Ed-[dZddlmZddlmZmZmZmZddlm Z ddl m Z ddl m Z d$dZd Zd Zd Zd Zd ZdZdZdZdZdZdZdZdZdZdZdZdZdZ dZ!dZ"dZ#dZ$dZ%d Z&d!Z'd"Z(d#Z)dS)%zGroebner bases algorithms. )Dummy) monomial_mul monomial_lcmmonomial_dividesterm_div)lex) DomainError)queryNc|td}ttd} ||}n #t$rt d|zwxYwjdc}|jr|js[ | cfd|D}n #t$rtd|zwxYw||}fd|D}|S) aa Computes Groebner basis for a set of polynomials in `K[X]`. Wrapper around the (default) improved Buchberger and the other algorithms for computing Groebner bases. The choice of algorithm can be changed via ``method`` argument or :func:`sympy.polys.polyconfig.setup`, where ``method`` can be either ``buchberger`` or ``f5b``. Ngroebner) buchbergerf5bzO'%s' is not a valid Groebner bases algorithm (valid are 'buchberger' and 'f5b'))domainc:g|]}|S)set_ring).0srings 9lib/python3.11/site-packages/sympy/polys/groebnertools.py zgroebner..)s%333AJJt$$333z'Cannot compute a Groebner basis over %scjg|]/}|d0S) clear_denomsr)rgorigs rrzgroebner...s6 = = =Qannq!**400 = = =r) r _buchberger_f5bKeyError ValueErrorris_Fieldhas_assoc_Fieldclone get_fieldr )seqrmethod_groebner_methods _groebnerrGrs ` @rr r sN#z""" u%f- uuujmsstttu;LFD ?4&"84 4tzz1A1A1C1CzDDJD$4333c333CC R R RG&PQQ Q R  #tA > = = = =! = = = Hs-A &+B B=c|j|j|j|jfd}fd}fd}sgSdd} |ddg}t t D]P}|}|d|}|r'||Q|krnit} t} t} tD]\}} || <| | | rOtfd| Dfd} | } | | || | | \} } | Od }| r|| \}}| ||ft|||} t| fd }|| |}|r|| | |d \} } n|d z }| t}| D]5}||| |hz }|r| |d 6fd |D}t|fd d}|S)aK Computes Groebner basis for a set of polynomials in `K[X]`. Given a set of multivariate polynomials `F`, finds another set `G`, such that Ideal `F = Ideal G` and `G` is a reduced Groebner basis. The resulting basis is unique and has monic generators if the ground domains is a field. Otherwise the result is non-unique but Groebner bases over e.g. integers can be computed (if the input polynomials are monic). Groebner bases can be used to choose specific generators for a polynomial ideal. Because these bases are unique you can check for ideal equality by comparing the Groebner bases. To see if one polynomial lies in an ideal, divide by the elements in the base and see if the remainder vanishes. They can also be used to solve systems of polynomial equations as, by choosing lexicographic ordering, you can eliminate one variable at a time, provided that the ideal is zero-dimensional (finite number of solutions). Notes ===== Algorithm used: an improved version of Buchberger's algorithm as presented in T. Becker, V. Weispfenning, Groebner Bases: A Computational Approach to Commutative Algebra, Springer, 1993, page 232. References ========== .. [1] [Bose03]_ .. [2] [Giovini91]_ .. [3] [Ajwa95]_ .. [4] [Cox97]_ c4t|fd}|S)Ncr|dj|djS)NrrLM)pairfrorders rz-_buchberger..select..ds1UU<<$q' qaz}+U+U%V%Vrkey)min)Pprr2rr3s rselectz_buchberger..selectas,VVVVVV W W W rc|fd|D}|sdS|}|vr't|<||j|fS)Nc g|] }| Srr)rjr2s rrz/_buchberger..normal..hs%%%QAaD%%%r)remmoniclenappendr0)rJhIr2s rnormalz_buchberger..normalgs} EE%%%%!