\h+SrSSKJrJrJrJrJrJr SSKJ r SSK J r SSK J r SSKJrJrJrJr SSKJrJrJrJrJrJr SSKJr SS KJr SS KJr SS K J!r!J"r" SS K#J$r$ SS K%J&r&J'r'J(r( \'S(Sj5r)\'S(Sj5r*\'SS.Sj5r+\'\S4Sj5r,\'S)Sj5r-Sr.Sr/Sr0Sr1Sr2Sr3SSK4J5r5 Sr6Sr7Sr8S r9S!r:S"r;S#rS&r?S'r@g)*zIFunctions for generating interesting polynomials, e.g. for benchmarking. )AddMulSymbolsympifyDummysymbols)Tuple)S) nextprime) dmp_add_termdmp_negdmp_muldmp_sqr)dmp_zerodmp_one dmp_grounddup_from_raw_dict dmp_raise dup_random)ZZ)dup_zz_cyclotomic_poly)DMP)PolyPurePoly) _analyze_gens)subsetspublic filldedentNFcUS::a[SU-5eUb [U5 O [S5nUS:a\SSKJn SSKJn SnU"S5/n[SUS-5H%n[U5nURU"U55 M' U"[U6XS 9$US:Xa US-S- nOAUS:XaUS -S US--- S-nO)US:Xa#US -S US--- SUS ---SUS--- S-nU(a [WU5$W$)zGenerates n-th Swinnerton-Dyer polynomial in `x`. Parameters ---------- n : int `n` decides the order of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz6Cannot generate Swinnerton-Dyer polynomial of order %sx)sqrt)minimal_polynomialpolys (i`ii@) ValueErrorrr(sympy.functions.elementary.miscellaneousr" numberfieldsr$ranger appendrr) nr r'r"r$paiexs P/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/specialpolys.pyswinnerton_dyer_polyr8s Av Dq HJ J } #J1uA4  !WIq!a%A! A HHT!W !"#q'1::Av TAX a TBq!tG^a  a TBq!tG^c!Q$h &QT 1C 7#8B?++cUS::a[SU-5e[[[U5[5[5nUb[ R "X15nO [R "U[S55nU(aU$UR5$)zGenerates cyclotomic polynomial of order `n` in `x`. Parameters ---------- n : int `n` decides the order of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz1Cannot generate cyclotomic polynomial of order %sr ) r-rrintrrnewrras_expr)r2r r'polys r7cyclotomic_polyr?Asv Av ?! CE E %c!fb12 6D}xx ||D%*-4,dlln,r9r&c>[U5nUS:dU[U5:dU(d[SU<SU<35eU(d[RnO2[ [ U[U55Vs/sH n[U6PM sn6nU(a [U/UQ76$U$s snf)z Generates symmetric polynomial of order `n`. Parameters ========== polys: bool, optional (default: False) Returns a Poly object when ``polys=True``, otherwise (default) returns an expression. rz.Cannot generate symmetric polynomial of order z for ) rlenr-r Onerrr;rr)r2r'gensr>ss r7symmetric_polyrE\s  D1uCI TVWY]^__ uugdCF&;<&;S!W&;<= %4 t /4/=s/Bc\[[XX45XS9nU(aU$UR5$)aGenerates a polynomial of degree ``n`` with coefficients