A fast algorithm for computing multiplicative relations between the roots of a generic polynomial

Abstract

Multiplicative relations between the roots of a polynomial in Q[x] have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial f∈Q[x] to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in Smyth (1986), Baron et al. (1995) and Dixon (1997). Based on the new condition, a subset E⊂Q[x] is defined and proved to be generic (i.e., the set Q[x]\\E is very small). We develop an algorithm deciding whether a given polynomial f∈Q[x] is in E and returning a basis of the lattice consisting of the multiplicative relations between the roots of f whenever f∈E. The numerical experiments show that the new algorithm is very efficient for the polynomials in E. A large number of polynomials with much higher degrees, which were intractable before, can be handled successfully with the algorithm.