Monomial summability and doubly singular differential equations

In this work, we consider systems of differential equations that are doubly singular, i.e. that are both singularly perturbed and exhibit an irregular singular point. If the irregular singular point is at the origin, they have the formεσxr+1dydx=f(x,ε,y),f(0,0,0)=0 with f   analytic in some neighborhood of (0,0,0)(0,0,0). If the Jacobian dfdy(0,0,0) is invertible, we show that the unique bivariate formal solution is monomially summable  , i.e. summable with respect to the monomial t=εσxrt=εσxr in a (new) sense that will be defined. As a preparation, Poincaré asymptotics and Gevrey asymptotics in a monomial are studied.