On the distribution of the number of roots of polynomials and explicit weak designs

Weak designs were defined in R. Raz, O. Reingold, and S. Vadhan [Extracting all the randomness and reducing the error in Trevisan's extractors, Proc 31st ACM Symp Theory of Computing, Atlanta, GA, May 1999, to appear in J Comput System Sci Special Issue on STOC 99] and are used there in constructions of extractors. Roughly speaking, a weak design is a collection of subsets satisfying some near-disjointness properties. Constructions of weak designs with certain parameters are given in Raz et al. These constructions are explicit in the sense that they require time and space polynomial in the number of subsets. However, the constructions require time and space polynomial in the number of subsets even when needed to output only one specific subset out of the collection. Hence, the constructions are not explicit in a stronger sense. In this work we provide constructions of weak designs (with parameters similar to the ones of Raz et al.) that can be carried out in space logarithmic in the number of subsets. Moreover, our constructions are explicit even in a stronger sense: Given an index to a subset, we output the specified subset in time and space polynomial in the size of the index. Using our constructions, we obtain extractors similar to some of the ones given in Raz et al. in terms of parameters, and that can be evaluated in logarithmic space. Our main construction is algebraic. In order to prove the properties of weak designs, we prove some algebro-combinatorial lemmas that may be interesting in their own right. These lemmas regard the number of roots of polynomials over finite fields. In particular, we prove that the number of polynomials (over any finite field) with k roots, vanishes exponentially in k. In other words, we prove that the number of roots of a random polynomial is not only bounded by its degree (a well-known fact), but, furthermore, it is concentrated exponentially around its expectation (which is 1). Our lemmas are proved by algebro-combinatorial arguments. The main lemma is also proved by a probabilistic argument. (C) 2003 Wiley Periodicals, Inc.