in ``[inf, sup]``. Parameters ---------- x `x` is the independent term of polynomial n : int `n` decides the order of polynomial inf Lower limit of range in which coefficients lie sup Upper limit of range in which coefficients lie domain : optional Decides what ring the coefficients are supposed to belong. Default is set to Integers. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. )domain)rrr=)r r2infsuprGr'r>s r7 random_polyrJts*,  13/ BD4,dlln,r9c [USS5n[U[5(a[U<SU<35nO#U(aU[ U6R -(aSn[U[5(a[U<SU<35nO#U(aU[ U6R -(aSnU(d[ [S55e/n[[U5Vs/sH oaX&- PM sn6n[U5HOnXqX&- - n[[U5V s/sHoU :wdM X&X)- PM sn 6n URX- 5 MQ [[XS5V V s/sH upX-PM sn n 6$s snfs sn fs sn n f)zConstruct Lagrange interpolating polynomial for ``n`` data points. If a sequence of values are given for ``X`` and ``Y`` then the first ``n`` values will be used. free_symbolsN:Fz~ Expecting symbol for x that does not appear in X or Y. Use `interpolate(list(zip(X, Y)), x)` instead.) getattr isinstancestrrr rLr-rrr0r1rzip) r2r XYokcoeffsr5numertnumerjdenomcoeffys r7interpolating_polyr\sK ND )B!S q!$ % UAY+++ !S q!$ % UAY+++  %>?@ @F U1X.Xqt8X. /F 1XAD!U1X@Xa{qtad{X@A ek" V8XU8 99/A9sE- E2  E2 E7 c [US-5Vs/sHn[S[U5-5PM nnUSUSpCU[USS6-nUS-[USSVs/sHofS-PM sn6-nUS-US--R"U6nUS-SU-US--US--S- -R"U6n [ S/UQ76n XU 4$s snfs snf)%Fateman's GCD benchmark: trivial GCD r#y_rNr%)r0rrPras_polyr) r2r5rSy_0y_1ur[vFGHs r7fateman_poly_F_1ris(-a!e 5 1s1v  A5tQqT c1QR5kA QQqrU+U!tU+,,A a%!a%!!1%A a%"S&a-#q&(1, -66:A Q  A 7N 6 ,s !B>C c\U"S5U"S5/n[U5Hn[X15U/nM U"S5U"S5U"S5/n[SU5Hn[X15[U5U/nM US- n[U[ U"S5U5SX5n[U[ U"S5U5SX5nU"S5*U"S5//U"S5U"S5U"S5*//n[U[ U"S5U5SX5n [ XSU5n [ XgX5n [ XX5n [X5n XU 4$)r^r#rr%r!)r0rrr rrr)r2Krdr5remUVfWrSrfrgrhs r7dmp_fateman_poly_F_1rqs0 1qt A 1X Q]A  1qtQqTA 1a[ Q]HQK + AAQ 1Q4+Q5AQ 1Q4+Q5A Q4%1QqT1Q4!A$/0AQ 1Q4+Q5A!1AaAaA A 7Nr9c [US-5Vs/sHn[S[U5-5PM nnUSn[USS6n[ X4-S-S-/UQ76n[ X4- S- S-/UQ76n[ X4-S-S-/UQ76nXV-XW-U4$s snf)7Fateman's GCD benchmark: linearly dense quartic inputs r#r_rNr%r0rrPrr)r2r5rSrbrdrhrfrgs r7fateman_poly_F_2rus(-a!e 5 1s1v  A5 A$C QqrU A cgkA ""A cgkA ""A cgkA ""A 3Q; 6s!B cU"S5U"S5/n[US- 5Hn[X15U/nM US- n[U[U"S5US- 