ICTP/SISSA Joint Colloquium (24 April 2007)

[Math]

ICTP/SISSA Joint Colloquium in Mathematics

Announcement



Tuesday, 24 April 2007, at 14.00 hrs.



Professor Enrico Bombieri

Institute for Advanced Study, Princeton, USA



Kahane polynomials and their derandomization
(Joint work with J. Bourgain)



Abstract:

In 1957 Erd\"os studied trigonometric polynomials with
all coefficients of absolute value 1 (called unimodular polynomials)
and was led to the conjecture that the the maximum modulus of a
unimodular polynomial of degree n is at least (1+c)\sqrt(n) for
some positive absolute constant c.  This conjecture was disproved
by Littlewood in 1966 and, on the basis of numerical evidence,
Littlewood conjectured that there are unimodular polynomials that
deviate by o(\sqrt{n}) from its mean-square value \sqrt{n+1}.
The existence of such polynomials, of any given degree, was proved
by Kahane in 1980 using probabilistic methods, obtaining a remainder
term of O(n^{1/2-1/17}\sqrt{log n}).

In this lecture it will be given a new construction of these
polynomials with the improved remainder term O(n^{1/2-1/9+epsilon}),
first using probabilistic methods, and then with an explicit
construction.  The explicit construction makes use of Deligne's
Riemann hypothesis for L-functions over varieties in positive
characteristic associated to mixed exponential sums in arbitrarily
many variables, as well as of sieves and recent results about gaps
between squarefree numbers.


Venue:  ICTP Main Lecture Hall, Main Building, entrance level
                (to be confirmed)