Rings of differentiable semialgebraic functions

Abstract

In this work we analyze the main properties of the Zariski and maximal spectra of the ring \({{\mathcal {S}}}^r(M)\) of differentiable semialgebraic functions of class \({{\mathcal {C}}}^r\) on a semialgebraic set \(M\subset {{\mathbb {R}}}^m\). Denote \({{\mathcal {S}}}^0(M)\) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in \({\text {Cl}}(M)\). This ring is the real closure of \({{\mathcal {S}}}^r(M)\). If M is locally compact, the ring \({{\mathcal {S}}}^r(M)\) enjoys a Łojasiewicz’s Nullstellensatz, which becomes a crucial tool. Despite \({{\mathcal {S}}}^r(M)\) is not real closed for \(r\ge 1\), the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring \({{\mathcal {S}}}^0(M)\). In addition, the quotients of \({{\mathcal {S}}}^r(M)\) by its prime ideals have real closed fields of fractions, so the ring \({{\mathcal {S}}}^r(M)\) 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 \({{\mathcal {S}}}^r(M)\) and \({{\mathcal {S}}}^0(M)\) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring \({{\mathcal {S}}}^r(M)\) is a Gelfand ring and its Krull dimension is equal to \(\dim (M)\). We also show similar properties for the ring \({{\mathcal {S}}}^{r*}(M)\) of differentiable bounded semialgebraic functions. In addition, we confront the ring \({\mathcal S}^{\infty }(M)\) of differentiable semialgebraic functions of class \({{\mathcal {C}}}^{\infty }\) with the ring \({{\mathcal {N}}}(M)\) of Nash functions on M.