bertini2 2.0.1

Software for numerical algebraic geometry

Quick links

• Wiki

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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:

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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!

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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.

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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.

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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.

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License

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

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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.