Distances from differences of roots of polynomials to the nearest integers

Abstract

We present tight bounds for distances from differences of roots of the polynomial f(x)∈ R[x] over a discrete normed commutative ring R without zero divisors to the nearest element of R . In the case most interesting for applications, R= Z , an algorithm for the determination of the existence of nonzero integers among these differences in time ( n 4log( H( f)+1)) 1+ ε is given. This problem arises when constructing algorithms for solving some systems of ODE.