Hom4PS-3 has its own project website now. Please visit here.






Solving systems of polynomial equations is an important problem in mathematics. It has a wide range of applications in many fields of mathematics, sciences, and engineering. By the Abel’s impossibility theorem and Galois theory, explicit formulae for solutions to such systems by radicals are unlikely to exist, as a result, numerical computation arises naturally in the solution to such systems. Homotopy continuation methods represent a major class of numerical methods for solving systems of polynomial equations. Hom4PS-3 is a software package that implements many homotopy continuation algorithms with which it could numerically approximate, identify, and classify solutions to systems of polynomial equations.

Hom4PS-3 is an ongoing project currently under active development. For the more stable version, you are encouraged to try out Hom4PS-2.0 which is more mature and stable.

The Hom4PS-3 team

The Hom4PS-3 team is led by Professor Tien-Yien Li.

  • Tianran Chen, Michigan State University
  • Tien-Yien Li, Michigan State University
  • Tsung-Lin Lee, National Sun Yat-sen University

We are supported by our research assistants

  • Nick Ovenhouse


Hom4PS-3 is has many new features.

Homotopy constructions

There are many different homotopy constructions for solving systems of polynomial equations each with its own strength. Hom4PS-3 currently supports the following homotopy constructions:

  • Total degree homotopy
  • Cheater’s homotopy
  • Polyhedral homotopy
  • Complex Newton’s homotopy

Other homotopy constructions are still under development.

Parallel computation

Hom4PS-3 is capable of performing computation in parallel on a wide range of parallel computer architectures, including:

  • Multi-core and Many-core architecture
  • Computer clusters
  • Distributed computation environments

Features that improve numerical robustness

  • Automatic multi-precision
  • Projective path tracking
  • Singular end game

Related software

Hom4PS-3 is directly related to earlier software packages in the Hom4PS family.

There are other software packages that also implement numerical homotopy continuation algorithms for solving systems of polynomial equations: