Introduction

In this note we will mainly be concerned with polynomial growth properties of C∞-differential forms related, to real analytic varieties. We announced basic results on analytic de Rham cohomology in [4]1–4. This note should be read as a continuation of these four notes. Details of the present note are too long to be included here, and will appear elsewhere.

The purpose of this note is as follows :

1. (I) To introduce the notion of p.g simpleprestratification for analytic varieties and for prestratified spaces of certain types (§1).

2. (II) To discuss relations between the notion of p.g. simple prestratification and that of normalized series of prestratified spaces introduced in [4]4 (§ 2).

To explain the notion in (I), let M be a C∞-manifold, and let be an open covering of M. Recall that is called simple if, for any such that is contractible (cf. A. Weil [8]). The existence of simple coverings for C∞-manifolds was used as a basic tool for the proof of the C∞-de Rham theorem (cf. [8]).

Now the notion of p.g. simple prestratification is a combination of the notions of prestratification (of a topological space), p. g. property of C∞-difFerential forms (related to an analytic variety) and simple covering (of a C∞-manifold).

Roughly, the role of p. g. prestratification in our study of p. g. properties of C∞– differential forms related to analytic varieties is similar to that in [8] of simple coverings in the C∞-de Rham theorem.