Nash Functions

Abstract

In the first section, we define an isomorphism between the ring of germs of Nash functions at the origin of R ⁿ, the ring of power series which are algebraic over the polynomials, and (for R = ℝ) the ring of germs of analytic algebraic functions. Prom the study of the ring of algebraic power series in the second section, we deduce the local properties of Nash functions. In the third section, we state a theorem of approximation of formal solutions of a system of Nash equations by Nash solutions. The study of global properties of Nash functions is greatly simplified by the Artin-Mazur description of Nash functions given in Section 4. We use this description in Section 5 to obtain the substitution theorem, from which we deduce a Nullstellensatz and a Positivstellensatz for Nash functions. In Section 6, we study the sets of zeros of Nash functions, and we prove that the germs of such sets are Nash equivalent to germs of algebraic sets. In Section 7, we prove that the ring of germs of Nash functions at the origin of R ⁿ is the henselization of the local ring of germs of regular functions, and we use this fact to prove that the ring of global Nash functions is noetherian. In Section 8, we consider Nash functions on the real spectrum, and we give a proof of Efroymson’s theorem of approximation of continuous semi-algebraic functions by Nash functions. Efroymson’s extension theorem is given in Section 9, where we also study the tubular neighbourhoods of Nash submanifolds. The last section is devoted to families of Nash functions, and we show that there is r = r(n, d) ∈ N, such that every semi-algebraic function of class C ^r on R ⁿ, satisfying a polynomial equation of total degree ≤ d, is Nash.

Throughout this chapter, R is a real closed field.