PHCpack in Macaulay

The Macaulay2 package PHCpack provides an interface to PHCpack, a generalpurpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The package also provides a fast mixed volume computation, the ability to filter solutions, homotopy path tracking, and a numerical irreducible decomposition method. As the size of many problems in applied algebraic geometry often surpasses the capabilities of symbolic software, this package will be of interest to those working on problems involving large polynomial systems. 1 Numerical homotopy continuation and PHCpack Many problems in applied algebraic geometry require solving, or counting the solutions of, a large polynomial or rational system. PHCpack is an interface to the program PHCpack, one of several efficient polynomial system solvers that use numerical homotopy continuation methods [5]. The basic idea behind homotopy continuation is simple: to solve a polynomial system f(x) = 0, one first constructs a system g(x) = 0 that is easy to solve and then constructs a homotopy, H(x(t)) = (1 − t)g(x) + tf(x), in order to numerically track paths from known solutions of g (with t = 0) to the solutions of the target system f (with t = 1). Available since release 1.4 of Macaulay2 [3], this package is motivated by [4] and uses the data types defined by Leykin in NAGtypes.m2. The main function of the package allows a Macaulay2 user to solve a system numerically through a blackbox solver, where the creation of the start system and homotopy continuation is done behind the scenes. The package also provides a fast mixed volume computation and allows the user to filter solutions, to track solution paths explicitly, and to perform numerical irreducible decompositions. EG and JV are with Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607. SP is with Department of Statistics, Pennsylvania State University, University Park, PA 16802. This material is based upon work partly supported by the National Science Foundation under Grants No. 0713018 and 1115777. The authors are grateful to Anton Leykin for his generous help with NAGtypes.m2. Work on this package was carried out while SP was in residence at the University of Illinois at Chicago, and latest contribution was made during the 2012 Macaulay2 workshop, supported by the US National Science Foundation.