On asymptotic expansions in several variables and applications to singularly perturbed first order PDEs

The goal of this talk is to describe some of the relations and properties between different approaches to asymptotic expansions for formal power series in several variables such as monomial, polynomial (as introduced by Mozo-Schäfke) and  Mayima's expansions. The Borel-Laplace analysis for each case is also described as a tool to prove monomial summability of solutions of singularly perturbed PDEs of the form $$x^\alpha\varepsilon^\beta\sum_{j=1}^n s_j/\alpha_j x_j \partial y/\partial x_j =F(x,\varepsilon,y),$$ where $x=(x_1,...,x_n), \varepsilon=(\varepsilon_1,...,\varepsilon_m)$  are tuples of complex variables, $\alpha=(\alpha_1,...,\alpha_n), \beta=(\beta_1,...,\beta_m)$ are tuples of positive integers, $s_1,...,s_n$ are non-negative real numbers such that $s_1+...+s_n=1$,   $F$ is analytic in a neighborhood of the origin and $\partial F/\partial y(0,0,0)$ is an invertible matrix. The results presented here are a first step on summability of formal solutions that are expected to be valid for more general equations of the same nature.