Existence and nonexistence of roots of functions involving one or more parameters has been the subject of numerous investigations. For a wide class of functions called quasi-polynomials, the above problems can be transformed into the existence and nonexistence of tangents of the envelope curves associated with the functions under investigation.In this book, we present a formal theory of the Cheng-Lin envelope method, which is completely new, yet simple and precise. This method is both simple — since only basic Calculus concepts are needed for understanding — and precise, since necessary and sufficient conditions can be obtained for functions such as polynomials containing more than four parameters.Since the underlying principles are relatively simple, this book is useful to college students who want to see immediate applications of what they learn in Calculus; to graduate students who want to do research in functional equations; and to researchers who want references on roots of quasi-polynomials encountered in the theory of difference and differential equations.Contents:PrologueEnvelopes and Dual SetsDual Sets of Convex-Concave FunctionsQuasi-PolynomialsC(0, ∞)-Characteristic Regions of Real PolynomialsC(0, ∞)-Characteristic Regions of Real ∇-PolynomialsCR-Characteristic Regions of ∇-PolynomialsReadership: Undergraduate science and engineering students, graduate mathematics students, researchers in the theory of equations.