5 x1 + x1*x2 + x2*x3 + x3*x4 - u5; x2 + x1*x3 + x2*x4 - 2*u5; x3 + x1*x4 - 3*u5; x4 - 4*u5; x1 + x2 + x3 + x4 + 1; TITLE : reduced 5-dimensional economics problem ROOT COUNTS : total degree : 8 2-homogeneous Bezout number : 12 with partition : {x1 x2 x3 }{x4 u5 } generalized Bezout number : 8 based on the set structure : {x1 x3 u5 }{x2 x4 } {x1 x2 u5 }{x3 x4 } {x1 x3 u5 }{x4 } {x4 u5 } {x1 x2 x3 x4 } mixed volume : 8 REFERENCE : This is the reduced economics problem (u5 = 1/x5). Alexander Morgan: `Solving polynomial systems using continuation for engineering and scientific problems', Prentice-Hall, Englewood Cliffs, New Jersey, 1987. (p 148). THE SOLUTIONS : 8 5 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 8.97399802625560E-01 2.18519562237977E+00 x2 : -2.46488856119036E+00 1.64476478851049E+00 x3 : -1.04157021643729E+00 -3.46184909267653E+00 x4 : 1.60905897500210E+00 -3.68111318213733E-01 u5 : 4.02264743750524E-01 -9.20278295534333E-02 == err : 6.004E-15 = rco : 3.043E-02 = res : 1.110E-15 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -1.47399802625560E-01 -2.18519562237977E+00 x2 : -1.67791479309413E+00 6.38326735148376E-01 x3 : 2.34742504765713E-01 1.41176117876408E+00 x4 : 5.90572090953981E-01 1.35107708467322E-01 u5 : 1.47643022738495E-01 3.37769271168304E-02 == err : 4.822E-15 = rco : 1.027E-01 = res : 9.930E-16 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 1.00000000000000E+00 1.04404871487976E-53 x2 : 1.00000000000000E+00 2.78412990634604E-53 x3 : 1.00000000000000E+00 4.17619485951906E-53 x4 : -4.00000000000000E+00 4.17619485951906E-53 u5 : -1.00000000000000E+00 0.00000000000000E+00 == err : 3.762E-37 = rco : 8.800E-02 = res : 1.218E-52 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 7.94799605251120E-01 -7.46761833343337E-60 x2 : -1.14417041381173E+00 7.15646756954031E-60 x3 : 3.05149904685739E-02 -3.49707676362942E-60 x4 : -6.81144181907962E-01 4.35611069450280E-60 u5 : -1.70286045476991E-01 1.08902767362570E-60 == err : 2.077E-15 = rco : 6.837E-02 = res : 5.551E-17 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -1.47399802625560E-01 2.18519562237977E+00 x2 : -1.67791479309413E+00 -6.38326735148376E-01 x3 : 2.34742504765713E-01 -1.41176117876408E+00 x4 : 5.90572090953981E-01 -1.35107708467322E-01 u5 : 1.47643022738495E-01 -3.37769271168304E-02 == err : 5.070E-15 = rco : 1.027E-01 = res : 9.486E-16 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -4.47996052511202E-02 4.34324265389982E-51 x2 : 1.67977712238073E+00 -7.35010295275354E-51 x3 : -1.16685956712542E+00 1.67047794380762E-51 x4 : -1.46811795000419E+00 -6.68191177523049E-52 u5 : -3.67029487501048E-01 -3.34095588761525E-52 == err : 4.160E-15 = rco : 7.638E-02 = res : 1.665E-16 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 8.97399802625560E-01 -2.18519562237977E+00 x2 : -2.46488856119037E+00 -1.64476478851049E+00 x3 : -1.04157021643729E+00 3.46184909267653E+00 x4 : 1.60905897500210E+00 3.68111318213733E-01 u5 : 4.02264743750524E-01 9.20278295534332E-02 == err : 6.125E-15 = rco : 3.043E-02 = res : 1.986E-15 == solution 8 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -2.50000000000000E-01 0.00000000000000E+00 x2 : -2.50000000000000E-01 0.00000000000000E+00 x3 : -2.50000000000000E-01 0.00000000000000E+00 x4 : -2.50000000000000E-01 0.00000000000000E+00 u5 : -6.25000000000000E-02 0.00000000000000E+00 == err : 0.000E+00 = rco : 9.055E-02 = res : 0.000E+00 ==