The database of polynomial systems

The demonstration database is an essential part of PHCpack. Motivations are plentiful:

Examples of data formats: As part of the user's guide, we provide a bounty of examples, which show the input formats and how the results of calculations can be preserved.
Capabilities of the program: We are proud to list some systems which are really hard for other solvers, although execution times are very transient thanks to rapid advances in hardware.
Relevance to practical applications: One important criterion for inclusion in the collection is the connection with an application field. We strived to provide accurate references to the literature, documenting the source of the application.

The database is tailored towards PHCpack. Besides the peculiar format of solution lists, this implies that the systems often have floating-point input coefficients and the focus is on approximating the isolated roots. Moreover, one criterion for entering the database is that it can be solved with PHCpack.

We tried to find original references, sometimes a piece of Maple code accompanies the derivation of the polynomial equations. An exciting part of the field of polynomial system solving is that one encounters applications as diverse as game theory, mechanical design, chemical reactions, control of linear systems, etc... We hope that the relevance of these applications will help to elevate our field to approximate the status currently enjoyed by areas such as Linear Algebra, Optimization, or Differential Equations.

We warn that a general-purpose solver will in general not be competitive against methods tailored for a specific application. Homotopies unable to exploit structure usually fail and end up squandering computer resources.

Acknowledgment. Ailing Zhao helped to set up the navigation tree.

This material is based upon work supported by the National Science Foundation under Grants No. 9804846, 0105739, and 0134611. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.