In this paper we have presented a package for solving polynomial systems by homotopy continuation. One of the main features of the software is the wide range of available homotopy methods, suitable to deal with various structures, such as dense, symmetric, product and sparse polynomial systems.
We mention two main directions for further research. The first direction consists of the elaboration of the package as solver based on homotopy continuation. Although the performance results are not that bad, the package needs further elaboration because not every feature uses the most optimal algorithm for its implementation. Furthermore, the package has been written in Ada 83, not using the full object-oriented capabilities of the new standard.
Another interesting direction could be the application of the developed methods to one application area. This would involve the construction of dedicated homotopies and the careful tune-up of continuation methods. Already for mechanical design problems, Charles Wampler illustrates in [25] the practical usefulness of polyhedral methods on some practical applications. It would be worthwhile to investigate whether the progress made in solving sparse polynomial systems could be useful to solve other engineering problems. In order to be effective, the problem formulations should be generated to be as sparse as possible, whereas previous formulations were more focussed on obtaining lower degree equations.
Acknowledgments. Influential suggestions to improve the program arose in discussions with Marc Beckers, Ioannis Emiris, Karin Gatermann, Birk Huber, Tien-Yien Li, Pierre Verlinden and Charles Wampler. We gratefully acknowledge their concern and advice.
We are indebted to Dirk Craeynest for his help with Ada, in particular, for introducing us to the gnu-ada compiler. Thanks go to Karel de Vlaminck and Yvan Barbaix for putting their Ada technology at our disposal.