Discriminants, resultants, and multidimensional determinants. Reprint of the 1994 edition.

*(English)*Zbl 1138.14001
Modern Birkhäuser Classics. Boston, MA: Birkhäuser (ISBN 978-0-8176-4770-4/hbk). x, 523 p. (2008).

The book under review is the unaltered reprint of the original edition first published in [Mathematics: Theory & Applications. Boston, MA: Birkhäuser (1994; Zbl 0827.14036)]. Therefore, as for its precise contents, we may refer to the exhaustive and enthusiastic review by W. Bruns, in which the book’s outstanding significance as an unique standard-text of modern combinatorial algebra and elimination theory was both predicted and profoundly substantiated. Reviewers of other leading journals have unanimously expressed the same overwhelming appreciation for the authors’ remarkable revival and expansion of the classical theory-discriminants, resultants, and determinants in the context of contemporary algebraic geometry, homological algebra; and combinatorial geometry. In fact, the book has developed into an indispensable modern classic, during the past 15 years, and it has initiated a true avalanche of related research activities due to its pioneering and inspiring impact. The steadily large number of quotations, which seems to be ever-flowing, be speaks the undiminished pacemaking role of this unrivalled source book in various areas of current mathematical research, ranging from algebraic geometry and the general theory of hypergeometric functions up to their recent applications in mathematical physics.

Now as before this utmost lucid and comprehensive monograph presents a unique blend of classical 19th century mathematics on the one hand, and very recent developments in algebraic geometry (Chow varieties, toric varieties, projective varieties, and hypersurfaces), combinatorial geometry and algebra (polytopes and their triangulations, aymmetric functions, rings of invariants, and special rings in local algebra), homological algebra (discriminantal complexes, resolutions, and differential complexes), and microlocal analysis (\(D\)-modules, perverse sheaves, and \(D\)-equivalence) on the other.

It is and remains this broader, unifying approach combining the classical heritage and the powerful abstract viewpoint of the subject that makes the book under review so timeless, attractive, enlightening, inspiring and virtually invaluable – a characteristic that will persist for further decades in the future. Besides, the comparatively inexpensive reprint of this classic in paperback form must be seen as another rewarding service to the mathematical community as a whole.

Now as before this utmost lucid and comprehensive monograph presents a unique blend of classical 19th century mathematics on the one hand, and very recent developments in algebraic geometry (Chow varieties, toric varieties, projective varieties, and hypersurfaces), combinatorial geometry and algebra (polytopes and their triangulations, aymmetric functions, rings of invariants, and special rings in local algebra), homological algebra (discriminantal complexes, resolutions, and differential complexes), and microlocal analysis (\(D\)-modules, perverse sheaves, and \(D\)-equivalence) on the other.

It is and remains this broader, unifying approach combining the classical heritage and the powerful abstract viewpoint of the subject that makes the book under review so timeless, attractive, enlightening, inspiring and virtually invaluable – a characteristic that will persist for further decades in the future. Besides, the comparatively inexpensive reprint of this classic in paperback form must be seen as another rewarding service to the mathematical community as a whole.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

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

14M12 | Determinantal varieties |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

13D25 | Complexes (MSC2000) |

13C40 | Linkage, complete intersections and determinantal ideals |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |