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数学家、作曲家、摄影师、小提琴手

2014年1月5日|类别:数学

计算Hirzebruch L-多项式

安德鲁·拉尼基最近让我计算希策布鲁赫L(左)-多项式 L(左)8用Pontrjagin数表示32维流形的特征。(赫泽布鲁克的书代数几何中的拓扑方法 仅列表L(左)1通过L(左)5第1.5条.)

这是数学软件计算它的代码,或乘法多项式序列的任何其他成员:

K[Q_,n_Integer]:=模[{z,x},对称还原[系列系数[产品[ComposeSeries[Series[Q[z],{z,0,n}],序列[x[i]z,{z,0,n}]],{i,1,n}],n],表[x[i],{i,1,n}],表[下标[c,i],{i,1,n}]][[1]]//FactorTerms]

的生成序列L(左)-序列是$\frac{\sqrt{z}}{\tanh\sqrt}$soL(左)8然后可以通过运行以下命令计算:

K[Sqrt[#]/Tanh[Sqrt[#]]&,8]/。c->p

产生:

$\hs空间{-30pt}L_8=\压裂{1}{488462349375}\左(355554717 p_8-63136017 p_1 p_7-18909191 p_2 p_6+15043799 p_1^2 p_6-5468617 p_3 p_5+8640058 p_1 p2 p_5-3750699 p_1 ^3 p_5-1334163 p_4^2+2843653 p_1 P3 p_4+1311182 p_2 ^2 p~4-307215 p_1^2p_2 p~4+972066 p_1^4 p_4+679224 p_2 p_3^2-825291 p_1^3 p_3^2 p_2-250470 p_2 p_2^2 p_3+1371584 p_1^3p_3 2 p_3-286966 p_1^5 p_3-68435 p_2^4+423040 p_1^2 p_2^3-407726 p_1^4 p_2^2+122508 p_1^6 p_2-10851 p_1^8\右)$

其他一些L(左)-多项式(现出版为OEIS序列A237111):

$\hs空间{-30pt}L_1=\压裂{1}{3} 第1页$

$\hs空间{-30pt}L_2=\压裂{1}{45}\左(7p_2-p_1^2\右)$

$\空间{-30pt}L_3=\压裂{1}{945}\左(62 p_3-13 p_1 p_2+2 p_1^3\右)$

$\hs空间{-30pt}L_4=\压裂{1}{14175}\左(381 p_4-71 p_1 p_3-19 p_2^2+22 p_1^2 p_2-3 p_1^4 \右)$

$\hs空间{-30pt}L_5=\压裂{1}{467775}\左(5110 p_5-919 p_1 p_4-336 p_2 p_3+237 p_1^2 p_3+127 p_1 p2^2-83 p_1^3 p_2+10 p_1^5 \右)$

$\hs空间{-30pt}L_6=\frac{1}{638512875}\left(2828954 p_6-503904 p_1 p_5-159287 p_2 p_4+12523 p_1 ^2 p_4-40247 p_3 ^2+860901 p_1 p_2 p_3-33863 p_1 ^3 p_3+8718 p_2 ^3-27635 p_1 ^2 p_2 ^2+12842 p_1 ^4 p_2-1382 p_1 ^6\right)$

$\hs空间{-30pt}L_7=\压裂{1}{1915538625}\左(3440220 p7-611266 p_1 p_6-185150 p_2 p_5+146256 p_1^2 p_5-62274 p_3 p_4+88137 p_1 p2 p_4-37290 p_1 ^3 p_4+22027 p_1 p_3^2+16696 p_2^2 p_3-39341 p_1^2p2 p_3+10692 p_1^4 p_3-7978 p_1 p _2^3+11880 p_1^3 p_2^2^2 p_2^5 p_2+420 p_1^7\右)$

