An algorithm is described that determines whether a given polynomial with integer coefficients has a cyclotomic factor. The algorithm is intended to be used for sparse polynomials given as a sequence of coefficient-exponent pairs. A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.
For P 2 Zx, let kPk denote the Euclidean norm of the coeecient v ector of
P. Turán asked if there exists an absolute constant C such that for every polynomial f ∈ Z[x] there exists an irreducible polynomial g ∈ Z[x] with deg(g) ≤ deg(f) and L(f − g) ≤ C, where L(·) denotes the sum of the absolute values of the coefficients. We show that C = 5 suffices for all integer polynomials of degree at most 40 by investigating analogous… (More)
A reciprocal polynomial g(x) ∈ Z[x] is such that g(0) = 0 and if g(α) = 0 then g(1/α) = 0. The non-reciprocal part of a monic polynomial f (x) ∈ Z[x] is f(x) divided by the product of its irreducible monic reciprocal factors (to their multi-plicity). This paper presents an algorithm for testing the irreducibility of the non-reciprocal part of a 0,… (More)
To better understand the distribution of gaps between k-free numbers, Erd˝ os posed the problem of establishing an asymptotic formula for the sum of the powers of the lengths of the gaps between k-free numbers. This paper generalizes the problem of Erd˝ os by considering moments of gaps between positive integers m for which f (m) is k-free. Here, f (x)… (More)