%%% & & 4 Az 1vv! 41: rcx|}|j|}t}|r|}|}|j}|fd|ks6t fd|Ds2t fd|Ds|||f|t} |rY|\}}|j}||ks| ||f|Yt} |r|\} } | j} | j}| |}|r | |ks||kr| | | f|| | z} t}|rD|}|j}|s|||D|||| fS)NcJ|j}|SNr/)ipmLCMhgr2mh monomial_divrs r lcm_dividesz0_buchberger..update..lcm_dividess, LQrUX..#|E1---rc3.K|]}|VdSrHr)ripxrNs r z._buchberger..update..s-66S C((666666rc3:K|]}|dVdSrNr)rr9rNs rrQz._buchberger..update..s1;;2KK1..;;;;;;r)r0copysetpopanyadd)r+BihrCCDigrmgEB_newig1ig2mg1mg2LCM12G_newrKrNrLr2rMrrs @@@rupdatez_buchberger..updateus bE T FFHH EE B"AB LR((E . . . . . . . . . |B##u, 6666A66666 ;;;;;;;;; r2h! $ EE UUWWFB2B LR((E<B''50 r2h  &uuwwHCC&)CC&)C Lc**E <r** & S"%%. & Lb))U2 & 3*%%% &   B2B<B''  "    " e|rNTc g|] }| Srr)rxr2s rrz_buchberger..s!!!!1!!!rc$|jSrHr/r2r3s rr4z_buchberger..sqtrr5rc0|jSrHr/)rr2r3s rr4z_buchberger..sUU1Q47^^rrc g|] }| Srr)rr]r2s rrz_buchberger..s   B!B%   rc$|jSrHr/rks rr4z_buchberger..s%%++rr6reverse)r3rrMrranger@r>rAr?rU enumeraterXr7removespolysorted)r2rr:rErgf1iprFr+CPrCrZreductions_to_zerorarbG1htGrr]rDrMrrr3s` @@@@@rrr2sSR JE$L$L$L      EEEEEEEEP   111B  qqqE s1vv % %A!Aae A % !''))$$$ 7    A A A B! 1! a " !!!!q!!!'<'<'<'< = = = qT  q"b!!2 " $6"::S 3* !C&!C&$ ' ' A33333 4 4 4 VAr]]  $F1b"Q%((EArr ! #  $ B VAbE1t8 $ $   FF2a5MMM    "   B ----t < < v otherwise rrr)uvr3s rsig_cmpr0sb tad{rtqt| 51;;qt $ 2 1rc6|d ||dfS)z Key for comparing two signatures. s = (m, k), t = (n, l) s < t iff [k > l] or [k == l and m < n] s > t otherwise rrr)rr3s rsig_keyrEs!qTE551;; rcVtt|d||dS)z Multiply a signature by a monomial. The product of a signature (m, i) and a monomial n is defined as (m * t, i). rr)rr)rrJs rsig_multrQs& |AaD!$$ad + ++rc,tt|t|t|jjdkr|}n|}t|t|z }t t||t |S)z Subtract labeled polynomial g from f. The signature and number of the difference of f and g are signature and number of the maximum of f and g, w.r.t. lbp_cmp. r)rrrrr3rr)r2rmax_polyrets rlbp_subr]sttAwwQq!4559 ((U1XX C tH~~sCMM 2 22rcttt||dt||t |S)z Multiply a labeled polynomial with a term. The product of a labeled polynomial (s, p, k) by a monomial is defined as (m * s, m * p, k). r)rrrrmul_termr)r2cxs r lbp_mul_termrnsD xQA''q):):2)>)>A G GGrctt|t|t|jjdkrdSt|t|kr"t |t |krdSdS)z Compare two labeled polynomials. f < g iff - Sign(f) < Sign(g) or - Sign(f) == Sign(g) and Num(f) > Num(g) f > g otherwise rr)rrrrr3r)r2rs rlbp_cmprxsvtAwwQq!455;r Aww$q'' q66CFF? 2 1rctt|t|jjt | fS)z4 Key for comparing two labeled polynomials. )rrrrr3rrs rlbp_keyrs2 DGGU1XX]0 1 1CFF7 ;;rc |j}t|j}t|j}t|d|d|jf}t |||}t |||}t tt|t| t||} t tt|t| t||} t| | dkr"t| ||t| ||fSt| ||t| ||fS)a7 Compute the critical pair corresponding to two labeled polynomials. A critical pair is a