Rings of differentiable semialgebraic functions

Abstract

In this work we analyze the main properties of the Zariski and maximal spectra of the ring Sr(M) of differentiable semialgebraic functions of class Cr on a semialgebraic set M ⊂ Rm. Denote S0(M) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in Cl(M). This ring is the real closure of Sr(M). If M is locally compact, the ring Sr(M) enjoys a Łojasiewicz’s Nullstellensatz, which becomes a crucial tool. Despite Sr(M) is not real closed for r ≥ 1, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring S0(M). In addition, the quotients of Sr(M) by its prime ideals have real closed fields of fractions, so the ring Sr(M) is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of Sr(M) and S0(M) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring Sr(M) is a Gelfand ring and its Krull dimension is equal to dim(M). We also show similar properties for the ring Sr∗(M) of differentiable bounded semialgebraic functions. In addition, we confront the ring S∞(M) of differentiable semialgebraic functions of class C∞ with the ring N (M) of Nash functions on M.