Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Parameter spaces: constructions and examples * Basic facts about moduli spaces of curves * Techniques * Construction of M_g * Limit Linear Series and the Brill-Noether Theory * Geometry of moduli spaces: Selected Results.

Moduli of curves

Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Noncommutative Geometry, Quantum Fields and Motives

We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution 
In dimension 3, the resolution is determined by a noncommutative potential Representation varieties of the Calabi-Yau algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential Numerical invariants, like ranks of cyclic homology groups, are expected to be given by `matrix integrals' over representation varieties 
We discuss examples of Calabi-Yau algebras involving quivers, 3-dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons Examples related to quantum Del Pezzo surfaces will be discussed in [EtGi]

Calabi-Yau algebras

© Gauthier-Villars (Editions scientifiques et medicales Elsevier), 1978, tous droits reserves. L’acces aux archives de la revue « Annales scientifiques de l’E.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

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Polar classes of singular varieties

The collected works of Niels Henrik Abel, edited by Sophus Lie and Ludwig Sylow in 1881, contain 29 papers. The first was written in 1823 when Abel was 21 years old, and the last was jotted down on his deathbed in January and February 1829.

The Legacy of Niels Henrik Abel

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On the hilbert scheme compactification of the space of twisted cubics

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basic Chern numbers. 
To justify the enumeration, we make a rudimentary classification of the types of singularities using Enriques diagrams, obtaining results like Arnold's. We show that the curves in question do, in fact, appear with multiplicity 1 using the versal deformation space, Shustin's codimension formula, and Gotzmann's regularity theorem. Finally, we relate our work to Vainsencher's work with up to seven nodes.

Enumerating singular curves on surfaces

We continue the development of methods for enumerating nodal curves on smooth complex surfaces, extending the range of validity. We apply the new methods in three important cases. First, for up to eight nodes, we prove

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Ragni Piene

Papers

Polar classes of singular varieties

The Legacy of Niels Henrik Abel

On the hilbert scheme compactification of the space of twisted cubics

Enumerating singular curves on surfaces

Node polynomials for families: methods and applications