Not too long ago, I was discussing with one of my freshman students what it is that we mathematicians do for research. He was convinced that the only thing mathematicians do all day is sit around trying to find solutions to big ugly equations. I (hopefully) talked him out of this notion, and tried to convince him that mathematics was about more than just looking at polynomials. After that long conversation, I hope that the student doesn't stumble across the new book Solving Polynomial Equations: Foundations, Algorithms, and Applications. If he does, he will find nine chapters by sixteen different authors on topics which are all related to finding the solutions of systems of polynomial equations all of which convince the reader that attempts to solve polynomials are every bit as much cutting-edge research as they were in the days of Cardano and Tartaglia.
The chapters of the book are independent and for the most part attempt to be self-contained, but on the other hand they often refer back and forth to one another, and different chapters assume different levels of understanding of different prerequisites. The book is based on course notes from a summer school held in Buenos Aires in July 2003 and sponsored by the International Centre For Pure And Applied Mathematics (CIMPA). While the chapters vary somewhat in the quality of their exposition, most of them are quite good and interesting introductions to the subjects at hand. The full table of contents can be found here, but this reviewer found the following chapters to be particularly interesting and representative of the topics in the whole book:
Darren Glass (email@example.com) is an Assistant Professor at Gettysburg College. His mathematical interests include Algebraic Geometry, Galois Theory, Number Theory, and Cryptography.
A.Dickenstein, I.Z.Emiris: Preface.
1 E.Cattani, A.Dickenstein: Introduction to Residues and Resultants.
2 D.A.Cox: Solving Equations via Algebras.
3 M.Elkadi, B.Mourrain: Symbolic-numeric Methods for Solving Polynomial Equations and Applications.
4 A.Kehrein, M.Kreuzer, L.Robbiano: An Algebraist's View on Border Bases.
5 M.Stillman: Tools for Computing primary Decompositions and Applications to Ideals Associated to Bayesian Networks.
6 J.Sabia: Algorithms and Their Complexities.
7 I.Z.Emiris: Toric Resultants and Applications to Geometric Modelling.
8 A.J.Sommese, J.Verschelde, Ch.W.Wampler: Introduction to Numerical Algebraic Geometry.
9 G.Chèze, A.Galligo: Four Lectures on Polynomial Absolute Factorization.