Polynomials And Polynomial Inequalities

For Theresa, Sophie, Alexandra, Jennifer, and Erika Prefa e Polynomials pervade mathemati s, and mu h that is beautiful in mathemati s is related to polynomials. Virtually every bran h of mathemati s, from algebrai number theory and algebrai geometry to applied analysis, Fourier analysis, and omputer s ien e, has its orpus of theory arising from the study of polynomials. Histori ally, questions relating to polynomials, for example, the solution of polynomial equations, gave rise to some of the most important problems of the day. The subje t is now mu h too large to attempt an en y lopedi overage. The body of material we hoose to explore on erns primarily polynomials as they arise in analysis, and the te hniques of the book are primarily analyti . While the onne ting thread is the polynomial, this is an analysis book. The polynomials and rational fun tions we are on erned with are almost ex lusively of a single variable. We assume at most a senior undergraduate familiarity with real and omplex analysis (indeed in most pla es mu h less is required). However, the material is often tersely presented, with mu h mathemati s explored in the exer ises, some of whi h are quite hard, many of whi h are supplied with opious hints, some with omplete proofs. Well over half the material in the book is presented in the exer ises. The reader is en ouraged to at least browse through these. We have been mu h in uen ed by Polya and Szeg}o's lassi \Problems and Theorems in Analysis" in our approa h to the exer ises. (Though unlike Polya and Szeg}o we hose to in orporate the hints with the exer ises.) viii Prefa e The book is mostly self- ontained. The text, without the exer ises, provides an introdu tion to the material, but mu h of the ri hness is reserved for the exer ises. We have attempted to highlight the parts of the theory and the te hniques we nd most att