Narrow Results

Let M be a simple 3-manifold with a toral boundary component. It is known that if two Dehn fillings on M along the boundary produce a reducible manifold and a toroidal mainfold, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperbolic manifolds admitting a reducible Dehn filling and a finite Dehn filling of every type: cyclic, dihedral, tetrahedral, octahedral and icosahedral.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2ⁿ + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xⁿ + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽_p, and in that situation, we show 1 ∈ S for all primes p.

In 1908, Schur raised the question of the irreducibility over ℚ of polynomials of the form f(x) = (x - a₁)(x - a₂)⋯(x - a_n) + 1, where the a_i are distinct integers. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the analogous question when replacing the linear polynomials with cyclotomic polynomials and allowing the constant perturbation of the product to be any integer d ∉ {-1, 0}. One interesting consequence of our investigations is that we are able to construct, for any positive integer N, an infinite set S of cyclotomic polynomials such that 1 plus the product of any k (not necessarily distinct) polynomials from S, where k ≢ 0(mod 2^N+1), is reducible over ℚ.