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Numerically Solving Polynomial Systems with Bertini

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by Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, Charles W. Wampler


2013 / xx + 352 pages / Softcover / ISBN 978-1-611972-69-6 / List Price $95.00 / SIAM Member Price $66.50 / Order Code SE25

Keywords: numerical algebraic geometry, polynomial continuation, homotopy methods, numerical analysis, computational algebraic geometry


This book is a guide to concepts and practice in numerical algebraic geometry—the solution of systems of polynomial equations by numerical methods. Through numerous examples, the authors show how to apply the well-received and widely used open-source Bertini software package to compute solutions, including a detailed manual on syntax and usage options. The authors also maintain a complementary web page where readers can find supplementary materials and Bertini input files.

Numerically Solving Polynomial Systems with Bertini approaches numerical algebraic geometry from a user's point of view with numerous examples of how Bertini is applicable to polynomial systems. It treats the fundamental task of solving a given polynomial system and describes the latest advances in the field, including algorithms for intersecting and projecting algebraic sets, methods for treating singular sets, the nascent field of real numerical algebraic geometry, and applications to large polynomial systems arising from differential equations.

Those who wish to solve polynomial systems can start gently by finding isolated solutions to small systems, advance rapidly to using algorithms for finding positive-dimensional solution sets (curves, surfaces, etc.), and learn how to use parallel computers on large problems. These techniques are of interest to engineers and scientists in fields where polynomial equations arise, including robotics, control theory, economics, physics, numerical PDEs, and computational chemistry.

The book is designed to serve the following audiences:

  • Scientists and engineers needing to quickly solve systems of polynomial equations, particularly those looking only for isolated roots, will find what they need in the "Bertini Quick Start Guide," part of the detailed "Bertini Users Manual" in Part IV of the book. These readers may also wish to refine their technique by reading Part I.
  • Engineers or scientists and senior undergraduate or beginning graduate students with a computational focus who have a knowledge of calculus, linear algebra, and undergraduate-level ODEs can become expert Bertini users by studying Parts I and II, including the examples and code, which is available online.
  • Those with a more mathematical bent can find details of the underpinnings of the methods in the more technical sections (marked with an asterisk throughout the book) and can read descriptions of the latest developments in Part III.

About the Authors
Daniel J. Bates is an Assistant Professor of Mathematics at Colorado State University. He is a member of the American Mathematical Society (AMS) and SIAM and is an active member of the SIAM Activity Group on Algebraic Geometry.

Jonathan D. Hauenstein is an Assistant Professor of Mathematics at North Carolina State University. He is a member of the American Mathematical Society (AMS) and SIAM. Hauenstein received a DARPA Young Faculty Award in 2013 and is an active member of the SIAM Activity Group on Algebraic Geometry.

Andrew J. Sommese is a Professor in the Department of Applied and Computational Mathematics and Statistics at the University of Notre Dame, where he has been the Vincent J. and Annamarie Micus Duncan Professor of Mathematics since 1994. Sommese received an Alfred P. Sloan Fellowship in 1979 and the Alexander von Humboldt Research Award for Senior U.S. Scientists in 1993 and became a Fellow of the American Mathematics Society in 2012. He is a member of SIAM and is currently on the editorial boards of Advances in Geometry, Milan Journal of Mathematics, and Journal of Algebra and Its Applications.

Charles W. Wampler has been employed at the General Motors Research and Development Center in Warren, Michigan, since 1985, rising to the rank of Technical Fellow in 2003. He is also an Adjunct Professor at the University of Notre Dame in the Department of Applied and Compuational Mathematics and Statistics. He is a Fellow of the American Society of Mechanical Engineers (ASME) and the Institute of Electrical and Electronics Engineers (IEEE). He won the ASME Mechanism and Robotics Award in 2007 and has won numerous awards within General Motors. He serves on the board of the International Journal of Robotics Research and is a member of ASME, IEEE, and SIAM.

ISBN 9781611972696

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