Polynomial systems occur in a wide variety of application domains. Homotopy continuation methods are reliable and powerful methods to compute numerically approximations to all isolated complex solutions. During the last decade considerable progress has been accomplished on exploiting structure in a polynomial system, in particular its sparsity. In this paper the structure and design of the software package PHC is described. The main program operates in several modes, is menu-driven and file-oriented. This package features a great variety of root-counting methods among its tools. The outline of one black-box solver is sketched and a report is given on its performance on a large database of test problems. The software has been developed on four different machine architectures. Its portability is ensured by the gnu-ada compiler.

D.3.2 Programming Languages Language Classification [Ada]
G.1.5 Numerical Analysis Roots of Nonlinear Equations
[Systems of equations, Polynomials, methods for]
G.2.1 Discrete Mathematics Combinatorics [Counting problems]
G.4 Mathematics of Computing Mathematical Software
Algorithms, Theory
homotopy continuation, polynomial systems, start system, root count,
Bézout number, mixed volume, Bernshtein's theorem, polyhedral homotopy,
enumerative geometry, Schubert calculus.

for polynomial systems by homotopy continuation

**Jan Verschelde
Mathematical Sciences Research Institute**

- Introduction
- Related Software
- Root Counts and Start Systems
- Polynomial Continuation and End Games
- The Four Stages of the Solver
- Execution Modes and Tools
- The Internal Design: the Libraries of PHCpack
- On Portability: Computers and Compilers
- Putting it all together: one Black-Box Solver
- The Test Database of Polynomial Systems
- Obtaining and Installing PHC
- Conclusions and Future Developments
- References
- Getting Started
- Reference Manual
- Index
- About this document ...