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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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##### References:
 [1] S. AKBULUT and H. KING , The topology of real algebraic sets , (L’Enseignement Math, Vol. 29, 1983 , pp. 221-261). MR 86d:14016b | Zbl 0541.14019 · Zbl 0541.14019 [2] S. AKBULUT and H. KING , Topology of Real Algebraic Sets , (MSRI Publ., Vol. 25, Springer-Verlag, New York, 1992 ). MR 94m:57001 | Zbl 0808.14045 · Zbl 0808.14045 [3] R. BENEDETTI and M. DEDÒ , The topology of two-dimensional real algebraic varieties , (Annali Math. Pura Appl., Vol. 127, 1981 , pp. 141-171). MR 83d:58009 | Zbl 0507.14018 · Zbl 0507.14018 · doi:10.1007/BF01811722 [4] R. BENEDETTI and J-J. RISLER , Real Algebraic and Semi-Algebraic Sets , (Hermann, Paris, 1990 ). MR 91j:14045 | Zbl 0694.14006 · Zbl 0694.14006 [5] J. BOCHNAK , M. COSTE and M-F. ROY , Géométrie Algébrique Réelle , (Springer-Verlag, Berlin, 1987 ). MR 90b:14030 | Zbl 0633.14016 · Zbl 0633.14016 [6] M. COSTE , Sous-ensembles algébriques réels de codimension 2 , (Real Analytic and Algebraic Geometry, Lecture Notes in Math., Springer-Verlag, 1990 , Vol. 1420, pp. 111-120). MR 91c:14069 | Zbl 0723.14040 · Zbl 0723.14040 [7] M. COSTE and K. KURDYKA , On the link of a stratum in a real algebraic set , (Topology, Vol. 31, 1992 , pp. 323-336). MR 93d:14088 | Zbl 0787.14038 · Zbl 0787.14038 · doi:10.1016/0040-9383(92)90025-D [8] A. DURFEE and M. SAITO , Mixed Hodge structures on the intersection cohomology of links , (Compositio Math, Vol. 76, 1990 , pp. 49-67). Numdam | MR 91m:32039 | Zbl 0724.14011 · Zbl 0724.14011 · numdam:CM_1990__76_1-2_49_0 · eudml:90054 [9] J. H. G. FU and C. MCCRORY , Stiefel-Whitney classes and the conormal cycle of a singular variety , (Trans. Amer. Math. Soc., Vol. 349 (2), 1997 , pp. 809-835). MR 97h:32009 | Zbl 0873.32006 · Zbl 0873.32006 · doi:10.1090/S0002-9947-97-01815-1 [10] R. HARDT , Semi-algebraic local triviality in semi-algebraic mappings , (Amer. Jour. Math, Vol. 102, 1978 , pp. 291-302). MR 81d:32012 | Zbl 0465.14012 · Zbl 0465.14012 · doi:10.2307/2374240 [11] S. HALPERIN and D. TOLEDO , Stiefel-Whitney homology classes , (Annals of Math, Vol. 96, 1972 , pp. 511-525). MR 47 #1072 | Zbl 0255.57007 · Zbl 0255.57007 · doi:10.2307/1970823 [12] K. KURATOWSKI , Topologie , (Polskie Towarzystwo Matematyczne, Warszawa, Vol. I, 1952 ). Zbl 0049.39703 · Zbl 0049.39703 · eudml:219345 [13] H. KING , The topology of real algebraic sets , (Proc. Symp. Pure Math, Vol. 40, Part I, 1983 , pp. 641-654). MR 85i:14013 | Zbl 0537.14018 · Zbl 0537.14018 [14] K. KURDYKA , Ensembles semi-algébriques symétriques par arcs , (Math. Ann., Vol. 282, 1988 , pp. 445-462). MR 89j:14015 | Zbl 0686.14027 · Zbl 0686.14027 · doi:10.1007/BF01460044 · eudml:164472 [15] M. KASHIWARA and P. SCHAPIRA , Sheaves on Manifolds , (Springer-Verlag, Berlin, 1990 ). MR 92a:58132 | Zbl 0709.18001 · Zbl 0709.18001 [16] S. ŁOJASIEWICZ , Sur la géométrie semi- et sous-analytique , (Ann. Inst. Fourier, Grenoble, Vol. 43, 5, 1993 , pp. 1575-1595). Numdam | MR 96c:32007 | Zbl 0803.32002 · Zbl 0803.32002 · doi:10.5802/aif.1384 · numdam:AIF_1993__43_5_1575_0 · eudml:75048 [17] C. MCCRORY and A. PARUSIŃSKI , Complex monodromy and the topology of real algebraic sets , (Compositio Math. (to appear)). arXiv | Zbl 0949.14037 · Zbl 0949.14037 · doi:10.1023/A:1000126025773 · minidml.mathdoc.fr [18] A. PARUSIŃSKI and Z. SZAFRANIEC , Algebraically Constructible Functions and Signs of Polynomials , (Université d’Angers, prépublication no. 18, juin 1996 ). Manuscripta Math. (to appear). · Zbl 0913.14019 [19] P. SCHAPIRA , Operations on constructible functions , (J. Pure Appl. Algebra, Vol. 72, 1991 , pp. 83-93). MR 92h:32012 | Zbl 0732.32016 · Zbl 0732.32016 · doi:10.1016/0022-4049(91)90131-K [20] M. SHIOTA and M. YOKOI , Triangulations of subanalytic sets and locally subanalytic manifolds , (Trans. Amer. Math. Soc, Vol. 286, 1984 , pp. 727-750). MR 86m:32014 | Zbl 0527.57014 · Zbl 0527.57014 · doi:10.2307/1999818 [21] D. SULLIVAN , Combinatorial invariants of analytic spaces , (Proc. Liverpool Singularities Symposium I, Notes in Math., Springer-Verlag, Vol. 192, 1971 , pp. 165-169). MR 43 #4063 | Zbl 0227.32005 · Zbl 0227.32005 [22] O. Y. VIRO , Some integral calculus based on Euler characteristic , Lecture Notes in Math, Springer-Verlag, Vol. 1346, 1988 , pp. 127-138). MR 90a:57029 | Zbl 0686.14019 · Zbl 0686.14019
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