Finiteness of semialgebraic types of polynomial functions
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 \textit{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 \textit{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 \(\leq m\).
number of semialgebraic equivalence classes of polynomial functions
scientific article; zbMATH DE number 54903
scientific article