Estimations for the Separation Number of a Polynomial System

This paper is devoted to the study of the separation number of a polynomial system f : C → C defining a zero-dimensional nonsingular variety. The separation number of f is defined by sep(f) = min ‖x− y‖ where the minimum is taken for x 6= y ∈ C with f(x) = f(y) = 0. In the case of a single variable polynomial f(x) = ad ∏d i=1(x − ri) various lower bounds for sep(f) have been proved and can be found in Mignotte’s (1989) book. These inequalities are based on the discriminant of the polynomial f and Hadamard’s inequality. For example sep(f) > d−(d+2)/2|Discr(f)|1/2M(f)1−d