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Algebraically constructible functions. (English) Zbl 0913.14018
Let \(X\) be a real algebraic set and let \(\varphi: X\to \mathbb{Z}\) be an integer valued function. The authors compare the following notions of constructibility for the function \(\varphi\):
(i) \(\varphi\) is called semi-algebraically constructible if \(\varphi\) admits a representation as a finite sum \(\varphi= \sum_{1\leq i\leq s} m_i\mathbf{1}_{X_i}\), where for any \(1\leq i\leq s\), \(X_i\) is a semialgebraic subset of \(X\) with characteristic function \(\mathbf{1}_{X_i}\) and \(m_1, \dots, m_s\) are integers. If all semialgebraic sets \(X_i\) are real algebraic, \(\varphi\) is called strongly algebraic constructible. The rings of semialgebraically constructible functions of \(X\) are denoted by \(F(X)\) and \(\mathbb{A}(X)\), respectively. Let \(\varphi\) be semialgebraically constructible. The value \(\int\varphi:=\sum_{1\leq i\leq s} m_i\chi (X_i)\) is called the Euler integral of \(\varphi\) (here \(\chi (X_i)\) denotes the Euler characteristic of \(X_i)\). The link of \(\varphi\) is the semialgebraically constructible function \(\Lambda\varphi\) defined by \((\Lambda \varphi) (x)= \int_{S(x, \varepsilon)} \varphi\) for any \(x\in X\), where \(S(x,\varepsilon)\) is the sphere of radius \(\varepsilon\) centered at the point \(x\), and \(\varepsilon>0\) is sufficiently small. The duality operator of semialgebraically constructible functions is defined by \(D\varphi= \varphi- \Lambda \varphi\). Moreover, let \(\Omega: F(X)\to F(X)\) defined by \(\Omega\varphi =\varphi+D \varphi\) for \(\varphi\in F(X)\). The function \(\varphi\in F(X)\) is called self-dual if \(D\varphi = \varphi\) (or equivalently \(\Lambda \varphi=0)\), antiself-dual if \(D \varphi= -\varphi\) (or equivalently \(\Omega\varphi =0)\). Generalizing these notions, \(\varphi\) is called Euler if \(\Lambda \varphi (x)\) is even for any \(x\in X\).
(ii) A semialgebraically constructible function \(\varphi \in F(X)\) is called Nash constructible if it admits a presentation as a finite sum \(\varphi= \sum_{1\leq i\leq s} m_if_i* \mathbf{1}_{T_i}\), where, for \(1\leq i\leq s\), \(Z_i\) is a real algebraic set, \(f_i: Z_i\to X\) is a regular, proper morphism, \(T_i\) is a connected component of \(Z_i\), \(f_i*\mathbf{1}_{T_i}: =\int_{T_i} f_i\) and \(m_i\) an integer.
The authors show the following results:
Let \(\varphi\in \mathbb{A}(X)\). Then \(\varphi\) is Euler and \(D\varphi\), \({1\over 2} \Lambda \varphi\) and \({1\over 2} \Omega \varphi\) belong to \(\mathbb{A} (X)\) (theorem 2.5).
Let \(d\) be the dimension of \(X\). Then \(2^dF(X) \subset \mathbb{A} (X)\). If \(\varphi\in F(X)\) is divisible by \(2^d\) then all functions obtained from \(\varphi\) by means of the operations addition/subtraction, multiplication and the operator \({1\over 2} \Lambda\), are integer valued (theorem 2.8 and corollary 2.10).
Let \(S\) be a closed semialgebraic subset of \(X\). Then \(\mathbf{1}_S\) is Nash-constructible, if and only if \(S\) is arc-symmetric (i.e. if and only if for any \(\gamma (-1,1)\to X\), \(\gamma ((-1,0)) \subset S\) implies \(\gamma ((-1,1)) \subset S)\). Every arc-symmetric semi-algebraic set of dimension \(\leq 3\) is homomorphic to an algebraic set.
Moreover, the numerical conditions of Akbulut and King, for a stratified semialgebraic set of dimension three to be algebraic, are reproved in the context of constructible functions.

MSC:
14P25 Topology of real algebraic varieties
14P20 Nash functions and manifolds
14P10 Semialgebraic sets and related spaces
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