The Minimum Root Separation of a Polynomial

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 semi-norm for a commutative ring is defined, and it is shown how any semi-norm can be extended to polynomial rings and matrix rings, obtaining a very general analogue of Hadamard's determinant theorem. (1) Computer Science Department, Stanford University, and Computer Sciences Department, University of Wisconsin-Madison. (2) Computer Science Department, Cornell University. This research is supported by Grants GJ-30125X and GJ-33169 from the National Science Foundation, the Wisconsin Alumni Research Foundation, and (in part) by the Advanced Research Projects Agency of the Office of the Secretary of Defense (SD-183). The views and conclusions contained in this document are those of the author and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the Advanced Research Projects Agency, NSF, or the U.S. Government. Reproduced in the USA. Available from the National Technical Information Service, Springfield, Virginia 22151. THE MINIMUM ROOT SEPARATION OF A POLYNOMIAL i L