Automatic computation of the complete root classification for a parametric polynomial

An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for non-realizable conditions. The improvement lies in the direct use of ‘sign lists’, obtained from the discriminant sequence, rather than ‘revised sign lists’. It is shown that the discriminant sequences, upon which the sign lists are based, are closely related both to Sturm-Habicht sequences and to subresultant sequences. Thus calculations based on any of these quantities are essentially equivalent. One particular application of complete root classifications is the determination of the conditions for the positive definiteness of a polynomial, and here the new algorithm is applied to a class of sparse polynomials. It is seen that the number of conditions for positive definiteness remains surprisingly small in these cases.