This chapter gives a high-level overview of analytic combinatorics in several variables. Stratified Morse theory reduces the derivation of coefficient asymptotics for a multivariate generating function to the study of asymptotic expansions of local integrals near certain critical points on the generating function’s singular set. Determining exactly which critical points contribute to asymptotic behavior is a key step in the analysis . The asymptotic behavior of each local integral depends on the local geometry of the singular variety, with three special cases treated in later chapters.

This first chapter motivates our detailed study of the behavior of multivariate sequences, and overviews the techniques we derive using the Cauchy Integral Formula, residues, topological arguments, and asymptotic approximations. Basic asymptotic notation and concepts are introduced, including the background necessary to discuss multivariate expansions.

This chapter derives asymptotics determined by a critical point where the singular variety is locally smooth: the generic situation which arises most commonly in practice. Several explicit formulae for asymptotics are given.

This chapter concludes the book. It contains a survey of the state of analytic combinatorics in several variables, including problems on the boundary of our current knowledge.

This chapter contains a variety of examples deriving asymptotics of generating functions taken from the research literature, illustrating the power of analytic combinatorics in several variables.

This appendix presents a collection of key results on Morse theory, intersection classes, and the computation of Leray residue forms, specialized to the most important local geometries treated in the book.

This appendix contains a compressed version of standard graduate topics in differential geometry such as vector fields, tangent and cotangent bundle, differential forms, and Stokes’s Theorem. Both real and complex manifolds are covered.

This chapter develops methods to compute asymptotics of univariate Fourier–Laplace integrals (which combine exponential decay and oscillation) and saddle point approximations. We illustrate both analytic and smooth methods for asymptotics.

This chapter derives asymptotics determined by a critical point near which the singular variety has a quadratic singularity. This necessitates introducing the theory of hyperbolic polynomials and cones of hyperbolicity, which guide advanced deformations of contours of integration on the way to computing asymptotics.

This chapter discusses assorted topics related to algebraic varieties and singular sets of multivariate rational functions. In particular, we cover Laurent expansions, polynomial amoebas, convex geometry, and bounds for generating function coefficients from so-called minimal points of singular sets.

This chapter covers standard material on generating functions in one and several variables. We describe how many common combinatorial constructions yield generating function specifications, often leading to rational or algebraic equations for generating functions. We also cover D-finite generating functions, which satisfy linear differential equations and arise both from linearly recurrent sequences and as diagonals of rational generating functions. Finally, we discuss labeled combinatorial constructions and exponential generating functions.