A complete discrimination system for polynomials with complex coefficients and its automatic generation
Write the polynomial \(f(x)\) as \(f_1(x)+ if_2(x)\), where \(f_1\) and \(f_2\) have real coefficients. The authors give a method of determining the numbers of real zeros, pairs of complex conjugate zeros, and complex zeros with no conjugate (which can arise if \(f_2\) has some non-zero coefficients) of \(f\). The method is based on constructing sequences of numbers starting from the signs of determinants of subresultants of various polynomials and their derivatives. It has been implemented in Maple, and some run times are given. The authors also consider dealing with polynomials with literal coefficients and the problem of investigating the positive definiteness of polynomials in two variables. Proofs are not given, but are referred to from papers inaccessible to this reviewer. Examples, for reasons of space, are not given in full.
scientific article; zbMATH DE number 1342527