**HOM4PS-2.0**

**HOM**otopy method **for**
solving **P**olynomial **S**ystems

**HOM4PS-2.0** is a software package which implements the polyhedral homotopy continuation method for solving polynomial systems in n equations with
n variables, where n > 1. By Bernshtein’s
combinatorial root count theorem [4], the polyhedral homotopies
are established to approximate all the isolated zeros of a polynomial system
using the continuation method [2][3]. Due to fewer homotopy paths, it yields a drastic improvement over the
classical linear homotopies for solving sparse
polynomial systems. Based on several sophisticated iteration criteria from
Numerical Analysis and some good properties in Algebraic Geometry, all
algorithms in the package are skillfully implemented, which plays an essential
ingredient for achieving remarkable efficiency and robustness as reported in
[1]. Consequently, it provides a useful tool for investigating solutions of
polynomial systems.

**References:**

[1] T. L. Lee, T. Y. Li and C. H. Tsai (2008), “** HOM4PS-2.0, A software package
for solving polynomial systems by the polyhedral homotopy
continuation method**”,

[2] B. Huber and B. Sturmfels
(1995), “** A polyhedral method for solving sparse polynomial systems**”,
Math. Comp., 64, pp. 1541-1555.

[3] T. Y. Li (2003), “** Solving polynomial systems by the homotopy continuation method**”, Handbook of
numerical analysis, Vol. XI, Edited by P. G. Ciarlet,
North-Holland,

[4] D. N. Bernshtein (1975), “** The
number of roots of a system of equations**”, Functional Analysis and
Appl., 9(3), pp. 183-185.

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