Math. Z. 208, 589 596 (1991) Mathematische Zeitschrift (C) Springer-Verlag 1991 Finiteness of semialgebraic types of polynomial functions Riccardo Benedetti 1 and Masahiro Shiota 2 i Dipartimento di Matematica, Universita di Pisa, Via Buonarroti, 2, 1-56100 Pisa, Italy 2 Department of Mathematics, College of General Education, Nagoya University, Chikusa-ku, Nagoya, 464 Japan Received February 6, 1991; in final form April 22, 1991 Introduction Triangulation theorems have been proved for sets of increasing order of generali- ty (semianalytic, subanalytic, Whitney stratified, etc.). In semialgebraic geometry, we have the much stronger result that triangulations of semialgebraic sets can be obtained in an effective way. In contrast to the triangulation of sets, the triangulation of mappings is often a more difficult problem (see [S]). The results of the present work are based on an effective triangulation theorem for semialge- braic mappings (Theorem 7). To be a little more precise let us introduce some notation. We write X~S(n,d,r) if X~IR ~ is a semialgebraic set with a given presentation of the form: k s i x = U N {/ij* jo) i-l j-1 where for each i and j, f~j is a polynomial function on IR" of degree <d and *ij means " = " or " > ", and sl + ... + Sk <= r. Similarly feS(n, n', d, r) means that f is a continuous semialgebraic map from a semialgebraic set X clR" to another Yc lR"' with graph F~ e S (n + n', d, r). The effectiveness of semialgebraic triangulation of semialgebraic sets implies that there exists an algorithm which, starting from any couple ((n, d, r), X)eN 3 x S(n, d, r) with compact X, produces (a) a triangulation of X, z: IK[ ~X, where z is a semialgebraic homeomorphism and K is a finite simplicial complex in lR ", and (b) a function N3~(n, d, r) -,(D, R, h)~N 3 such that zES(n, n, D, R) and # K, the number of simplexes of K, is not larger than h. For such an algorithm see [D-K], [B-C-R] and [B-R]. It is ultimately based on Tarski-Seidenberg theorem, that is, on what we could call the "projec- tion method", it actually holds over any real closed field R.