From Covering Problems to a Conjecture of Turan
by
Michael Filaseta
University of South Carolina

A covering of the integers is a system of congruences x \equiv a_j mod m_j, where a_j and m_j denote integers with m_j > 0 for each j, such that every integer satisfies at least one of the congruences. A conjecture of Turán's asserts loosely that every polynomial with integer coefficients is close to an irreducible polynomial with integer coefficients. More precisely, he has conjectured that there is an absolute constant C such that if f(x) = \sum_j=0^r a_jx^j has integral coefficients, then there is an irreducible g(x) = \sum_j=0^s b_jx^j with integer coefficients such that s <= r and such that the sum of the absolute values of the coefficients of f(x)-g(x) is bounded by C. The conjecture remains open, though Schinzel has established the conjecture with the condition s <= r replaced by a more relaxed condition on s. This talk will be a survey of some recent investigations beginning with some old results and questions on coverings, moving to connected questions about polynomials and their reducibility, and ending with some recent progress on the conjecture of Turán.

Home Page of Michael Filaseta

Date received: March 16, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacu-02.