Tel Aviv University Dear all, This week at the Horowitz seminar on Probability, Ergodic Theory and Dynamical Systems at Tel Aviv University we are happy to have Fedor Nazarov from Kent State University. NOTE THE SPECIAL END TIME! The talk will be longer than usual. Speaker: Fedor Nazarov, Kent State University Title: Littlewood-Offord-Turan estimate for the number of real zeroes of a random polynomial with i.i.d. coefficients Date: Monday, December 31 Time: from 14:30 to 16:00! Place: Schreiber 309 Abstract: We show that the average number of real zeroes of any random polynomial of degree $n$ with independent identically distributed coefficients does not exceed $C\log^4 n$ with some absolute $C>0$. The proof follows closely the paper by Littlewood and Offord published in 1942, where the case of $\pm 1$ coefficients was considered. Our main deviation from their scheme is using the Turan lemma from Turan's 1953 book in place of the pointwise "anticoncentration" estimate used by Littlewood and Offord. The result is on par with the original Littlewood-Offord bound $C\log^2 n$ but still short of the would be optimal estimate $C\log n$, which is widely believed to hold and whose validity is known for all sufficiently regular distributions. This is a joint work with Mikhail Sodin. Best regards, Ron Seminar webpage: <http://www.math.tau.ac.il/~peledron/Horowitz_seminar/Horowitz_seminar.html> --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Ron Peled <peledron@post.tau.ac.il>