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Dimension of graphoids of rational vector-functions. (English) Zbl 1275.54022

Let \(X\), \(Y\) be spaces and \(f\) a map whose domain \(\mathrm{dom}(f)\) is contained in \(X\) and whose range is \(Y\). Such \(f\) will be called a partial function on \(X\). The closure \(\overline\Gamma(f)\) of the graph of \(f\) in \(X\times Y\) is called the graphoid of \(f\). The graphoid determines a function \(\overline f:X\to\mathcal{P}(Y)\) (the power set of \(Y\)), assigning to a point \(x\in X\), \(\overline f(x)= \{y\in Y\,|\,(x,y)\in\overline\Gamma(f)\}\). This \(\overline f\) is called the graphoid extension of \(f\). The domain, \(\mathrm{dom}(f)\), equals \(\{x\in X\,|\,\overline f(x)\neq\emptyset\}\).
The authors are interested in the graphoids of rational functions of \(k\)-variables, \(f(x_1,\dots,x_k)=\frac{p(x_1,\dots,x_k)}{q(x_1,\dots,x_k)}\), where \(p\) and \(q\) are relatively prime polynomials. The domain of such a map is the open dense subset \(\mathrm{dom}(f)=\mathbb{R}^k\setminus (p^{-1}(0)\cap q^{-1}(0))\) of \(\mathbb{R}^k\). Using \(\overline{\mathbb{R}} =\mathbb{R}\cup\{\infty\}\) to denote the projective real line, one may treat \(\overline{\mathbb{R}}^k\) as a \(k\)-dimensional torus; so \(\mathrm{dom}(f)\) is an open dense subset of \(\overline{\mathbb{R}}^k\). If \(\mathbb{R}(x_1,\dots,x_k)\) denotes the field of such rational functions, then the elements of \(\mathbb{R}(x_1,\dots,x_k)\) will take values in \(\overline{\mathbb{R}}\). A subset \(\mathcal{F}\subset\mathbb{R}(x_1,\dots,x_n)\) is called a rational vector-function.
In this setting, in case \(\mathcal{F}\) is countable, then one can define \(\mathrm{dom}(\mathcal{F})=\bigcap\{\mathrm{dom}(f)\, |\,f\in\mathcal{F}\}\); the latter is a dense \(\mathrm{G}_\delta\)-set in \(\overline{\mathbb{R}}^k\). One may think of \(\mathcal{F}:\mathrm{dom}(\mathcal{F})\to\overline{\mathbb{R}}^\mathcal{F}\) via \(x\mapsto(f(x))_{f\in\mathcal{F}}\). Its graphoid is a closed subset of the compact Hausdorff space \(\overline{\mathbb{R}}^k\times \overline{\mathbb{R}}^\mathcal{F}\), and it has a graphoid extension \(\overline{\mathcal{F}}\) with \(\mathrm{dom}(\overline{\mathcal{F}})= \overline{\mathbb{R}}^k\), i.e., \(\overline{\mathcal{F}} :\overline{\mathbb{R}}^k\to\mathcal{P}(\overline{\mathbb{R}}^\mathcal{F})\).
For uncountable families \(\mathcal{F}\), a different approach is needed, and such is described by the authors. One then similarly obtains \(\overline{\mathcal{F}}:\overline {\mathbb{R}}^k\to\mathcal{P}(\overline{\mathbb{R}}^\mathcal{F})\) as just indicated. The good properties that such a “set-valued” function has are listed as (1)–(4) on page 25.
The paper is devoted to:
{ Problem 1.1.} Given a family of rational functions \(\mathcal{F}\subset\mathbb{R}(x_1,\dots,x_k)\), study topological (and dimension) properties of the graphoid \(\overline{\Gamma}(\mathcal{F}) \subset\overline{\mathbb{R}}^k\times\mathbb{R}^\mathcal{F}\) of \(\mathcal{F}\). A precise question: Has \(\overline{\Gamma}(\mathcal{F})\) the topological dimension \(k\)?
The main result of the paper is,
{ Theorem 1.2.} For any family of rational functions \(\mathcal{F}\subset\mathbb{R}(x,y)\), its graphoid \(\overline{\Gamma}(\mathcal{F})\subset\overline{\mathbb{R}}^2\times \overline{\mathbb{R}}^\mathcal{F}\) has covering dimension \(2\).

MSC:

54F55 Unicoherence, multicoherence
55M10 Dimension theory in algebraic topology
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
14P05 Real algebraic sets
26C15 Real rational functions
55M25 Degree, winding number
54C50 Topology of special sets defined by functions
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