEGOUR(dXR5p[XU5nOSnUR(a+UR5n[XU5up[XU5nOUnUR (a[#X5upEOUR$(aj['USU5up [)X UR*5upE[-U5Hun up [/X U5U 4XZ'M UR1XGR*5nO[3SU-5eUR(a[-U5Hun up [XU5U 4XZ'M UR1XG5nUR5UW5nU(ar[-U5HPun up [7X5n [9X U5n[XU5nX 4XZ'UR;XAR=X55nMR UR1XA5nUnU(a*UR?SUR@URB/U45 XC-[EU54$);Factor univariate polynomials into irreducibles in `K[x]`. Nr#factorization not supported over %s)#r&rIis_FiniteFieldrg is_Algebraicr_is_GaussianRingrUis_GaussianFieldrLis_Exact get_exactris_FieldrrBrris_Polyr$rWr]r r%rr^quor;r@mulpowrrr2r[) rlrr"rrrm K0_inexactrndenomrxrrqmax_norms rtrrsH  DAA"GD &q-w '.w  (/w  (/w{{A2.AJ ;; A'q1HEA1%AA 77*10NE7 YYaA&DA,Q1559NE&w/ 6A'a0!4 0IIeUU+ECbHI I ;;&w/ 6A)!3Q7 0JJu(EFF5%(E!*7!3IAv+A2H&qB7A#A:6A"#GJFF5&&*=>E "4#**55qBFFBGG,a01 :}W- --rvc[X5up#U(d[U/5S4/$[USSX!5nXCSS4/USS-$)rlrhrN)rrr>)rlrnrrmrs rtr\r\sY$Q*NE E7#Q'(( 71:a=% 3AJqM"#gabk11rvc U(d [X5$[XU5up0[XU5up@UR(a[ XU5upVGOUR (a[ XU5upVGOcUR(a[XU5upVGOBUR(a[XU5upVGO!UR(dX"R5p'[XXr5nOSnUR(a+UR5n[!XX(5up[XX(5nOUnUR"(aE[%XU5upn ['X U5upV[)U5Hun up [+X X5U 4Xl'M OUR,(ai[/XU5up [1X UR25upV[)U5Hun up [5X U5U 4Xl'M UR7XXR25nO[9SU-5eUR(a[)U5Hun up [XX5U 4Xl'M UR7XX5nUR;UW 5nU(as[)U5HQun up [=XU5n[?XX5n[XX'5nX 4Xl'URAXRRCX55nMS UR7XR5nUn[)[EU55HHupU(dMSX- -S-SU --URF0nURIS[KUX5U45 MJ XT-[MU54$)=Factor multivariate polynomials into irreducibles in `K[X]`. Nrm)r)rhr)'rr'rJrnrjrorerprZrqrXrrrsrrtrrCrr"r1r r#rur$rWr]r%rr^rvr<rArwrxrrrrr[)rlrxrrrrrmryrnrzlevelsrrrqr{r"terms rtrWrWs q%% r "DA"1,GD &qR0w 'b1w  (r2w  (r2w{{A*1AJ ;; A'b4HEA"(AA 77&qQ/LFq*13NE&w/ 6A)!Q:A> 0 YYaA&DA,Q1559NE&w/ 6A'a0!4 0IIeUU+ECbHI I ;;&w/ 6A)!6: 0JJu(EFF5%(E!*7!3IAv+A"5H&qA:A#A"9A"#GJFF5&&*=>E "4#**55(1+& ae t#d1f,bff5q=q5q9: ' :}W- --rvcU(d [X5$[XU5up4U(d[X15S4/$[USSX1U5nXTSS4/USS-$)r~rhrN)r\rWr r?)rlrxrnrrmrs rtrcrcMsi &q,,$Q1-NE E%q)** 71:a=%A 6AJqM"#gabk11rvc[USU5$)zS Returns ``True`` if a univariate polynomial ``f`` has no factors over its domain. r)dmp_irreducible_p)rlrns rtdup_irreducible_pr[s Q1 %%rvcf[XU5up4U(dg[U5S:agUSup5US:H$)zU Returns ``True`` if a multivariate polynomial ``f`` has no factors over its domain. TrhFr)rWr)rlrxrnrrmrqs rtrrcs; !q)JA  W qzAv rv)F)NN)__doc__sympy.external.gmpyrsympy.core.randomrsympy.polys.galoistoolsrrrrr r r r r rrsympy.polys.densebasicrrrrrrrrrrrrrrrrr r!r"r#r$r%r&r'sympy.polys.densearithr(r)r*r+r,r-r.r/r0r1r2r3r4r5r6r7r8r9r:r;r<r=r>r?r@rAsympy.polys.densetoolsrBrCrDrErFrGrHrIrJrKrLrMrNrOrPsympy.polys.euclidtoolsrQrRrSsympy.polys.sqfreetoolsrTrUrVrWrXrYrZsympy.polys.polyutilsr[sympy.polys.polyconfigr\sympy.polys.polyerrorsr]r^r_r`sympy.utilitiesramathrbrrcrrdrrerfruryrrrrrrrrrrrrrr rrrrr*r5r1rLrUrXrZr_rergrjrr\rWrcrrr~rvrtrs@,&""""""""$$$$$$$$&&&&&""0(FF$::7I!8!8:x%6r75rfR  Pf    )X8:Qh(*43l-`A H1hK\@(F,,_DK\ ,M ?.D2J.Z 2& rv