##
**Algorithms in real algebraic geometry.
2nd ed.**
*(English)*
Zbl 1102.14041

Algorithms and Computation in Mathematics 10. Berlin: Springer (ISBN 3-540-33098-4/hbk). x, 662 p. (2006).

This is the second edition of the monograph of the same authors [Algorithms in Real Algebraic Geometry, Springer, Berlin (2003; Zbl 1031.14028)], which appears due to high interest and large demand by researchers working in the field and related areas. Main changes made by the authors in the second edition, intend to improve presentation as a teaching source. In particular, the cylindrical decomposition algorithm and the real root counting have been made independent on some advanced technical polynomial computation, and respectively, these topics appear in the book earlier.

Among new topics included in the book are normal polynomials and virtual roots, discriminants of symmetric matrices, computation of the first Betti number of a semi-algebraic set in a single-exponential time. The updated bibliography reflects the last years achievements in computational real algebraic geometry.

Among new topics included in the book are normal polynomials and virtual roots, discriminants of symmetric matrices, computation of the first Betti number of a semi-algebraic set in a single-exponential time. The updated bibliography reflects the last years achievements in computational real algebraic geometry.

Reviewer: Eugenii I. Shustin (Tel Aviv)

### MathOverflow Questions:

Can one compute the fundamental group of a complex variety? Other topological invariants?How can I prove that \((n-1)\)-dimensional manifold is not contained in a \((n-2)\)-dimensional affine variety?

### MSC:

14P10 | Semialgebraic sets and related spaces |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14Q99 | Computational aspects in algebraic geometry |

68W30 | Symbolic computation and algebraic computation |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |