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Finiteness of semialgebraic types of polynomial functions. (English) Zbl 0744.14034

This paper shows an effective semialgebraic triangulation of a semialgebraic function over \(\mathbb{R}\) and any real closed field \(R\), which is a refinement of a result by M. Shiota in Real analytic and algebraic geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1420, 247-307 (1990; Zbl 0716.32006). As a corollary a computable function \(\psi\) on \(\mathbb{N}^ 2\) is found so that the number of semialgebraic equivalence classes of polynomial functions on \(R^ n\) of degree \(\leq m\) is bounded by \(\psi(n,m)\). — Here two polynomial functions \(f_ 1\) and \(f_ 2\) on \(R^ n\) are called semialgebraically equivalent if there exist semialgebraic homeomorphisms \(\tau_ 1\) of \(R^ n\) and \(\tau_ 2\) of \(R\) such that \(f_ 1\circ\tau_ 1=\tau_ 2\circ f_ 2\). This corollary is a generalization of a result of T. Fukuda [Publ. Math., Inst. Hautes Etud. Sci. 46, 87-106 (1976; Zbl 0341.57019)], which showed finiteness of the number of topological equivalence classes of polynomial functions on \(\mathbb{R}^ r\) of degree \(\leq m\).
Reviewer: R.Benedetti

MSC:

14P10 Semialgebraic sets and related spaces
32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions
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References:

[1] [B-C-R] Bochnak, J., Coste, M., Roy, M.-F.: Géométrie algébrique réelle. Berlin Heidelberg New York: Springer 1987
[2] [B-R] Benedetti, R., Risler, J.-J.: Real algebraic and semialgebraic sets. Paris: Hermann 1990
[3] [D-K] Delfs, H., Knebusch, M.: On the homology of algebraic varieties over real closed fields. J. Reine Angew. Math.335, 122–163 (1982) · Zbl 0484.14006 · doi:10.1515/crll.1982.335.122
[4] [F] Fukuda, T.: Types topologiques des polynômes. Publ. Math. Inst. Hautes Étud. Sci.46, 87–106 (1976) · Zbl 0341.57019 · doi:10.1007/BF02684319
[5] [L] Lojasiewicz, S.: Triangulations of semi-analytic sets. Ann. Sc. Norm. Super. Pisa, Cl. Sci.18, 449–474 (1964)
[6] [R-S] Rourke, C.-P., Sanderson, B.-J.: Introduction to piecewise linear topology. Berlin Heidelberg New York: Springer 1976 · Zbl 0254.57010
[7] [S] Shiota, M.: Piecewise linearization of subanalytic functions II. In: Galbiati, M., Tognoli, A. (eds.) Real analytic and algebraic geometry. (Lect. Notes Math., vol. 1420, pp. 247–307) Berlin Heidelberg New York: Springer 1990
[8] [T] Thom, R.: La stabilité topologique des applications polynomiales. Enseign. Math.8, 24–33 (1962) · Zbl 0109.40002
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