$\hs空间{-30pt}L_8=\压裂{1}{488462349375}\左(355554717 p_8-63136017 p_1 p_7-18909191 p_2 p_6+15043799 p_1^2 p_6-5468617 p_3 p_5+8640058 p_1 p2 p_5-3750699 p_1 ^3 p_5-1334163 p_4^2+2843653 p_1 P3 p_4+1311182 p_2 ^2 p~4-307215 p_1^2p_2 p~4+972066 p_1^4 p_4+679224 p_2 p_3^2-825291 p_1^3 p_3^2 p_2-250470 p_2 p_2^2 p_3+1371584 p_1^3p_3 2 p_3-286966 p_1^5 p_3-68435 p_2^4+423040 p_1^2 p_2^3-407726 p_1^4 p_2^2+122508 p_1^6 p_2-10851 p_1^8\右)$

$\hs空间{-30pt}L_9=\压裂{1}{194896477400625}\左(57496915570 p_9-10208138209 p_1 p_8-3048553630 p_2 p_7+2429800789 p_1^2 p_7-844282766 p_3 p_6+1377913993 p_1 p2 p_6-602296505 p_1^3 p_6-271389940 p_4 p_5+388346645 p_1 p _5+194787260 p_2^2 p.5-503353722 p_1^1 p_2 p_2 5+151701815 p_1 ^4 p_5+93946261 p_1 p_4^2+130062108 p_2 p_3 p _4p_3 4p_1^4 p_3 p_4-150462162 p_1 p_2^2 p_4+176155388 p_1^3 p_2 p4-39714485 p_1^5 p_4+12765448 p_3^3-78021660 p_1 p_2 p_3^2+44101890 p_1^3 p_3^2-20200592 p_2^3 p~3+100397374 p_1^2 p_2^2 p_3-70683311 p_1^4 p_2 p~3+12038553 p_1^6 p_3+11098737 p_1 p2^4-29509334 p_1^3p_2^3+20996751 p_1 ^5 p_2^2-5391213 p_1^7 p_2+43870 p_1^9右)$

$\hs空间{-30pt}左_{10} =\frac{1}{32157918771103125}\left(3844943648994 p_{10}-682613292644p_1 p_9-23707792197 p_2 p_8+162436741673 p_1^2 p_8-55782134394 p_3 p_7+91819476941 p_1 p2 p_7-40205725073 p_1^3 p_7-15408421872 p_4 p_6+24755004136 p_1 P3 p_6+12733594519 p_2^2 p-6-312800250 p_1^ p_2 p_6+10051904387 p_1^4 p_6-368943247 p_5^2+786084666 p_5 p_5+7156900191 p_2 p2 p_3 5p_5-92068432 59 p_1^2 p_3 p_5-9177195410 p_1 p_2^2p5+11001659873 p_1^3 p_2 p_5-2541992887 p_1^5 p_5+1745258589 p_2 p~4^2-2219530950 p_1^2 p_4^2+1300052223 p_3^2 p4-6094105875 p_1 p_2 p_3 p_4+3572462273 p_1^ p_3 p_4-932098945 p_2^3 p_4+489605 p_1^2p_2^2 p_2^2p_4-369438754 p_1^4 p_2p_4+671189161 p_1 ^6 p_4-595253 p_1 3p_3^3-7505097 15 p_2^2 p_3^2+2548895877 p_1^2 p_2 p_3^2-923938550 p_1^4p_3^2+1322085815 p_1 p_2^3 p_3-2812761078 p_1^3 p_2^2 p_3+1451051181 p_1^5 p_2 p_3-208588561 p_1^7 p_3+46251222 p_2^5-471455695 p_1^2 p_2^4+768892622 p_1^4 p_2^3-43317957 p_1^6 p_2^2+97463470 p_1^8 p_2-733662 p_1 ^{10}右)$