tuple (um, f, vm, g), where um and vm are terms such that um * f - vm * g is the S-polynomial of f and g (so, wlog assume um * f > vm * g). For performance sake, a critical pair is represented as a tuple (Sign(um * f), um, f, Sign(vm * g), vm, g), since um * f creates a new, relatively expensive object in memory, whereas Sign(um * f) and um are lightweight and f (in the tuple) is a reference to an already existing object in memory. rr) rrLTronerrrr leading_termrr) r2rrrltfltgltumvmfrgrs r critical_pairrs.[F ((+C ((+C s1vs1v & & 3B "c6 " "B "c6 " "B c$q''588#8#8#:#:CFFCCR H HB c$q''588#8#8#:#:CFFCCR H HBr2"2R"ab2q11R"ab2q11rct|djj}t|d|t |d}t|d|t |d}t ||}|dkrdS|dkrlt|d|t |d}t|d|t |d}t ||}|dkrdSdS)a} Compare two critical pairs c and d. c < d iff - lbp(c[0], _, Num(c[2]) < lbp(d[0], _, Num(d[2])) (this corresponds to um_c * f_c and um_d * f_d) or - lbp(c[0], _, Num(c[2]) >< lbp(d[0], _, Num(d[2])) and lbp(c[3], _, Num(c[5])) < lbp(d[3], _, Num(d[5])) (this corresponds to vm_c * g_c and vm_d * g_d) c > d otherwise rrrr)rrzerorrr)cdrc0d0ryc1d1s rcp_cmprs 1;;  D QqT4QqT # #B QqT4QqT # #BBABwrAv 1tS1YY ' ' 1tS1YY ' ' BOO 7 2 1rc tt|d|jt|dtt|d|jt|dfS)z+ Key for comparing critical pairs. rrrr)rrrr)rrs rcp_keyrs] C!diQqT33 4 4gc!A$ SVWXYZW[S\S\>]>]6^6^ __rctt|d|dt|d|dS)z Compute the S-polynomial of a critical pair. The S-polynomial of a critical pair cp is cp[1] * cp[2] - cp[4] * cp[5]. rrr)rr)cps rs_polyrs8 <1r!u--|BqE2a5/I/I J JJrc`|D]}|dt|dkr+tt|j|drdS|dt|dkr?|t |kr,tt|d|drdSdS)a Check if a labeled polynomial is redundant by checking if its signature and number imply rewritability or comparability. (sign, num) is comparable if there exists a labeled polynomial h in B, such that sign[1] (the index) is less than Sign(h)[1] and sign[0] is divisible by the leading monomial of h. (sign, num) is rewritable if there exists a labeled polynomial h in B, such thatsign[1] is equal to Sign(h)[1], num < Num(h) and sign[0] is divisible by Sign(h)[0]. rrTF)rrrr0r)signnumrYrCs ris_rewritable_or_comparablers    7T!WWQZ  a T!W55 tt 7d1ggaj  SVV| #DGGAJQ88 44 5rct|jj}t|jj}t|s|S |}|D]}t|rt t|jt|jrt t|jt|j|}ttt||dt||dkr"t||}t||}n||kst|s|S)a@ F5-reduce a labeled polynomial f by B. Continuously searches for non-zero labeled polynomial h in B, such that the leading term lt_h of h divides the leading term lt_f of f and Sign(lt_h * h) < Sign(f). If such a labeled polynomial h is found, f gets replaced by f - lt_f / lt_h * h. If no such h can be found or f is 0, f is no further F5-reducible and f gets returned. A polynomial that is reducible in the usual sense need not be F5-reducible, e.g.: >>> from sympy.polys.groebnertools import lbp, sig, f5_reduce, Polyn >>> from sympy.polys import ring, QQ, lex >>> R, x,y,z = ring("x,y,z", QQ, lex) >>> f = lbp(sig((1, 1, 1), 4), x, 3) >>> g = lbp(sig((0, 0, 0), 2), x, 2) >>> Polyn(f).rem([Polyn(g)]) 0 >>> f5_reduce(f, [g]) (((1, 1, 1), 4), x, 3) Tr) rrr3rrr0rrrrrrr)r2rYr3rrrCthps r f5_reducers6 !HHM E 1XX] !F 88   AQxx #E!HHKq== qeAhhk6BBAxQ166QGG!K*!Q//#ArNN 6 q Hrcj gttD]>}|}|d|}|r|?krngfdttDfddfdttD}|fddt}d}t|r|}t|dt|d rOt|d t|d rzt|} t| }tt|t||d z}t|rg} t|D]\}}t|dt|d |gr| |Ft|d t|d |gr| |t!