M. Mignotte, D. Stefanescu. Polynomials. An Algorithmic Approach, Springer-Verlag, Singapore, 1999. Approx. 320pp. ISBN: 981-4021-51-2. US$49 softcover.

This textbook gives a well-balanced presentation of the classic procedures of polynomial algebra which are computationally relevant and some algorithms developed during the last decade. The first chapter discusses the constrcution and the representation of polynomials. The second chapter focuses on the computational aspects of the analytical theory of polynomials. Polynomials with coefficients in a finaite field are then described in chapetr three, and the final chapter is devoted to factorization of polynomials with integral coefficients.

The book is primarily aimed at graduate students taking courses in Polynomial Algebra, with a prerequisite knowledge of set theory, usual fields and basic algebra. Fully worked out examples, hints and references complement the main text, and details concerning the implementation of algorithms as well as indicators of their efficiency are provided. The book is also useful as a supplementary text for courses in scientific computing, analysis of algorithms, computational polynomial factorization, and computational geometry of polynomials.

Contents: 1. An Introduction to Polynomials: Construction and representation of polynomials; Complexity and cost; Polynomial division; Polynomial factorization; Polynomial roots. Eliminations. Resultants; Symmetric functions; Polynomial interpolation; Irreducinility criteria. 2. Complex Polynomials: Polynomial size; Geometry of polynomials; Stable polynomials; Polynomial roots inside the unit disk; Bounds for the roots; Applications to integer polynomials; Separation of roots. 3. Polynomials with Coefficients in a Finite Field: Finite fields; Cyclotomic polynomials; Fast Fourier transform; Number of irreducible polynomials over a finite field; Constrcution of irreducible polynomials over a finite field; Roots of polynomials over finite fields; Squarefree polynomials; Berlekamp's algorithm; Niederreiter's algorithm. 4. Integer Polynomials: Kronecker's factorization method; The berlekamp-Zassenhaus algorithm; The LLL factorization algorithm. Bibliography; Notation; List of Algorithms; Index.