Computation of Minkowski Values of Polynomials over Complex Sets

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