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 $\text{f(x)∈}\text{R}\text{[x]}$ over a discrete normed commutative ring $\text{R}$ without zero divisors to the nearest element of $\text{R}$. In the case most interesting for applications, $\text{R}\text{=}\text{Z}$ , an algorithm for the determination of the existence of nonzero integers among these differences in time (n⁴log(H(f)+1))^1+ε is given. This problem arises when constructing algorithms for solving some systems of ODE.