A test for nonnegativity of real polynomials

A rigorous method to determine whether a polynomial f(x) is nonnegative for all x in an interval is proposed. Conventionally, a method which determines whether the multiplicity of every zero of f(x) is even has been used for this problem. The conventional method is, however, somewhat complicated because the Sturm function sequence must be generated repeatedly and the number of variations of sign must be investigated for each sequence. The method proposed in this paper determines whether f(x) + ε (ε is a sufficiently small positive number) is positive for all x in the interval. This method is simpler than the conventional one in the sense that only two Sturm function sequences are generated. Moreover, the determination can easily be made with the number of variations of sign of the function sequence made by concatenating two Sturm function sequences.