$\hs空间{-30pt}左_{11} =\压裂{1}{2218896395206115625}\左(107522567547780页_{11}-19088863620918第1页_{10}-5695530073310p_2 p_9+4542143981708 p_1^2 p_9-1555173021402 p_3 p_8+2565423874181 p_1 p_2 p_8-1123840932470 p_1^3 p_8-411527701200 p_4 p_7+683711631540 p_1 p_3 p_7+354034647890 p_2^2 p_7-922678415929 p_1^2 p_7+280446917930 p_1^4 p_7-129278298520 p_5 p_6+186412296664 p_1 p_4 p_6+189226541336 p_3 p_6-244992687206 p_1^2 p_6-250388515289 p_ 1个p_2^2p_6+302166509321 p_1^3 p_2 p_6-70283925270 p_1^5 p_6+44421264839 p_1 p_5^2+60174415980 p_2 p~4 p_5-77314873002 p_1^2 p_5+27308760472 p_3^2 p_0-5-139409773867 p_1 p2 p_5+8331979706 p_1^3p_3 p_5-23080275470 p_2 ^3 p_5+123981810568 p_1^2p_2^2 p_2_5-95102983271 p_1^4 p_2 p+5+17820178690 p_1^6 p_5+13901765658 p_3 p_4^2-33935234091 p_1 p_2p4^2+20066965081 p_1^3 p_4^2-25211090659 p_1 p_3^2 p_4-23432247682 p_2^2 p_3 p_4+8201241803 p_1^2 p_2 p_3 p2 p_4-30872129848 p_1^4 p_3 p_4+25074247417 p_1 p2^3 p~4-56265244569 p_1^3p_2^2^2 p~4+30821702183 p_1^5 p_2 p_4~4-4743504310 p_1^7 p_4701152064 p_3^3+80796657 p_1^2p_3^3 p_3+20273980429 p_1 p_2^2 p_3^2-29454532487 p_1^3 p_2 p_3^2+7815215137p_1^5 p_3^2+2684318680 p_2^4 p_3-22975754625 p_1^2 p_2^3 p_3+29852017156 p_1^4 p_2^2 p_3-12130331601 p_1^6 p_2 p_3+1509537376 p_1^8 p_3-1620540574 p_1 p_2^5+674992170 p_1^3 p_2^4-7829491106 p_1 ^5 p_2^3+366298844 p_1^7 p_2^2-732652806 p_1~^9 p_2+51270780 p_1^{11}右)$

$\hs空间{-30pt}左_{12} =\压裂{1}{3028793579456347828125}\左(59482964049337110 p{12}-10560195815097210 p_1 p{11}-3150694647926016 p_2 p{10}+2512728757752768 p_1^2 p{10{-859683903320810 p3 p9+1418912367890776 p1 p2 p9-621655204655468 p1 ^3 p9-224971539271761 p4 p8+377051717854661 p1 p3 p8+195571661859765 p2 ^2 p8-509919773356306 p1 ^2 p8+155056838434481 p1 ^4 p8-60744246118110 p5 p7+98470681343871 p1 p4 P7+103098432426 p2 p3 p7-133798852640129 p1 ^2 p3 p7-137649995502241 p1 p2 ^2p7+166399979251074 p1^3 p2 p7-38770694005481 p1^5 p7-14410087794673 p6^2+30742795880856 p1 p5 p6+28042926014435 p2 p4 p6-36219418521 p1^2 p4 p2409762108 p3^2 p6-72790761663197 p1 p2 p6+4379115251039 p1 p3 p3 p6-123548777 p2 ^3 p6+667838642637 p1 p1 ^2 p2 ^2 p_6-51565502511583 p_1^4 p_2 p_6+9726168419869p_1^6 p_6+6690315663313 p_2 p_5^2-8610718021212 p_1^2 p_5,2+9009316465271 p_3 p_4 p_5-23053514535432 p_1 p_2 p_0 p_5+13772029163489 p_1^3 p_4 p _5-10424737212619 p_1 p _3^2 p_1-5-10349150762991 p_2 ^2 p_3 p _5+36980302520182 p_1^ 2 p_2 p_3 p_5~14155214042120 p_1^4 p_3 p2p_5+1 2199190282918 p_1 p2^3 p_27983499 6067711 p_1^3 p_2^2 p_5+15683170260164 p_1^5p_2 p_5-2471593910869 p_1^7 p_5+871756041250 p_4^3-5299281214631 p_1 p_3 p_4 ^2-2544205275373 p_2 p_2 ^2 p_4+9000337394702 p_1^2 p_2 p~4^2-34104361584 p_1^4 p_4|2-3803158206 p_2 p_3^2 p~4+66705750121 p_1^2p_3^2p_2^2p_4+12382204834839 p_1 p2^2 p_3p_4+4-185641153047 p_1^3 p_2p_3p~4+50999 150967499 p_1^5 p_3 p_4+945948441582 p_2^4 p_4-8478203154675p_1^2 p_2^3 p_4+11595957601472 p_1^4 p_2^2 p_4-4991726989557 p_1^6 p_2 p_4+663128304922 p_1^8 p_4-206345908947 p_3^4+2489517554874 p_1 p_2 p_3^3-1830324992020 p_1^3 p_3^3+1052197012244 p_2^3 p_3^2-6891343720499 p_1^2 p_2^2 p_3^2+6109565322130 p_1^4 p_2 p_3^2-12754655567049 p_1^6 p_3^2-182897746 6670 p_1 p_2^4 p_3+6374277369326 p_1^3 p_2^3p_3-587105059546 p_1^5 p_2^2 p_3+1966494037756 p_1^7 p_2 p_3-215864759622 p_1^9 p_3-44720298010 p_2^6+6799766780 p_1^2 p_2^5-1676297291469 p_1^4 p_2^4+1502303537032 p_1^6 p_2^3-600842521854 p_1^8 p_2^2+108419277660 p_1 ^{10}p_2-70902922730 p_1^12}右)$

