The minimum root separation of a polynomial

Abstract

The minimum root separation of a complex polynomial A is defined as the minimum of the distances between distinct roots of A. For polynomials with Gaussian integer coefficients and no multiple roots, three lower bounds are derived for the root separation. In each case, the bound is a function of the degree n of A and the sum d of the absolute values of the coefficients of A. The notion of a seminorm for a commutative ring is defined, and it is shown how any seminorm can be extended to polynomial rings and matrix rings, obtaining a very general analogue of Hadamard’s determinant theorem.