Wiki

Overview

The solution of arbitrary polynomial systems is an area of active research, and has many applications in math, science and engineering. This software, Bertini 2, is a complete re-implementation of Bertini 1 from C into C++/Python.

The theoretical basis for the solution of polynomials with Bertini is a theorem which bounds the number of solutions a system may have. It sits together with the numerical computational tool of "homotopy continuation". the act of "continuing" from one system into another through a "homotopy", as depicted in the below diagram:

homotopy continuation

Current capabilites

Bertini 2 currently has implemented the foundations of Numerical Algebraic Geometry. Development is ongoing, and here's what we have so far:

C++ functions and types, with Python bindings.
Through Python, runtime scriptable construction of systems and interactivity with their zero-dimensional solutions.
Construction of multivariate polynomial and non-polynomial systems.
Evaluation of systems and their Jacobians in double and arbitrary multiple precision, using two different methods.
Construction of the Total Degree and Multihomogeneous start systems.
Construction of homotopies (they're just systems with path variables defined).
Tracking of a start point x_0, corresponding to a particular time $t_0 \in \mathbb{C}^n$ in a homotopy $H$, from $t_0$ to $t_1$.
Running of the Power Series and Cauchy endgames, in double, multiple, and adaptive precision.

Development is ongoing, and we want your help!

Missing functionality

Parallel solving
Numerical irreducible decomposition
Membership testing
and other algorithms

Users wanting a more developed implementation are recommended to use Bertini 1 or homotopycontinuation.jl, or one of the other packages implementing the theory.

Installation

Pre-built wheels – the way to go!

The Python package bertini2 provides pre-built wheels for Linux, macOS, and Windows.

pip install bertini2

Once it's installed, you import bertini

Linux: Python 3.9-3.13
MacOS (Apple Silicon): Python 3.9-3.13
MacOS (Intel): not supported
Windows: Python 3.9-3.11

Building from source

Please see the Wiki compiling section for instructions on compiling Bertini 2.

Other information

The official project repository is hosted on GitHub at github.com/bertiniteam/b2.

Please note that this is a long-term project, and is under active development. If you want to help, please see the wiki for contact information. We have opportinuties for all skill levels and interests.

License

Bertini 2 is Free and Open Source Software. Source is available under GPL Version 3, with additional terms as permitted under Section 7.

Thank yous

A huge thank you to:

HongKee Moon, for his help in getting this package to be pip-installable with a comprehensive CI build system.
Jack Hagen, for helping get away from the autotools and replacing CMake.
Mike Mumm, for helping with straight-line-programs
Jeb Collins, for writing much of the parser system, implementing the predictors, and so much more
Tim Hodges, for contributing to the endgame implementations
Alan Liddel, for tons of help with Python parts
Dan Bates, Jon Hauenstein, for critical advise support, and guidance
Andrew Sommese, Charles Wampler, Dan Bates, Jon Hauenstein, for writing Bertini 1

And to all the other people who have contributed to this package over the years.