$\hs空间{-30pt}左_{13} =\压裂{1}{9086380738369043484375}\左(72322615121247036 p{13}-12839651079926 p_1 p{12}-3830739593311176 p_2 p{11}+3055097425061946 p_1^2 p{11{-1045049756196264 p_3 p{10}+1725094701581424 p_1 p2 p{10/-755821122963534 p_1^3 p{10{-272712748239906 p_4 p_9+45807860111560 p_1 p_3 p_9+2376998426306 p_2^2 p_9-619830309829310 p_1^2 p_2 p_9+188499057034344 p_1^4 p_9-70565079148746 p_5 p_8+118306288116891 p_1 p _8+124969620711936 p_2 p_3 p2p_8~162151396044763 p_1^2p_3p_8-167097801278477 p_1 p2^2 p_08+20208584507 p_1^3 p_2 p_2 p_28-477 105612724954 p_1^5 p_8-21922060368516 p_6p7+31742893399152 p1 p5 p7+32491662944022 p2 p4 p7-42035081558097 p1 p4 p7+16441304584404 p3 p3 ^2 p7-87179794392340 p1 p2 p3 p7+5252247311857 p1 ^3 p3 p7-14890478221466 p2 ^3 p7+80649517416487 p1 ^2 p2 ^2 p7 p7-623768431291 p1 p1 ^4 p2 p7+1175293691794 p1 ^6 p7+7511972573838 43 p_1 p_6^2+10136863360022 p_2 p_5 p_6-13080453013002p_1^2 p_5 p_6+8993483945286 p_3 p_4 p_6-23579084234895 p_1 p_2 p_6+14158404892491 p_1^3 p_4 p_6-11593881767582 p_1p_3^2 p_1781330593106 p_2^2 p_3 p_6+42359270842707 p_1^ 2 p_2 p_3 p _6-16315258400629 p_1^4 p_3p_6+14320243955795 p_1 p2^3 p_33061727161314 p_1^3p_2^2p_6+1 8650010728593 p_1 ^5第2页p6-2958448000074第1页^7第6页+2170310001860p_3 p_5^2-5616857698569 p_1 p_2 p_5*2+3362197563453 p_1^3 p_5|2+1598325532596 p_4^2 p_5-7521438548927 p_1 p2p_4 p_5-3747037194988 p_2^2 p~4 p_5+13387399017752 p_1^2 p_2p_4 p_5-5124542468415 p_1^4 p_5 p_5-3386244167460 p_2p_3^2 p_05+6038413635675 p_1^2p_3^2p_5+11951705360023 p_1 ^2第3页p_5-18248868443891第1页^3第2页p_3第5页+5106691054163p_1^5 p_3 p_5+983844985516 p_2^4 p_5-9004069487482 p_1^2 p_2^3 p_5+12584826999985 p_1^4 p_2^2 p_5-5540530920427 p_1^6 p_2 p_5+553423126204 p_1^8 p_5-726569491346 p_1 p_4^3-17384753736 p_2 p_3 b_4^2+3069125281035 p_1^2p_3 p2 p_4 ^2+2936673351 p_1 p2^2 p~4444873416929 p_1^3 p_3 2 p4^2+1231788304752 p1^5 p4^2-444806806572 p3^3p_4+4390275986442 p_1 p_2 p_3^2 p_4-3296411478022 p_1^3 p_3^ 2 p_4+1358265174044 p_2^3 p~3 p_4-9151640683451 p_1^2 p_2^2 p_3 p_4+83607711912 p_1 ^4 p_2 p_3 p_4-1806365634680 p_1^6 p_3 P4-1396161717978 p_1 p2^4 p_4+5087384629390 p_1^3p_2^3p_4p_4-923620079719 p_1^5 p_2^2p_4+1 743200 1389284 p_1^7 p_2 p_4-20372180174 p_1^9 p_4+238460897625 p_1p_3^4+418208554632 p_2^2 p_3^3-1845509565666 p_1^2 p_2 p_3^3+827899012917 p_1^4 p_3^5-15606938696 p_1 p_2 ^3 p_3^2+4159635712346 p_1^3 p_2^2p_3^2-2612096050645 p_1^5 p_2 p_2 ^2+44899445468 p_1^7 p_3^2-126439832080 p_2 ^5 p_3+1660385531550 p_1^2p_2^4 p_3-3418480202739 p_1 ^4 p_2^3 p_3+2422033713280 p_1^6 p_2^2 p_3-689846708628 p_1^8 p_2p_3+6770375018 p_1^{10}p_3+81719534070 p_1 p_2^6-49024209730 p_1^3 p_2^5+838678611984 p_1^5 p_2^4-610634946312 p_1^7 p_2^3+2134336190 p_1^9 p_2^2-3511088158 p_1^}p_2+2155381956 p_1^[13}右)$

