Generalized Hardy fields in several variables (Q1111611)

We extend the notion of a Hardy field (of real valued functions) to the context of germs of functions of several variables defined over a filter of sets and subject to specified conditions of smoothness, which we formalize using Palais' notion of a smoothness category. In this context we show that for any Hardy field of germs of a given smoothness category \({\mathcal C}\), its relative algebraic closure in the ring G\({\mathcal C}\) of all germs in the same smoothness category is a real closed field, which is then the unique real closure of the given Hardy field inside G\({\mathcal C}.\) This generalizes a result of Robinson concerning the usual Hardy fields on \({\mathbb{R}}\) with respect to the \(C^{\infty}\)-category (where the germs are taken at \(+\infty)\). Moreover for any smoothness category \({\mathcal C}\) we define the class of \({\mathcal C}\)-fields and we note that the same result on the real closure is valid in this case. In a following paper: ``Exponentially and logarithmically closed Hardy fields in several variables'' (to appear in J. Pure Appl. Algebra) the same author presents a method of construction for generalized Hardy fields.