The number of irreducible factors of a polynomial, II
Christopher G. Pinner, Jeffrey D. Vaaler · Acta Arithmetica · 1996
: Given a polynomial f 2 k[x], k a number field, we consider bounds on the number of cyclotomic factors of f appropriate when the number of non-zero coefficients of the polynomial, N(f ), is substantially less than than its degree. In particular we obtain bounds which (apart from a small degree dependence) are only polynomial in N(f ). These results arise from variants of Mann's theorem on linear relations between roots of unity. 1. Introduction Let k be an algebraic number field and F (x) a polynomial in k[x] with degree @(F ) and F (0) 6= 0. In [5] we considered the problem of estimating the number of irreducible factors of F in k[x] in terms of @(F ) and of the height H(F ) of the vector of coefficients of F . As is already clear from earlier work of Schinzel [6] and Dobrowolski [1], it is natural in problems of this type to give separate estimates for the number of cyclotomic factors and for the number of noncyclotomic factors. In the present paper we estimate the number of irredu...