In the fall of 1987, I taught a graduate computer science course entitled
“Symbolic Computational Algebra” at New York University. A rough set
of class-notes grew out of this class and evolved into the following final
form at an excruciatingly slow pace over the last five years. This book also
benefited from the comments and experience of several people, some of
whom used the notes in various computer science and mathematics courses
at Carnegie-Mellon, Cornell, Princeton and UC Berkeley.
The book is meant for graduate students with a training in theoretical
computer science, who would like to either do research in computational
algebra or understand the algorithmic underpinnings of various commercial
symbolic computational systems: Mathematica, Maple or Axiom, for
instance. Also, it is hoped that other researchers in the robotics, solid
modeling, computational geometry and automated theorem proving communities
will find it useful as symbolic algebraic techniques have begun to
play an important role in these areas.
The main four topics–Gr¨obner bases, characteristic sets, resultants and
semialgebraic sets–were picked to reflect my originalmotivation. The choice
of the topics was partly influenced by the syllabii proposed by the Research
Institute for Symbolic Computation in Linz, Austria, and the discussions
in Hearn’s Report (“Future Directions for Research in Symbolic Computation”).