Computation of Minkowski Values of Polynomials over Complex Sets

As a generalization of Minkowski sums, products, powers, and roots of complex sets, we consider the Minkowski value of a given polynomial P over a complex set X. Given any polynomial P(z) with prescribed coefficients in the complex variable z, the Minkowski value P(X) is defined to be the set of all complex values generated by evaluating P, through a specific algorithm, in such a manner that each instance of z in this algorithm varies independently over X. The specification of a particular algorithm is necessary, since Minkowski sums and products do not obey the distributive law, and hence different algorithms yield different Minkowski value sets P(X). When P is of degree n and X is a circular disk in the complex plane we study, as canonical cases, the Minkowski monomial value P m (X), for which the monomial terms are evaluated separately (incurring $$ \frac{1}{2} $$ n(n+1) independent values of z) and summed; the Minkowski factor value P f (X), where P is represented as the product (z−r 1)⋅⋅⋅(z−r n ) of n linear factors – each incurring an independent choice z∈X – and r 1,...,r n are the roots of P(z); and the Minkowski Horner value P h (X), where the evaluation is performed by “nested multiplication” and incurs n independent values z∈X. A new algorithm for the evaluation of P h (X), when 0∉X, is presented.