5SX5n[ [XA5[ XTU5/X5n[ [XA5U/X5n[U[US- U5SX5n[ [XA5U/X5n[ XhX5[ XxX5U4$)rsr#rr%)r0rr rrr r) r2rkrdr5rlreroghs r7dmp_fateman_poly_F_2rys 1qt A 1q5\ Q]A  AAQ 1Q4Q/A9Aa 0118A"A)AQAq)1a3A"A)A 1 a 3Q 66r9c v[US-5Vs/sHn[S[U5-5PM nnUSn[USSVs/sH oDUS--PM sn6n[ X0S--U-S-S-/UQ76n[ X0S--U- S- S-/UQ76n[ X0S--U-S-S-/UQ76nXg-Xh-U4$s snfs snf)8Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) r#r_rNr%rt) r2r5rSrbr[rdrhrfrgs r7fateman_poly_F_3r|s(-a!e 5 1s1v  A5 A$C !AB% (%Q!a%j% ()A cElQ"Q &++A cElQ"Q &++A cElQ"Q &++A 3Q; 6)s !B1B6c d[US-UR0U5n[SUS- 5H"n[U/[ X15US-US-U5nM$ [U[ U"S5US- 5SX5n[ [[X@S- U5/[ US- U5US-X5X5n[ [U/[ US- U5US-X5X5n[U[ US- U5SUS- U5n[ [U/[ US- U5US-X5X5n[XWX5[XgX5U4$)r{r#rr%) roner0r rrrr r)r2rkrdr5rerorwrxs r7dmp_fateman_poly_F_3rs31q5!%%.!,A 1a!e_ !gamQUAE1 = Q 1Q4Q/A9AgaQ*+WQUA->AqLa TA aS'!a%"3QUAA1HAQAq)1a!eQ7A aS'!a%"3QUAA1HA 1 a 3Q 66r9)ringc[S[5upp#US-U-US--SUS--U-U--SUS--U--SUS---SU--SUS--US---SUS--U--SUS---X#S---SU-U--U-S-$)Nx,y,zr%r!r(r,r#rrRr r[zs r7_f_0r,sgr"JA! a46!Q$;1a4! #a1fQh .1a4 7!A# =!Q$q!t KaPQSTPTfUVh VYZ[\^_[_Y_ _bcghdhbh hklmnknopkp pst twx xxr9c0[S[5upp#US-U-U-US-US--US---US-US---SUS--U-U--SUS--U--US-US---SUS--U--XS--U--SU-US--U--SU-US---X-US---SU-U-US---X-U--SU-U--SU-US---SU-U--S U--US-US---SUS--U--SU-US---S U-U--S U--S U--S -$)Nrr!r%r)ibi,i@iXiprrs r7_f_1r0sZgr"JA! a46!8ad1a4i1n $q!tAqDy 02ad719Q; >AqD JQPQTRSUVRVY VY[\]_`\`Y`abYb befjkgkeklmem mprstptuvxyuypyz{p{ {ABCCDEGHDHH HKLKNOPRSOSKS SVXYZVZ[\V\]^`a]aVa adedghidi ilopqlqrsls svxyzvz{|~{v BEFGBGHIBI ILOPQLQ QTUWXTXYZ\]Y]T] ]`bcdfgcg`ghi`i ilnoplpqrtuqulu ux{|}x}~x BEFGBG GJMNOJO ORV VVr9c[S[5upp#US-US--US-US--U--US-U-US---US-US---US-US---US-U-U--SUS--U--SUS--U--US-US--U--SUS--US--- US-US---SUS--US--- X#--SU-- SU--S- $)Nrrr!r%Z irrs r7_f_2r4sgr"JA! a419q!tAqDy{ "QT!VAqD[ 01a419 Aad JQPQTRSUVRVYWXZ[W[^ [^_ab^bcd^defhiei^i ilmoplpqrlrstlt twx|}y}w}~BCCxC CFGKLHLFL LOPORSTVWSWOW WZ[Z]^_ab^bZb befehijlmimem mpqstptuvpv vyz~{y r9c[S[5upp#US-*US--U-US-US--US--- US-US--US--- SUS--US--- US -US--US --- US -US--US---SUS --US--U--SUS --US--US--- US -US --US--- US-US --US---SUS--US --US---US-U-US--- US -US --US ---SUS --US --US---US -US--US --- SUS --US--US---S US --US---SUS --US --US--- US-US--US ---SUS--US--US ---US-US--US --- SUS--US --US ---S US--US --US---US-US--US---SUS--US--US---SUS--U-US --- US-US---SUS--US---XS--US ---SU-US--US ---SU-US--US ---S U-US--US---US -US---SUS --US ---SUS ---S US ---$) Nr r*rr!r r%r,r(rrrs r7_f_4r<s gr"JA! qD5A:a Y\]_`\`abdeae\efgijfj\j jmnoprsosmstuwxtxmxyzmz z}~@ACD@D~DEFHIEI~IJKMNJN~N NQRTUQUVWYZVZQZ[\^_[_Q_ _bcefbfghjkgkbklmoplpbp pstuvxyuysyz{}~z~s~@BCCsC CFGIJFJKLFLMNPQMQFQ QTUWXTXYZ\^Y^T^_`bc_cTc cfghiklhlflmnprmrfrstvwswfw wz{}~z~@BCCzCDEGHDHzH HKLMNPQMQKQRSUVRVKVWXZ[W[K[ [^`abdeae^efgijfj^j jmnoprsosmstuwxtxmxyz|}y}m} }@ACD@DEFHIEI@IJKMNJN@N NQRSTVWSWQWXY[\X\Q\]^`a]aQa adeghdhijlmimdmnoqrnrdr ruvwxz{w{u{|}@|@u@ABDEAEuE EHJKLNOKOHOPQSTPTHTUVXYUYHY Y\]_`\`abdeae\efgijfj\j jmnoprsosmstuwxtxmxyz|}y}m} }@ABCEFBF@FGH@HIJLMIM@M MPQSTPTUVXYUYPY Y\]^_ab^b\bcdfgcg\g gjkoplpjpqrtuquju uxyz{x{|}@|@x@ABDEAExE EHIJKHKLMOPLPHPQRTUQUHU UXZ[\X\]^`a]aXabcefbfXf fijlmimnoqrnrir ruvwxz{w{u{|}@ |@ u@ @ C D E F H I E I C I I L N O P R S O S L S S S r9c[S[5upp#US-*SUS--U-- SUS--U--SU-US--- SU-U-U--SU-US--- US-- SUS--U--SU-US--- US--$)Nrr!r%r,rrs r7_f_5r@sgr"JA! qD51QT6!8 a1fQh &1QT 1AaCE!G ;ac!Q$h FA MPQRSUVRVPVWXPX X[\]^[^_`bc_c[c cfgijfj jjr9c[S[5upp#nSUS--U-SUS--US--US---SUS--US--- SU-US--- SU-US--- S U-U-US---S U-U-U-U--S US--US--US--- S US--US---US-US--US--- US-US---SUS --US---SUS--US---SUS--US--- SU-US--- $) Nzx,y,z,tiCr(-r!r%i/^rr,r)rr r[rts r7_f_6rDsB'MA! 19Q;AqDAad* *R1WQT\ 9CE!Q$J FAaQRd RUXYZUZ[\U\]^`a]aUa adfghdhijdjkldlmndn nqrstvwswqwxy{|x|q|}~AB~BrB BEFGHJKGKEKLMOPLPEP PSTVWSWXY[\X\S\]^`a]aSa adeghdhijlmimdm mpqrsuvrvpvwxz{w{p{ {~@ACD@D~DEFHIEI~I ILMNOQRNRLRSTVWSWLW WZ[\]Z]^_ab^bZb bbr9c[S[5upp#SUS--US--US--SUS--US--US---SUS--US--US--- SUS--U-US--- US-US--US---SUS--US--U--US-US--US--- SUS--US--US---SUS--U-US--- SUS--US--- SUS--US---SUS--US--US---SUS--US--U--SUS--US--US--- SUS--US--US---SUS--US--US--- SUS--U-US--- SUS--U-US--- SUS--U-US--- SUS--US--U--US-US--US---US-US--US--- SUS--US--US--- S US--US--U--SUS--U-US--- SUS--U-US---SUS--US--- SUS--US---SUS--US---SUS--US--US--- SUS--US--U--SUS--U-US--- SUS--U-US--- SUS--U-US---SU-US--U--SU-US--US--- SU-U-U-- SU-US---SUS--- SU-US---$) Nrr(r,r%r!rrr*rrrs