An explicit criterion to determine the number of roots of a polynomial on an interval

Abstract. In a recent article, a complete discrimination system was given for a polynomial with symbolic/literal coefficients. Such a system is a set of explicit expressions in terms of the coefficients, which is sufficient for determining the numbers and multiplicities of the real and imaginary roots. Sometimes, however, one wants to know the number of roots in a certain range, especially, in a interval. We give here an explicit criterion to determine the number of positive roots of a real polynomial, and hence an efficient algorithm is introduced for determining the number of roots in a general interval. 1