Algebraic geometry, central to pure mathematics, has important applications in such fields as engineering, computer science, statistics and computational biology, which exploit the computational algorithms that the theory provides. Users get the full benefit, however, when they know something of the underlying theory, as well as basic procedures and facts. This book is a systematic introduction to the central concepts of algebraic geometry most useful for computation. Written for advanced undergraduate and graduate students in mathematics and researchers in application areas, it focuses on specific examples and restricts development of formalism to what is needed to address these examples. In particular, it introduces the notion of Gröbner bases early on and develops algorithms for almost everything covered. It is based on courses given over the past five years in a large interdisciplinary programme in computational algebraic geometry at Rice University, spanning mathematics, computer science, biomathematics and bioinformatics.
Contents
Introduction; 1. Guiding problems; 2. Division algorithm and Gröbner bases; 3. Affine varieties; 4. Elimination; 5. Resultants; 6. Irreducible varieties; 7. Nullstellensatz; 8. Primary decomposition; 9. Projective geometry; 10. Projective elimination theory; 11. Parametrizing linear subspaces; 12. Hilbert polynomials and Bezout; Appendix. Notions from abstract algebra; References; Index.
Reviews
"Yet another introduction to algebraic geometry? No! This is a book
that has been missing from our textbook arsenal and that belongs on
the bookshelf of anyone who plans to either teach or study algebraic
geometry."
Sándor Kovács, University of Washington
"This is a commonsense introduction with examples and relations
to computational algebra. Hassett is in touch with current
thinking in algebraic geometry itself, and has a light touch
with the computational aspects."
Miles Reid, University of Warwick
"... A nice introduction to algebraic geometry. The book is clearly written and should be an important reference for elementary courses in algebraic geometry and commutative algebra."
D.-M. Popescu, Mathematical Reviews
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