$\hs空间{-30pt}左_{14} =\压裂{1}{3952575621190533915703125}\左7167238 p4 p{10}+80742799399504690 p1 p3 p{10{+41902383906298496 p_2^2 p_{10}-109268418713940708 p_1^2 p_{10}+3332309583418836400 p_1^4 p_{10}-12295169263985832 p5 p9+20794596976366700 p1 p4 p9+22011574730657134 p2 p3 p9-28563786972400662 p1 p1 ^2 p3 p 9-29447481485141260 p1 p2 ^2 p9+35617257910457980 p1 ^3 p2 p9-8303170596462900 p1 ^5 p9-32837774068218 p6 p8+5347010561485880 p1 p5 p8+5657331509067973 p2 p2p4 p8-732226363498567 9 p_1^2 p_4 p_8+2881472301806105 p_3^2p_8-15311255840106597 p_1 p_2 p_3 p_8+9228451340334101 p_1^3 p_3 p_8-2619556019278058 p_2^3 p_8+14194112098851433 p_1^2 p_2^2 p_8-10982871479970858 p_1^4 p_2 p_8+2075954705219238 p_1^6 p_8-77555476256166 p_7^2+1656188545730 p_1 p_6 p_7+15151968231834 p_2 p_5 p_7+7-19538195748202 p_1^ p_5p_5+14931834 8643293045570 p3 p4 p7-3959880502362187 p1p2 p4 p7+2381631835724001 p1 ^3 p4 p7-1996126430113095 p1 p3 ^2 p7-204151434526030 p2 ^2 p3 p7+7352710942428867 p1 ^2 p2 p3 p2 p7-2836758879081301 p1 ^4 p3 p7+2501988111691518 p1 p2 ^3 p7-5786202450321413 p1 ^3p2 p7+3269478592431558 p1 ^5 p2 p7-519509387477238 p1 ^7 p7 p7+35829951 749599 p_2 p_6^2-462741237149047 p_1^2p6^2+470192290021650 p3 p5 p6-1232603833449492 p1 p2 p5 p6+739752938870166 p1 p5 p6+21125827690392 p4 ^2 p6-1087286066381534 p1 p3 p4 p4 p6-552091543508825 p2 ^2 p4 p6+1982493137575544 p1 p1 2 p3^2 p6+1969271080274147 p1 p2^2 p3p_6-302530402558420 p_1^3 p_2 p_3p_6+851815550847629 p_1^5 p_3 p_6+166069974113788 p_2^4 p_6-1529551857415271 p_1^2 p_2^3 p_6+2151552906683779 p_1^4 p_2^2 p_6-9533580668833460 p_1^6 p_2 p_6+1 30486561787496 p_1^8 p_6+0659517148035 p_4 p_5 ^2-262056899985 p_1 p_3p_3^2-1318113665 p_2 ^2 p5^2+471919294342151 p1^2 p2 p5^2-181038396627908p_1^4 p_5^2-192614659760042 p_1 p_4^2 p_5-353063374913277 p_2 p_3 p_5+6295412510225 p_1^2 p_3 p2p_4 