| D]}||=D]} t| r~t#|| }t|dt|d |grNt|d t|d |grz|||fd dt|j} | tdjkr|nWtD]G\}} | t| jkr||nH|d z }n|d z }t|dD}t)|}t+|fddS)a Computes a reduced Groebner basis for the ideal generated by F. f5b is an implementation of the F5B algorithm by Yao Sun and Dingkang Wang. Similarly to Buchberger's algorithm, the algorithm proceeds by computing critical pairs, computing the S-polynomial, reducing it and adjoining the reduced S-polynomial if it is not 0. Unlike Buchberger's algorithm, each polynomial contains additional information, namely a signature and a number. The signature specifies the path of computation (i.e. from which polynomial in the original basis was it derived and how), the number says when the polynomial was added to the basis. With this information it is (often) possible to decide if an S-polynomial will reduce to 0 and can be discarded. Optimizations include: Reducing the generators before computing a Groebner basis, removing redundant critical pairs when a new polynomial enters the basis and sorting the critical pairs and the current basis. Once a Groebner basis has been found, it gets reduced. References ========== .. [1] Yao Sun, Dingkang Wang: "A New Proof for the Correctness of F5 (F5-Like) Algorithm", http://arxiv.org/abs/1004.0084 (specifically v4) .. [2] Thomas Becker, Volker Weispfenning, Groebner bases: A computational approach to commutative algebra, 1993, p. 203, 216 TNc pg|]2}ttj|dz||dz3Sr)rr zero_monom)rrwrzrs rrz_f5b..ps=NNN1S!a% ( (!A$A 6 6NNNrc>t|jSrH)rr0rks rr4z_f5b..qsuQxx{++rrocg|]B}t|dztD]}t|| CSr)rqr@r)rrwr=rYrs rrz_f5b..tsU _ _ _a%PQTUPUWZ[\W]W]J^J^ _ _Q-!adD ) ) _ _ _ _rc$t|SrHrrrs rr4z_f5b..us6"d++rrrrrrc$t|SrHrrs rr4z_f5b..s6"d#3#3rrcPg|]#}t|$Sr)rr?)rrs rrz_f5b..s(%%%aq  %%%rc$|jSrHr/rks rr4z_f5b..s55;;r)r3rqr@r>rAsortrVrrrrrrrr?rrreversedrr0insert red_groebnerru)rzrrwrxryr{kr|rrindicesrrJqHrYr3s`` @@rr r ;s_D JE A   s1vv  A!Aae A  6    ONNNNc!ff NNNAFF++++TF::: ` _ _ _ _5Q== _ _ _BGG++++TG::: AA b''7$ VVXX 'r!uc"Q%jj! < <   &r!uc"Q%jj! < <   2JJ aOO Qq))1q5 1 1 88( $G"2 & &2.r!uc"Q%jj1#FF&NN1%%%%0ABqE QCHH&NN1%%%g&&  qEE " "88"&q!T22B22a5#be**qcJJ! 4RUC1JJLL! IIbMMM GG3333TG B B Ba AuQxx55qu111  %aLLDAquQxx%%a "4"44A FAA ! # o b''7$t &%1%%%AQA !.... = = ==rcd}|}g}|rI|tfd||zDs||I||S)z Compute reduced Groebner basis, from BeckerWeispfenning93, p. 216 Selects a subset of generators, that already generate the ideal and computes a reduced Groebner basis for them. cg}t|D]G\}}||d|||dzdz}|r||Hd|DS)z1 