r7_w_1rHsL gr"JA! QT6!Q$;q!t a1fQTk!Q$. .1a41QT1A AAadF1HQPQTM QTUWXTXYZ\]Y]T]^_ab^bTb beghiklhlelmnpqmqeqrses svwyzvz{|~{vABDEAEwE EHJKLNOKOHOPQSTPTHTUVXYUYHY Y\^_`bc_c\cde\efgijfj\j jmopqstptmtuvxyuymy y|}~AB~B|BCDFGCG|G GJKLMOPLPJPQRTUQUJUVWYZVZJZ Z]^_`bc_c]cdeghdh]hij]j jmnoprsosmstuwxtxmxyz|}y}m} }@ABCEFBF@FGHJKGK@KLMOPLP@P PSTUVXYUYSYZ[]^Z^S^_`bc_cSc cfghiklhlflmnfnoprsosfs svwxy{|x|v|}~v~@BCCvC CFGHIKLHLFLMNFNOPRSOSFS SVWXY[\X\V\]^`a]aVabcVc cfgijfjklnokofopqstptft twxz{w{|}@|@w@ABDEAEwE EHIJKMNJNHNOPRSOSHSTUWXTXHX X[\]^`a]a[abcefbf[fgh[h hkmnoqrnrkrstktuvxyuyky y|~@BCC|CDE|EFGIJFJ|J JMOPQSTPTMTUVXYUYMY Y\]^_ab^b\bcdfgcg\g gjklmoplpjpqrtuquju uxyz{}~z~x~@BCCxCDEGHDHxH HKLMNPQMQKQRSUVRVKVWXKX X[\]^`a]a[abc[cdeghdh[h hklmnpqmqkqrskstuwxtxkx x{|}~@ A }A {A B C {C D E G H D H {H H K L M N K N O P R S O S K S T U K U U X Y Z [ X [ \ ] _ ` \ ` X ` a b d e a e X e e h i j k h k l m h m n o h o o r s t u r u v w y z v z r z z } ~  @ B C  C } C C F G H I F I J K M N J N F N N N r9cB[S[5upnSUS--US--SUS--US---SUS--US---S US--US--- S US --US ---SUS --US---S US --U--SUS ---US-US ---US-US---S US--- US -US---US -US --- SUS --US--- SUS --US---US-US --- SUS--US---US-US--- S US--US---SUS---S US--- $)Nzx,yr*r!0r%rrHr,r(ri$r)rr r[s r7_w_2rLs5"oGA! ad71a4<"QT'!Q$, &AqDA 51a41 Dr!Q$wqRSt| SVWXY[\X\V\]^`a]aVa adefgijfjdjkldl lopqrtuquou uxy{|x|}~AB~ByB BEFHIEIJKMNJNEN NQSTUWXTXQX X[\^_[_`acd`d[d dghjkgklmoplpgp pstuvxyuysyz{}~z~s~ ~ADEFHIEIAIJKMNJNAN NQRTUQUVWYZVZQZ Z]^_`bc_c]cdeghdh]h hklnokopqstptkt twyz{}~z~w~@BCCwC CFHIJLMIMFM MPRSTVWSWPW WWr9c[5[5[5[5[ 5[ 5[ 54$N)rrrrrrrr9r7f_polysrPs% 6464646464646 AAr9c*[5[54$r)rrrr9r7w_polysrSs 646>r9)NF)r r[)A__doc__ sympy.corerrrrrrsympy.core.containersr sympy.core.singletonr sympy.ntheoryr sympy.polys.densearithr r rrsympy.polys.densebasicrrrrrrsympy.polys.domainsrsympy.polys.factortoolsrsympy.polys.polyclassesrsympy.polys.polytoolsrrsympy.polys.polyutilsrsympy.utilitiesrrrr8r?rErJr\rirqruryr|rsympy.polys.ringsrrrrrrrrrrrrrr9r7rsOA@'"##:'0/77&,&,R--4#(00.')--4::B"<  7*  7*#yV~S kbN WBr9