"[1] The book was written by Lenore Blum, Felipe Cucker, Michael Shub and Stephen Smale, with a foreword by Richard M. Karp, and published by Springer-Verlag in 1998 (doi:10.1007/978-1-4612-0701-6, ISBN 0-387-98281-7).
The ring of integers is studied as a particular example, as are the algebraically closed fields of characteristic zero, which are shown from the point of view of NP-completeness within their computational models to all be equivalent to the complex numbers.
Other topics considered in this section include finding the roots of polynomials and the intersection points of algebraic curves, the condition number of systems of equations, and the time complexity of linear programming with rational coefficients.
A key tool in this area is the use of the number of connected components of a semialgebraic set to provide a lower bound on the time complexity of an associated computational problem.
[2] The book is aimed at the level of a graduate student or researcher in these topics,[2][3] and in places it assumes background knowledge of classical computational complexity theory, differential geometry, topology, and dynamical systems.