Computational Algebraic Geometry

The interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. Advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. This opens wonderful new vistas and allows us to pose, study and solve problems that were previously out of reach. Suitable for graduate students, the objective of this 2003 book is to bring advanced algebra to life with lots of examples. The first chapters provide an introduction to commutative algebra and connections to geometry. The rest of the book focuses on three active areas of contemporary algebra: Homological Algebra (the snake lemma, long exact sequence inhomology, functors and derived functors (Tor and Ext), and double complexes); Algebraic Combinatorics and Algebraic Topology (simplicial complexes and simplicial homology, Stanley-Reisner rings, upper bound theorem and polytopes); and Algebraic Geometry (points and curves in projective space, Riemann-Roch, Cech cohomology, regularity).

• Concise snapshots of several different areas of advanced algebra - algebraic combinatorics, algebraic topology, commutative algebra and algebraic geometry
• Introduction to homological algebra in a concrete setting
• Lots of examples and exercises to help reader develop facility for computation by hand and by computer
• Appendices on abstract algebra and complex analysis to provide quick refreshers to the reader
• Code interspersed with text to encourage the reader to experiment as progressing through the book