Branching and Connectivity

Abstract

In the first section we derive the Milnor-Thom bound [366, 516] which gives a quantitative estimate on the number of connected components of a semi-algebraic set in Rⁿ described by polynomial equalities and inequalities. This estimate depends on the number of variables n, the number of inequalities and the maximum of the degrees of the occurring polynomials. Its proof is based on Bézout’s inequality and on the Morse-Sard theorem. In the next section we investigate computation trees over R which solve the membership problem for a semi-algebraic subset W of Rⁿ. Ben-Or’s lower bound [37] on the multiplicative branching complexity of such membership problems in terms of the number of connected components of W is deduced from the Milnor-Thom bound. Then we discuss applications to the real knapsack problem and to several problems of computational geometry (such as the computation of the convex hull of a finite set of points in the plane).