A note on positive roots discrimination system

It is of theoretical and practical significance to determine the number of roots in a certain range of a polynomial by an explicit criterion. Positive roots discrimination system for polynomial is a set of explicit expressions in terms of the coefficients, which is sufficient for determining the numbers of the positive roots. We give a new explicit criterion to determine the number of positive roots of a real polynomial with symbolic coefficients. The advantage is that the corresponding determination matrix has lower order, which facilitates symbolic computation.