We d escribe new methods for the e s t imation of the b o u nds of the coeecients of proper divisors of integer polynomials in one v ariable. There exist classes of poly-nomials for which our estimates are better than those obtained using t he polynomial measure or the 2-weighted norm.

We consider the problem of derandomizing random walks in the Euclidean space R. We show that for k = 2, and in some cases in higher dimensions, such walks can be simulated in Logspace using only poly-logarithmically many truly random bits. As a corollary, we show that the random walk can be deterministically simulated in space O(logn √ log log n), where 1/n is the desired precision of the simul...

In 1957 W.M. Hirsch conjectured that every d-polytope with n facets has edge-diameter at most n ? d. Recently Holt and Klee constructed polytopes which meet this bound for a number of (d; n) pairs with d 13 and for all pairs (14; n). These constructions involve a judicious use of truncation, wedging, and blending on polytopes which already meet the Hirsch bound. In this paper we extend these te...