+ terms with positive powers of t Theorem (Huber and Sturmfels): For almost all choice of the (complex) coefficients and the (rational) powers of t, the polyhedral homotopy “works”.
PHCpack : Jan Verschelde HOM4PS : T. Gao, T.Y. Li, and M. Wu (MixedVol) HOM4PS-2.0 : T.L. Lee, T.Y. Li, C.H. Tsai (MixedVol-2.0) Software : homotopy methods for polynomial systems PHoM : T. Gunji, S. Kim, M. Kojima, A. Takeda, K. Fujisawa, T. Mizutani Bertini : D. Bates, J. Hauenstein, A. Sommese, C. Wampler (1999) (2004) (2006) (2008) (2003)
n Total degreeMixed Volume# of isolated zero eco- 153,188,6468,192 169,565,93816,384 noon- 919,68319,665 1059,04959,029 cyclic- 9362,88011,0166,642 103,628,80035,94034,940
n PHomPHCpackHOM4PSHOM4PS-2.0 eco- 15>12h3h55m33m15s2m25s 16>12h 2h55m6m36s noon- 95h01m33m28s21m41s22s 10>12h2h33m3h20m1m27s cyclic- 9>12h3h50m8m37s44s 10>12h11h00m57m44s2m47s Running on a Dell PC with Pentium4 CPU of 2.2GHz
# of CPUs CPU time (second) ratio 1445.321.00 2223.491.99 3150.692.96 591.314.88 768.706.48 # of CPUs CPU time (second) ratio 11475.391.00 2734.962.00 3494.192.99 5295.904.99 7212.876.93 eco-16 cyclic-11 The scalability of solving systems by polyhedral homotopy method
PHCpackHOM4PS-2.0 # of complex solutions 26,53327,235 # of real solutions 133135 # of physical solutions 3132 running time 11h51m7m20s 4-body Albouy-Chenciner equation with m1=m2=m3=m4 Running on a Dell PC with Pentium D CPU of 3.2GHz
5-body Albouy-Chenciner equation (1) 20 equations in 20 variables (2) total degree = 61,917,364,224 (3) mixed volume = 439,631,712
5-body Albouy-Chenciner equation with equal masses (1)HOM4PS-2.0 (2)Call subroutines in MPI library (3)32 CPUs from SHARCNET (4)6 days (1)101,062,826 solutions. (2) 8775 are real solutions. (3) 258 are physical solutions (rij>0)
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