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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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