Bounding the number of connected components of a real algebraic set

Abstract

For every polynomial map f=(f1,..., fk): ℝn →ℝk, we consider the number of connected components of its zero set, B(Zf) and two natural "measures of the complexity of f," that is the triple (n, k, d), d being equal to max(degree of fi), and the k-tuple (Δ1,...,Δ4), Δk being the Newton polyhedron of fi respectively. Our aim is to bound B(Zf) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom's bound μd (n)=d(2 d-1)n-1. Considered as a polynomial in d, μd (n) has leading coefficient equal to 2n-1. We obtain a bound depending on n, d, and k such that if n is sufficiently larger than k, then it improves μd (n) for every d. In particular, it is asymptotically equal to 1/2(k+1)nk-1 dn, if k is fixed and n tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections. © 1991 Springer-Verlag New York Inc.