The Automatic Computation of the Complete Root Classification for a Parametric Polynomial

A new algorithm for the automatic computation of the complete root classification of a real parametric polynomial is described. The improvement lies in the fact that previous algorithms required ‘revised sign lists’ derived from the discriminant sequence. This is replaced here by the direct use of ‘sign lists’ derived from the discriminant sequence without the need for revision of the lists. As well as offering greater efficiency, the new algorithm offers a better filter for eliminating non-realizable conditions. The new algorithm is applied to the problem of determining the conditions for the positive definiteness of a polynomial. For a class of sparse polynomials, the number of conditions remains surprisingly small.