Monomial multisummability through Borel-Laplace transforms and ODEs

The aim of this text is to explain in detail the theory of monomial asymptotic expansions in two variables and its applications in the theory of summability of formal power series, solution of singularly perturbed differential equations in the complex domain. The work is divided in four chapters. In the first one we recall the theory of asymptotic expansions and k-summability in one variable and then the corresponding notions in a monomial. In the second one monomial summability is studied through Borel-Laplace like integral transformations. The behaviour under point blow ups in included. The third chapter is devoted to applications: monomial summability of singularly perturbed linear differential equations, a family of PDEs and Pfaffian systems with normal crossings. Finally in the last chapter we recall the basic multisummability theory in one variable and we propose an analogous notion for the monomial case in a particular case. The book is aimed to researchers interested in asymptotic expansions and differential equations in the complex domain. Being a self-contained text it is also adequate to non-specialists interested in these topics.