Separating families for semi-algebraic sets
Let \(R\) be the real closure of an ordered field \(F\) and let \(V\) be an algebraic subset of \(\mathbb{R}^ N\) defined over \(F\), i.e., the set of zeros in \(\mathbb{R}^ N\) of some polynomial in \(F[T_ 1, \dots,T_ N]\). For \(a \in \mathbb{R}\) define \(\text{sgn} (a)\) to be 1 (resp. 0, resp. \(-1)\) if \(a>0\) (resp. \(a=0\), resp. \(a<0)\). Obviously, each semialgebraic subset \(S\) of \(V\) defined over \(F\) admits a separating family \(\{f_ 1, \dots, f_ p\}\) of polynomials over \(F\), (that means, for every pair of points \(x \in S\), \(y \in V \backslash S\), there exists an index \(i\) such that \(\text{sgn} (f_ i(x)) \neq \text{sgn} (f_ i(y))\). The paper under review provides estimates of a new invariant \(p_ F(V)\) defined as the least integer such that each semialgebraic set \(S \subset V\) defined over \(F\) admits a separating family consisting of \(\leq p_ F (V)\) polynomials. The main result states the inequality: \[ p_ F(V) \leq \sum^ d_{i = 0} \max \{1,1 + 4^{e + i - 1} - 2^{e + i - 1}\}\tag{1} \] where \(d = \dim V\) and \(e = e_ F \geq 0\) is an invariant of the ordered field \(F\) (e.g., \(e = 1\) if \(F = \mathbb{Q}\), and \(e = 0\) if \(F\) is hereditarily euclidean, for instance if \(F = R)\). On the other hand, for a large class \({\mathcal F}\) of fields -- but distinct of the one consisting of all ordered fields -- the author gets a lower bound for \(p_ F (\mathbb{R}^ d)\). In fact, \({\mathcal F}\) contains \(\mathbb{Q}\) and hereditarily euclidean fields, and if \(\alpha (n) = {n (n + 1) \over 2}\), and \(\beta (n) = 2 \alpha (n)\), it is proved that: \[ \begin{aligned} p_ F (\mathbb{R}^ d) &= \log_ 2 \bigl( \beta^{d - 1} (2) \bigr) + d \quad \text{if} \quad e = 0 \qquad \text{and}\\ p_ F (\mathbb{R}^ d) &= \log_ 2 \biggl( \alpha^{e - 1} \bigl( \beta^ d (2) \bigr) \biggr) + d + e - 1 \quad \text{if} \quad e = 1,\;F \in {\mathcal F}. \end{aligned} \tag{2} \] There are three special cases: \(d=1\), \(e \leq 1\); \(d=2\), \(e=0\), where this lower bound agrees with the upper bound stated in (1). This way are obtained the equalities: \(p_ F (\mathbb{R}) = 2\), \(p_ F (\mathbb{R}^ 2) = 5\) if \(e_ F = 0\); \(p_ F (\mathbb{R}) = 4\) if \(e_ F = 1\) and \(\mathbb{F} \in F\). The quoted results are obtained as corollaries of the corresponding ones in the setting of real algebra, i.e., spaces of orderings. For example, inequality (1) is obtained for constructible sets in the real spectrum of noetherian rings and then applied to the coordinate ring, over \(F\), of the variety \(V\). The main ingredient in the proof is a separation theorem -- via signature of quadratic forms -- due to \textit{L. Bröcker} [Expos. Math. 9, No. 4, 289-334 (1991; Zbl 0783.14035)]. The lower bound (2) is obtained as a consequence of the study of the corresponding invariant \(p(X,G)\) for finite spaces or orderings. As the author observes the kind of involved arguments makes conceivable to get similar results in other geometric contexts, e.g. substituting polynomials by analytic functions, as \textit{C. Andradas}, \textit{L. Bröcker} and \textit{J. M. Ruiz} did concerning another important invariant, the stability index [Invent. Math. 92, No. 2, 409-430 (1988; Zbl 0655.32011)].
scientific article; zbMATH DE number 492807
scientific article