The actual reduction algorithm. Nrc6g|]}|Sr)r?)rrxs rrz3red_groebner..reduction..s %%%a %%%r)rrr>rA)r8QrwrxrCs r reductionzred_groebner..reductionsx aLL  DAqaeaAi'((A  %%1%%%%rc3LK|]}t|jjVdSrH)rr0)rr2f0s rrQzred_groebner..s2@@Q#AD"%00@@@@@@r)rVrWrA)r+rrrzrrs @rrrs & & & A A  UUWW@@@@!a%@@@@@  HHRLLL  9Q<<rctt|D]]}t|dzt|D]:}t|||||}||}|rdS;^dS)z) Check if G is a Groebner basis. rFT)rqr@rtr>)r+rrwr=rs r is_groebnerrs3q66]]q1uc!ff%%  AadAaD$''AaA uuu   4rc|j|j}|fdt|D]Q\}}|j|jkrdS|d|||dzdzD] }t |j|jrdS!RdS)z2 Checks if G is a minimal Groebner basis. c$|jSrHr/rr3s rr4zis_minimal..qtrr5FNrT)r3rrrrLCrrr0)r+rrrwrrCr3s @r is_minimalrs JE [FFF$$$$F%%%! 1 46:  552A21q566"  Aad++ uuu   4rcJ|j|j}|fdt|D]j\}}|j|jkrdS|D]=}|d|||dzdzD]"}t|j|drdS#>kdS)z2 Checks if G is a reduced Groebner basis. c$|jSrHr/rs rr4zis_reduced..rrr5FNrrT) r3rrrrrrtermsrr0)r+rrrwrtermrCr3s @r is_reducedrs JE [FFF$$$$F%%%! !!1 46:  55GGII ! !DrrUQq1uvvY& ! !#AD$q'22! 5555! ! ! 4rc|j|jkrtd|j}|j}|r|s|jSt |dkrct |dkrPt |j|j}||j|j}| ||S| \}}| \}}|||d| D}d| Dd| Dz} td} | | f|jzt} | |} | | } t#| | g| }dfd |D}fd |d  D}||}|S) a Computes LCM of two polynomials using Groebner bases. The LCM is computed as the unique generator of the intersection of the two ideals generated by `f` and `g`. The approach is to compute a Groebner basis with respect to lexicographic ordering of `t*f` and `(1 - t)*g`, where `t` is an unrelated variable and then filtering out the solution that does not contain `t`. References ========== .. [1] [Cox97]_ Values should be equalrc"g|] \}}d|z|f Srrrmonomcoeffs rrz groebner_lcm..4%EEE,%u%EEErc"g|] \}}d|z|f S)rrrs rrz groebner_lcm..5rrc$g|] \}}d|z| fSrrrs rrz groebner_lcm..6s'EEE,%uf%EEErr)symbolsr3c`tfd|D S)Nc3(K|] }|V dSrHr)rrr=s rrQz7groebner_lcm..is_independent..As'88EuQx888888r)rWmonoms)rCr=s `ris_independentz$groebner_lcm..is_independent@s18888QXXZZ8888888rc,g|]}|d|Srr)rrCr s rrz groebner_lcm..Cs*444nnQ224!444rc4g|]\}}|dd|zfSrSr)rrrlcms rrz groebner_lcm..Es.III<5%qrrE#I&IIIrr)rr"rrr@rr0rrterm_new primitiverrr%rr from_termsr )r2rrrrrfcgcf_termsg_termsrt_ringrzr+basisrh_termsrCr rs @@r groebner_lcmrs  v31222 6D [F Ay 1vv{+s1vv{+QT14(( 14&&}}UE*** KKMMEB KKMMEB **R  CEE!''))EEEGEE!''))EEEEE!''))EEEFG c A ZZt| 33Z ? ?F'""A'""A aVV $ $E999 5444U444AIIII1Q4::<<IIIG   A Hrc|j|jkrtd|jj}|jsD|\}}|\}}|||}||zt||g}t|dkrtd|d}|js||zS| S)z6Computes GCD of two polynomials using Groebner bases. rrzLength should be 1r) rr"rr#rgcdquorr@r?)r2rrrrrrrCs r groebner_gcdrJsv31222 V]F ?! A AjjR   1 <1%%&''A 1vv{/-... !A ?1u wwyyrrH)*__doc__sympy.core.symbolrsympy.polys.monomialsrrrrsympy.polys.orderingsrsympy.polys.polyerrorsr sympy.polys.polyconfigr r rrtrrrrrrrrrrrrrrrrrrr rrrrrrrrrr%sC!!$#####XXXXXXXXXXXX%%%%%%......((((((& & & & PRRRh    (+++    *    ,,,333"HHH   *<<< 2 2 2F! ! ! H```KKK6000f}>}>}>@@   ((7 7 7 rr