The polynomial roots repartition and minimum roots separation

It is known that, if all the roots of a are real, they can be localised, using a set of intervals, which contain the arithmetic average of the roots. The aim of this paper is to present an original method for giving other distributions of the roots/ modules of the roots on real axis, a method for evaluating and improving the polynomial minimum root separation results, a method for the complex polynomials and for polynomials having all roots real. We use the discriminant, Hadamard's inequality, Mahler's measure and new original inequalities. Also we will make some considerations about the cost for isolate the real roots. Our method is based on the successive splitting for the interval which contains all roots.