Numerically solving polynomial systems with Bertini.

*(English)*Zbl 1295.65057
Software - Environments - Tools 25. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611972-69-6/pbk). xx, 352 p. (2013).

Several problems in science and engineering require the solution of multivariate polynomial systems. Many times, such systems are solved by means of numerical algorithms. In particular, the authors of the book under review and colleagues have contributed a series of techniques aim to numerically compute and manipulate the solution sets of polynomial systems by homotopic continuation methods.

The purpose of the present book is twofold. On one hand, it provides an introduction to the methodology developed by the authors for the solution and description of the geometry of the sets of solutions of polynomial systems. On the other hand, the authors show how to use an open-source software package, called Bertini, where the corresponding algorithms are implemented.

The first part of the book, which comprises Chapters 1–7, focusses on isolated solutions of polynomial systems. Chapters 1 introduces polynomial systems and their solution sets, including a first approach to singularities and multiplicity. Chapters 2 and 3 discuss the basic components of homotopy continuation methods, together with two critical features which allow Bertini to deal with singularities: adaptive precision and endgames. In Chapter 4, the projective space is introduced as a means to numerically deal with solution paths diverging to infinity. Chapter 5 covers the main alternatives to the standard total-degree homotopy, namely multihomogeneous, linear-product and polyhedral homotopies. An equation-by-equation approach, called the regeneration procedure, is also discussed. Chapter 6 is concerned with the solution of parameterized families of polynomial systems, by means of parameter homotopies. Finally, a number of advanced topics in connection with the computation of isolated solutions is treated in Chapter 7.

The second part of the book, which consists of Chapters 8–11, is devoted to the numerical treatment of positive-dimensional systems. Chapter 8 introduces the fundamental tool for representing positive-dimensional irreducible sets, namely witness sets. Numerical irreducible decompositions are also discussed. Chapter 9 and 10 are devoted to describe the algorithms for computing witness sets and a numerical irreducible decomposition. Chapter 11 includes advanced topics about positive-dimensional algebraic sets, including the concept of multiplicities and the deflation method for reducing multiplicity.

In the third part of the book, namely Chapters 12–17, further applications and algorithms are discussed. Chapter 12 considers the intersection of irreducible algebraic sets and, more generally, fiber products, by means of witness sets. Chapter 13 is devoted to singularities on positive-dimensional sets, including solution sets with a common singularity structure, called isosingular sets. Chapter 14 treats a critical point for applications, namely the computation of real solutions. Chapter 15 briefly discusses applications of homotopy continuation methods to algebraic geometry, such as the computation of the geometric genus of a curve and characteristic classes. Chapter 16 explains how to numerically manipulate images of algebraic sets under algebraic maps without requiring explicit defining equations for the image. Finally, Chapter 17 includes a brief overview to methods for solving certain large polynomial systems which arise from discretizations of ordinary and partial equations.

The book ends with a number of appendices, where the authors give a thorough users manual for Bertini.

Many interesting examples from engineering and science are used throughout the book, which are completely solved by means of Bertini. In this way, the readers are provided with clear and detailed illustrations of the use of Bertini. Two authors of the book have published a monograph on the subject, which focusses on theory behind the algorithms [A. J. Sommese and C. W. Wampler II, The numerical solution of systems of polynomials. River Edge, NJ: World Scientific (2005; Zbl 1091.65049)]. As the authors state, the present book concentrates on the practical usage of the algorithms. In this sense, the book is a good introduction to numerical methods for solving polynomial systems suitable for engineers, scientists or numerical analysts not acquainted with an algebraic geometry background.

The purpose of the present book is twofold. On one hand, it provides an introduction to the methodology developed by the authors for the solution and description of the geometry of the sets of solutions of polynomial systems. On the other hand, the authors show how to use an open-source software package, called Bertini, where the corresponding algorithms are implemented.

The first part of the book, which comprises Chapters 1–7, focusses on isolated solutions of polynomial systems. Chapters 1 introduces polynomial systems and their solution sets, including a first approach to singularities and multiplicity. Chapters 2 and 3 discuss the basic components of homotopy continuation methods, together with two critical features which allow Bertini to deal with singularities: adaptive precision and endgames. In Chapter 4, the projective space is introduced as a means to numerically deal with solution paths diverging to infinity. Chapter 5 covers the main alternatives to the standard total-degree homotopy, namely multihomogeneous, linear-product and polyhedral homotopies. An equation-by-equation approach, called the regeneration procedure, is also discussed. Chapter 6 is concerned with the solution of parameterized families of polynomial systems, by means of parameter homotopies. Finally, a number of advanced topics in connection with the computation of isolated solutions is treated in Chapter 7.

The second part of the book, which consists of Chapters 8–11, is devoted to the numerical treatment of positive-dimensional systems. Chapter 8 introduces the fundamental tool for representing positive-dimensional irreducible sets, namely witness sets. Numerical irreducible decompositions are also discussed. Chapter 9 and 10 are devoted to describe the algorithms for computing witness sets and a numerical irreducible decomposition. Chapter 11 includes advanced topics about positive-dimensional algebraic sets, including the concept of multiplicities and the deflation method for reducing multiplicity.

In the third part of the book, namely Chapters 12–17, further applications and algorithms are discussed. Chapter 12 considers the intersection of irreducible algebraic sets and, more generally, fiber products, by means of witness sets. Chapter 13 is devoted to singularities on positive-dimensional sets, including solution sets with a common singularity structure, called isosingular sets. Chapter 14 treats a critical point for applications, namely the computation of real solutions. Chapter 15 briefly discusses applications of homotopy continuation methods to algebraic geometry, such as the computation of the geometric genus of a curve and characteristic classes. Chapter 16 explains how to numerically manipulate images of algebraic sets under algebraic maps without requiring explicit defining equations for the image. Finally, Chapter 17 includes a brief overview to methods for solving certain large polynomial systems which arise from discretizations of ordinary and partial equations.

The book ends with a number of appendices, where the authors give a thorough users manual for Bertini.

Many interesting examples from engineering and science are used throughout the book, which are completely solved by means of Bertini. In this way, the readers are provided with clear and detailed illustrations of the use of Bertini. Two authors of the book have published a monograph on the subject, which focusses on theory behind the algorithms [A. J. Sommese and C. W. Wampler II, The numerical solution of systems of polynomials. River Edge, NJ: World Scientific (2005; Zbl 1091.65049)]. As the authors state, the present book concentrates on the practical usage of the algorithms. In this sense, the book is a good introduction to numerical methods for solving polynomial systems suitable for engineers, scientists or numerical analysts not acquainted with an algebraic geometry background.

Reviewer: Guillermo Matera (Buenos Aires)

##### MSC:

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

12Y05 | Computational aspects of field theory and polynomials (MSC2010) |

14Q15 | Computational aspects of higher-dimensional varieties |

65H04 | Numerical computation of roots of polynomial equations |

65H10 | Numerical computation of solutions to systems of equations |

65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |

65Y15 | Packaged methods for numerical algorithms |

68W30 | Symbolic computation and algebraic computation |