13 + 1*x11*x22 -1*x21*x12 +( 2.99079826308040E-01 - 9.54228095109106E-01*i)*zz1 +( 8.83228497194909E-01 + 4.68942876844102E-01*i)*zz2 +(-3.44281457711880E-01 - 9.38866485649468E-01*i)*zz3 +( 8.87683769919208E-01 - 4.60453607459017E-01*i)*zz4 +( 4.47211881899738E-01 + 8.94428047797974E-01*i)*zz5; -1*x22*x13 + 1*x12*x23 +( 4.22087187634768E-01 + 9.06555241579118E-01*i)*zz1 +(-4.56960728887527E-01 - 8.89486870197970E-01*i)*zz2 +(-8.43362826308347E-01 - 5.37344529330295E-01*i)*zz3 +(-4.06392962359337E-02 - 9.99173882565716E-01*i)*zz4 +( 9.92159338379809E-01 + 1.24979387363434E-01*i)*zz5; -1*x23*x14 + 1*x13*x24 +( 9.22479210350733E-01 + 3.86046767206627E-01*i)*zz1 +(-9.71344601741428E-01 + 2.37675544951065E-01*i)*zz2 +(-9.24611619933040E-01 + 3.80911213125578E-01*i)*zz3 +( 9.78495715492418E-01 + 2.06267144167414E-01*i)*zz4 +( 2.93402476683710E-01 - 9.55989009704539E-01*i)*zz5; + 1*zz1; + 1*zz2; + 1*zz3; + 1*zz4; + 1*zz5; +(-3.05088812597533E-01 + 3.77847263055096E-02*i)*x11 +(-4.46878131801796E-02 - 2.82884047717594E-02*i)*x22 +( 8.22962348492065E-02 + 3.39066950518597E-01*i)*x21 +(-1.96721738265849E-02 + 1.56280921461042E-01*i)*x12 +( 6.71931816298008E-02 + 2.77401007989914E-01*i)*x23 +( 1.13783728501105E-01 + 2.34078581108962E-01*i)*x13 +(-3.58983728413844E-01 + 9.79937642331424E-02*i)*x24 +(-2.17742125458329E-01 + 8.39352621179128E-02*i)*x14 +(-2.73607810768922E-01 - 1.52408000773383E-01*i)*zz1 +(-9.16230511920813E-02 - 2.47402255533774E-01*i)*zz2 +(-5.16409991613015E-02 + 3.36238841587146E-01*i)*zz3 +( 1.17179017573981E-01 + 5.01091567707116E-02*i)*zz4 +(-9.85892208197771E-02 - 5.32837245133843E-02*i)*zz5 -3.19053610969360E-01; +(-2.83240704329335E-01 - 2.21160857495297E-02*i)*x11 +(-2.20291398578109E-01 + 2.50041285419785E-01*i)*x22 +(-2.75460513692308E-01 + 2.02713734690814E-01*i)*x21 +(-2.20944308513094E-01 - 3.49418221187933E-02*i)*x12 +(-2.07425528690503E-01 + 2.49356312571882E-01*i)*x23 +( 7.73855683061981E-02 - 2.05083296833359E-01*i)*x13 +(-4.73799645570721E-03 + 3.87494792168632E-01*i)*x24 +( 9.51528828707092E-03 - 1.96375603147221E-01*i)*x14 +( 1.65453982235084E-01 - 7.85918454616573E-02*i)*zz1 +(-2.14051707672799E-01 + 1.04610857126611E-01*i)*zz2 +( 1.71872551551816E-01 - 1.39639759583117E-01*i)*zz3 +(-7.73446622209079E-02 - 7.30633134825698E-02*i)*zz4 +(-2.23591627813240E-01 + 2.01822818020280E-01*i)*zz5 +( 2.30770011999714E-01 - 6.70385808360583E-02*i); +( 2.35670908121771E-01 + 3.05994946607059E-01*i)*x11 +(-3.16208118758180E-01 + 1.93840827460823E-02*i)*x22 +(-1.20790971877393E-01 - 2.10252139483074E-01*i)*x21 +( 2.19670588498508E-01 + 1.91539904868371E-01*i)*x12 +(-1.07867911004679E-01 + 5.01063896615595E-02*i)*x23 +(-2.11766736470355E-01 + 1.45782560324510E-01*i)*x13 +(-5.85292965555109E-02 - 9.79429114148796E-02*i)*x24 +(-1.69451082490203E-01 - 2.04713440816488E-01*i)*x14 +(-2.15858127740136E-01 - 1.21084664710855E-01*i)*zz1 +(-3.89925930720371E-01 - 2.08122357884896E-01*i)*zz2 +(-1.03330167313090E-01 - 2.46860925861091E-01*i)*zz3 +( 2.15937290847134E-01 + 9.29134174965203E-02*i)*zz4 +( 6.10291688254758E-02 + 1.51750027821995E-01*i)*zz5 +( 1.51551746451145E-01 + 9.81177072971600E-02*i); +(-1.73758263219059E-01 + 2.23226147475596E-01*i)*x11 +( 1.71613264238212E-01 - 1.64478587858706E-01*i)*x22 +(-2.60525555346979E-01 + 8.10625503413346E-02*i)*x21 +(-3.15176358703842E-02 + 1.63516795503259E-01*i)*x12 +(-3.82800341179796E-01 - 1.08319295709545E-01*i)*x23 +( 5.07195797347653E-02 - 2.56011600732736E-01*i)*x13 +( 2.53914553011985E-01 + 7.42477475610203E-02*i)*x24 +(-5.47414244166746E-02 + 2.42101069236192E-01*i)*x14 +(-2.85705838502890E-02 + 1.47192110010791E-01*i)*zz1 +(-9.33366074354516E-02 + 7.05661168169716E-02*i)*zz2 +(-3.58952267802719E-01 - 8.91291562754540E-02*i)*zz3 +( 2.91316199611846E-01 - 1.14964898363308E-01*i)*zz4 +(-6.55091035207087E-02 - 2.65130993252555E-01*i)*zz5 +(-1.78352002283125E-01 + 1.60867498869669E-01*i); +(-1.26020563032239E-01 + 1.61720344532255E-02*i)*x11 +(-2.98896109268916E-01 - 3.07720442723131E-01*i)*x22 +(-6.51121417811955E-02 + 3.97810880303650E-01*i)*x21 +(-2.04281507598212E-01 + 6.00895016047942E-02*i)*x12 +( 1.52546790887937E-01 - 1.56913887785863E-01*i)*x23 +(-1.01281875068304E-02 + 2.55393833449420E-01*i)*x13 +(-3.59723592065775E-02 - 1.16290632465234E-01*i)*x24 +( 1.17756893075022E-01 + 1.98538987479121E-01*i)*x14 +( 1.55216451842971E-01 + 2.06980287749358E-01*i)*zz1 +( 2.80309710243232E-02 - 9.92013233794279E-03*i)*zz2 +( 1.18371506754300E-01 - 2.49486727772149E-01*i)*zz3 +( 2.50485287548154E-01 - 1.79657706999126E-01*i)*zz4 +( 3.46774471983711E-01 + 1.84997250592251E-01*i)*zz5 +( 1.06957111541383E-01 - 7.52333383560282E-02*i); TITLE : All adjacent 2-by-2 minors of a general 2-by-4 matrix REFERENCES : Persi Diaconis, David Eisenbud, and Bernd Sturmfels: "Lattice Walks and Primary Decomposition", pages 173-193, in Bruce E. Sagan, Richard P. Stanley (eds): "Mathematical Essays in Honor of Gian-Carlo Rota". Progress in Mathematics, Vol 161, Birkh"auser, 1998 Serkan Hosten and Jay Shapiro: "Primary Decomposition of Lattice Basis Ideals", Journal of Symbolic Computation Vol 29, No 4 and 5, pages 625-639, 2000. The original system before the embedding is 3 8 x11*x22 - x21*x12; x12*x23 - x22*x13; x13*x24 - x23*x14; THE SOLUTIONS : 8 13 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -7.81835106284084E-01 7.81248496480405E-02 x22 : -3.51662633063401E-02 -5.23553243566960E-02 x21 : 5.47530470927815E-01 3.94057301107774E-01 x12 : 7.10678068142400E-02 1.85945512463326E-02 zz1 : 3.83320602048486E-32 -3.21523146063693E-32 zz2 : 7.80058054492775E-33 1.16602570490245E-31 zz3 : 4.55450608709025E-33 4.46567366310180E-32 zz4 : 3.17498786143002E-32 -3.49578687051646E-32 zz5 : 0.00000000000000E+00 0.00000000000000E+00 x13 : 4.57720220966937E-01 1.80885251599913E-02 x23 : -2.84274398188580E-01 -2.71772022700639E-01 x14 : -6.85851953187770E-01 -5.30950237874352E-01 x24 : 1.39613283684569E-01 7.31463278657349E-01 == err : 1.698E-15 = rco : 1.009E-03 = res : 1.027E-15 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -1.23673644334083E-01 4.96160159612145E-01 x22 : 5.95996895407253E-32 2.37007982223940E-31 x21 : 1.16208079190208E+00 -2.80709655087489E-01 x12 : -9.39753213075002E-32 2.34522420492634E-31 zz1 : -1.84399183162354E-31 -7.03596145170822E-32 zz2 : 1.24809130501704E-31 -2.23979043246270E-31 zz3 : -2.40274926101279E-31 -4.57625753178842E-32 zz4 : -5.08858518908591E-32 1.18050199922464E-32 zz5 : 0.00000000000000E+00 0.00000000000000E+00 x13 : -1.27880157896229E+00 7.70410053422843E-01 x23 : -6.76258877568251E-01 6.95075407454036E-01 x14 : -1.15627859653324E+00 -8.76704296190128E-01 x24 : -8.71134121989943E-01 -3.59953209097017E-01 == err : 3.023E-15 = rco : 1.006E-02 = res : 2.314E-15 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -8.05723785214199E-02 -2.15893712634767E-01 x22 : 8.12453342046895E-01 1.81323071528954E-01 x21 : 2.15063292619610E-01 -4.67150976615442E-01 x12 : 3.14219594917265E-01 -2.00988315333157E-01 zz1 : 2.33294073605501E-32 -9.25352762830068E-33 zz2 : 2.03452750153742E-32 2.28638624787291E-33 zz3 : 3.23349795706836E-33 1.00258160509371E-33 zz4 : 1.18362739411855E-32 -5.69231157440432E-33 zz5 : 1.58952733222296E-32 -8.34062671595941E-33 x13 : 8.26558486532186E-02 1.91690450412833E-01 x23 : -1.73467864327626E-01 4.32378972591246E-01 x14 : -2.21498689221761E-01 2.28807207246373E-01 x24 : -7.10650434164044E-01 9.22897246354485E-03 == err : 4.918E-16 = rco : 1.180E-02 = res : 1.051E-15 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -6.91211541198169E-01 6.61372088957494E-02 x22 : 3.69008177344594E-31 -5.74902125461742E-31 x21 : 5.59044564557793E-01 3.78780953316725E-01 x12 : 1.82656399631901E-31 6.53698238944254E-31 zz1 : 3.22809119278863E-32 -4.61839384994122E-33 zz2 : 2.01442302549423E-32 2.15865225414852E-32 zz3 : 1.65328410534143E-32 8.87163360859510E-33 zz4 : 1.78422117715697E-32 2.08462144007022E-32 zz5 : 0.00000000000000E+00 0.00000000000000E+00 x13 : 5.05471742251980E-01 -4.80494096526234E-03 x23 : -3.06177932312845E-01 -2.88780339964590E-01 x14 : -6.95504469221639E-01 -4.85130836995601E-01 x24 : 1.37543579653477E-01 6.92512030172235E-01 == err : 8.464E-16 = rco : 1.023E-03 = res : 1.317E-15 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : 1.49692618205883E-01 1.00534540518242E-01 x22 : 8.78349850757110E-01 -1.24473258656925E-01 x21 : -7.50619870801432E-02 