The number of polytopes, configurations and real matroids

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/ oo. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thorn from real algebraic geometry. §1. Introduction. Let c(n, d) denote the number of (combinatorial types of) d-polytopes on n labelled vertices and let cs(n, d) denote the number of simplicial rf-polytopes on n labelled vertices. The problem of determining or estimating these two functions (especially for 3-polytopes) was the subject of much effort and frustration of nineteenth-century geometers. Although it follows from Tarski's Theorem on the decidability of first order sentences in the real field that the problem of computing c(n, d) is solvable (cf. (Gr. pp 91-92)), it seems extremely difficult actually to determine this number even