10 + 1*x11*x22 -1*x21*x12 +( 7.00979162564722E-01 + 7.13181753587443E-01*i)*zz1 +( 9.96720212126009E-01 + 8.09247721002832E-02*i)*zz2 +(-9.14520302965104E-01 + 4.04540004776553E-01*i)*zz3 +(-8.07470126885620E-01 + 5.89908462549336E-01*i)*zz4; -1*x22*x13 + 1*x12*x23 +(-4.31191710639030E-01 - 9.02260333094716E-01*i)*zz1 +( 9.96120998645843E-01 - 8.79940683046823E-02*i)*zz2 +(-4.97895122273411E-01 - 8.67237249670668E-01*i)*zz3 +( 7.67569247697673E-01 - 6.40966028732278E-01*i)*zz4; + 1*zz1; + 1*zz2; + 1*zz3; + 1*zz4; +(-1.12063699620649E-01 - 2.79912161465742E-01*i)*x11 +(-2.44497437126605E-01 - 1.76437224381967E-01*i)*x22 +( 1.48916117912133E-01 + 2.62169946284983E-01*i)*x21 +(-1.48211795779852E-01 + 2.62568761471738E-01*i)*x12 +( 2.96522735211724E-01 + 5.46201282646684E-02*i)*x23 +( 9.87405163307863E-02 + 2.84884891392683E-01*i)*x13 +( 4.34401076310947E-02 - 2.98365627976967E-01*i)*zz1 +(-1.02446585273946E-01 - 2.83573249928125E-01*i)*zz2 +( 2.43750972775824E-01 - 1.77467050969836E-01*i)*zz3 +(-1.37830266114658E-01 + 2.68163958525103E-01*i)*zz4 -3.01511344577764E-01; +(-1.93991976681824E-01 - 7.05917645167077E-02*i)*x11 +(-2.00483628779831E-01 + 4.70131461778525E-02*i)*x22 +(-2.43972932773597E-02 - 2.81414497317197E-01*i)*x21 +( 2.16533683022135E-01 + 2.16443039006691E-02*i)*x12 +(-2.62415852641674E-01 - 1.40274344283176E-01*i)*x23 +( 1.98474077057217E-01 + 6.81064672561189E-04*i)*x13 +(-1.89768269048465E-01 - 5.95473456821662E-02*i)*zz1 +( 1.45070047962431E-01 - 3.36894108362742E-01*i)*zz2 +( 3.63199055640695E-01 + 2.01986346045358E-01*i)*zz3 +( 7.82440017131932E-02 - 3.67893396655775E-01*i)*zz4 +(-1.78575312088296E-01 - 3.73810439089624E-01*i); +( 2.89000838635064E-01 - 8.50547078593371E-02*i)*x11 +(-3.85274753377001E-01 - 4.30178913863514E-02*i)*x22 +(-1.99764715124441E-01 - 1.18016585642271E-02*i)*x21 +( 2.64982711978452E-01 - 1.85777213347526E-01*i)*x12 +(-1.37517738534168E-01 + 9.64570264383342E-02*i)*x23 +( 2.05691441518706E-01 + 3.39791702166995E-01*i)*x13 +( 2.96104345368854E-01 + 1.42071116937982E-01*i)*zz1 +( 1.07216427586946E-01 + 2.35804684224362E-01*i)*zz2 +( 1.27830843360944E-01 - 3.86909972664540E-01*i)*zz3 +( 2.92954511175483E-01 + 1.54215291548217E-02*i)*zz4 +( 3.35637205342807E-02 + 3.52261198253576E-03*i); +(-3.13660040110270E-01 - 2.59912845309725E-01*i)*x11 +( 2.57499692370116E-01 - 2.16660085333904E-01*i)*x22 +(-2.48925039938868E-01 - 3.98824162025051E-02*i)*x21 +( 3.08772972388443E-01 - 3.65421111597284E-02*i)*x12 +(-1.93670318127096E-02 - 4.28134050364957E-01*i)*x23 +(-5.67188419549648E-02 + 2.03642874598384E-01*i)*x13 +( 2.16093843264145E-01 - 9.50333332297792E-02*i)*zz1 +(-4.16627974441252E-03 + 2.31522188527959E-01*i)*zz2 +(-2.32971526192318E-01 - 5.85933413960365E-02*i)*zz3 +(-2.96774083470275E-01 - 1.48890755413144E-01*i)*zz4 +(-2.29912864705795E-01 + 4.54251102450386E-02*i); TITLE : All adjacent 2-by-2 minors of a general 2-by-3 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 2 6 x11*x22 - x21*x12; x12*x23 - x22*x13; THE SOLUTIONS : 4 10 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -1.14983794944269E+00 -3.13718956353879E-02 x22 : -2.89233048354529E-02 -8.34375178145198E-01 x21 : -9.82132190039170E-01 -6.00231738072066E-02 x12 : -6.67176560299574E-02 -9.73696847573085E-01 zz1 : -4.86351308835838E-32 1.38252097512100E-32 zz2 : -1.09259227319259E-32 1.68725192195309E-32 zz3 : -1.05752065961400E-32 -1.36980995125038E-32 zz4 : 0.00000000000000E+00 0.00000000000000E+00 x13 : -2.88631477831556E-01 -8.46006426315723E-01 x23 : -2.22332592575923E-01 -7.31615900433522E-01 == err : 1.358E-15 = rco : 2.593E-02 = res : 1.396E-15 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -1.51540407780692E+00 -1.61355990745372E-01 x22 : 6.06575205742999E-32 1.36728808177497E-32 x21 : -6.69203026448905E-01 9.60647396382458E-01 x12 : 1.29737121633557E-32 1.54521450250898E-32 zz1 : 2.46151560532688E-32 -5.14490923236620E-32 zz2 : 3.54738132153542E-32 -7.23124254422673E-33 zz3 : 2.78309593471004E-35 3.09645508597812E-32 zz4 : -2.23647638363484E-32 -1.14090495735248E-32 x13 : 4.22783534987393E-03 -7.65192972921765E-01 x23 : 8.09402615355541E-01 -1.28061583951889E+00 == err : 1.343E-15 = rco : 2.311E-02 = res : 6.106E-16 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -1.05249673784490E+00 -4.83817258892493E-01 x22 : -9.34190897342867E-01 1.04952753778098E+00 x21 : -2.06620548439213E+00 1.29536996278851E+00 x12 : -6.60170245818440E-01 -9.80144897163359E-02 zz1 : 2.79667364935024E-32 -2.49257543376289E-33 zz2 : -1.54181574225645E-32 7.76857446048649E-33 zz3 : 0.00000000000000E+00 0.00000000000000E+00 zz4 : 0.00000000000000E+00 0.00000000000000E+00 x13 : 5.85937791301550E-01 6.56779943720218E-01 x23 : 1.83257395304603E+00 -2.74200184632055E-01 == err : 1.888E-15 = rco : 1.927E-02 = res : 2.280E-15 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x11 : -1.03564593637362E+00 1.34937006216365E+00 x22 : -1.16738916418956E+00 9.09884980344596E-01 x21 : 1.09831288855712E-01 -1.01782420829222E+00 x12 : 2.44303409988259E+00 -2.82063729669101E-01 zz1 : 0.00000000000000E+00 0.00000000000000E+00 zz2 : -1.09963523584478E-31 -1.35448710724225E-31 zz3 : 9.67208936990035E-32 8.44632109660740E-32 zz4 : 2.28948729285679E-32 1.49094264337348E-31 x13 : -1.52338074076645E+00 2.11982446978267E+00 x23 : 1.19296740907002E-01 -1.56654059306176E+00 == err : 3.537E-15 = rco : 1.195E-02 = res : 3.495E-15 ==