1.04896309046187E-01 x12 : -2.10388142927594E-01 -1.22219933023348E+00 zz1 : -1.21319725522088E-32 -1.72620562197088E-32 zz2 : 8.83514916218953E-33 1.82545670357777E-33 zz3 : -9.31320373956874E-34 3.50646783566463E-36 zz4 : -7.04275943048300E-33 -5.24943722920283E-33 zz5 : 1.85694092563793E-33 9.70126906899303E-34 x13 : 8.98538385009676E-33 -2.73949570083539E-32 x23 : 1.64493633552735E-32 -1.95469743996143E-32 x14 : 3.37739393081029E-01 -3.90585839080483E-01 x24 : -8.04979502593136E-01 -9.87317358082179E-02 == err : 5.900E-16 = rco : 8.792E-03 = res : 1.072E-15 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -9.56182118457782E-01 2.48732989520003E-01 x22 : 7.76716306580131E-01 7.57561274849502E-02 x21 : -4.21320797355075E-01 -1.74803788593237E-01 x12 : 1.44057730618487E+00 -8.84306418532794E-01 zz1 : 9.12272622070129E-32 -1.73653100364153E-31 zz2 : 2.67111963963136E-31 1.36392878727691E-31 zz3 : -8.48125266465811E-32 5.06214725852642E-32 zz4 : 8.62321461597352E-32 -1.84202596534452E-33 zz5 : -7.33589044963939E-32 8.84351303885622E-32 x13 : 6.60340931594921E-01 -2.79653718395665E-01 x23 : 3.21018025002783E-01 8.10033366466635E-02 x14 : 5.18410601992180E-01 4.65352801043994E-01 x24 : 6.12179395295897E-02 3.15745155390277E-01 == err : 3.774E-15 = rco : 1.046E-02 = res : 2.026E-15 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -1.68421547240849E+00 6.36963220319880E-01 x22 : -1.82095539220343E-01 -2.80150593578571E-01 x21 : 2.79972834748812E-01 3.07934540533483E-01 x12 : 1.41679841409610E+00 -2.87297035455739E-01 zz1 : 4.57271814438215E-32 3.47954245703111E-32 zz2 : -1.09499732133453E-31 8.91557163668298E-32 zz3 : 1.97562687319355E-32 1.13312816523879E-31 zz4 : -1.23147776649505E-32 -5.34638456763254E-32 zz5 : -1.77729574176619E-32 -4.17961235000356E-33 x13 : 3.11986576332608E-31 -1.74077034969376E-31 x23 : -1.55758160268201E-32 9.34491722936017E-32 x14 : -2.86391770032142E-01 -5.36477449999023E-01 x24 : 4.13309929980160E-01 6.69449316810031E-01 == err : 3.940E-15 = rco : 1.004E-02 = res : 2.276E-15 == solution 8 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -9.55883216190089E-01 1.51052859549073E+00 x22 : -3.39497841713970E-01 -3.00321204064696E-01 x21 : 3.42650160300522E-01 2.63239656935553E-01 x12 : 1.10984028058086E+00 -1.51146339416282E+00 zz1 : 2.49664313413681E-32 6.24780139700468E-32 zz2 : 1.62595863516463E-31 -5.78455979220136E-32 zz3 : 5.32329085045368E-32 3.39862950290465E-33 zz4 : 1.73885731595991E-32 5.96876100720629E-34 zz5 : -8.83928099291515E-32 8.14553470562608E-32 x13 : -8.62541752851413E-01 8.79503087083428E-01 x23 : 1.92795292826678E-01 2.26927432719800E-01 x14 : -3.91514320096083E-01 -1.15602584933586E+00 x24 : -2.86870891319890E-01 6.88869228228266E-02 == err : 4.296E-15 = rco : 1.116E-02 = res : 2.172E-15 ==