p_5+625456789776172 p_1 p2^2 p~4 p_5-9552047966125 p_1 ^3 p_2 p_5+2673900465882279 p_1^5 p_5 p_5-53363251821705 p_3^3 p_5+564108245216523 p_1 P2p_3^2 p5-43007009637136 p_1^3 p_3 ^2 p_5+185786489726488 p_2^3 p_3 p_5-1273653276229747 p_1^2p_2^2 p_3 p_5+1185467415261029 p_1^4 p_2 p_3 p2 p_5-260700468307217 p_1^6 p_3 p_5-209198940940456 p_1 p_2^4 p_5+778136940573984 p_1^3 p_2^3 p_5-769326245733787 p_1 ^5 p_2^ 2 p_5+278476291878048 p_1^7 p_2 p_5-333018333996 p_1^9 p_5-34286351316642 p_2 p_2^3+6074217415757 p_1^2 p_4^3 p_3-4259838414 2751 p_3^2 p_4^2+289343407856985 p_1 p_2 p_3p_4^2-218728278954802 p_1^3 p_3 p_4*2+46158975519344 p_2^3 p_4|2-313545353277647 p_1^2 p_2^2 p_4~28912928422879 p_1^4 p_2 p_4 ^2-62985004440353 p_1^6 p_4$2+74063033154267 p_1 p_3^3 p~4+104089783176396 p_2^2p_3^2 p~4-46828685435998 p_1^2p_2p_4+21445846172387 p_1_4 p_3^2 p_4-289335556881588 p_1 p_2^3 p_3 p_4+792778821013802 p_1^3 p_2^2p_3 p_4-51299483378385 p_1^5 p_2 p_3 p2 p_4+91106047648244 p_1^7 p_3 P4-13272078302266 p_2^5 p_4+181225720113282 p_1^2 p_2^4 p_4-389536969484841 p_1^4 p_2^3 p_4+28951185106240 p_1^6 p_2 ^2 p_4-86977563536574 p_1^8 p_2 p_4+40739733574034 p_1 p_1^{10}p_4+11475032305464 p_2 p_3^4-25480617652053 p_1^2 p_3^4-89349671976612 p_1 p_2^2 p~3^3+160436593965288 p_1^3 p_2 p~3^3-50969939652942 p_1^5 p_3^3-19070069854338 p_2 ^4 p_3^2+203682123751428 p_1^2p_2^3p_3^2-320494896445 p_1^4 p_2^2p_3^2+1546907155096 p_1^6 p_2 p_2 p_2^2-2 2586488151310 p_1^8 p_3^2+33078442359342 p_1 p_2^5 p_3-171075425779366p_1^3 p_2^4 p_3+243830587870408 p_1^5 p_2^3 p_3-139882802086008 p_1^7 p_2^2 p_3+34654263473350 p_1^9 p_2 p_3-3081957515434 p_1^{11}p_3+599122543748 p_2^7-12711483862714 p_1^2 p_2^6+44064808661200 p_1^4 p_2^5-570910838739788 p_1^6 p_2^4+34963111228740 p_1^8 p_2^3-10859561841202 p_1^{10}p_2^2+164 2500201244 p_1^{12}p_2-94997844116 p_1^{14}\右)$