Polynomials and Polynomial Inequalities

By P. Borwein and T. Erd\'elyi

Graduate Texts in Mathematics 161, Springer, Berlin, 1995, 480 pp.,
hardcover DM 98, ISBN 0-387-94509-1

Polynomials pervade mathematics, virtually every branch of mathematics
from algebraic number theory and algebraic geometry to applied analysis
and computer science, has a corpus of theory arising from polynomials. The
material explored in this book primarily concerns polynomials as they
arise in analysis, focusing on polynomials and rational functions of a
single variable. The book is self-contained and assumes at most a
senior-undergraduate familiarity with real and complex analysis. After an
introduction to the geometry of polynomials and a discussion of
refinements of the Fundamental Theorem of Algebra, the book turns to a
consideration of various special polynomials. Chebyshev and Descartes
systems are then introduced, and M\"untz systems and rational systems are
examined in detail.  Subsequent chapters discuss denseness questions and
the inequalities satisfied by polynomials and rational functions.
Appendices on algorithms and computational concerns, on the interpolation
theorem, and on orthogonality and irrationality conclude the book. 

Contents: 
   Introduction and Basic Properties 
   Some Special Polynomials 
   Chebyshev and Descartes Systems 
   Denseness Questions  
   Basic Inequalities
   Inequalities in M\"untz Spaces  
   Inequalities in Rational Function Spaces

Appendices: 
   Algorithms and Computational Concerns 
   Orthogonality and Irrationality 
   An Interpolation Theorem 
   Inequalities for Generalized Polynomials 
   Inequalities for Polynomials with Constraints.  

Wolfram Koepf