Is there a way to determine if a polynomial has only real roots?

[Klaus_Hoffmann]

given a Polynomial or a trigonometric Polynomial

$$K(z)= \sum_{n=0}^{N}a_{n}x^{n}$$ and

$$H(x)= \sum_{n=0}^{N}b_{n}e^{inx}$$

is there a criterion to decide or to see if K(z) or H(x) have ONLY real roots

 

[mathman]

For the ordinary polynomial there is a procedure involving generating a Sturm sequence (gets messy for large N) which can be used to determine the number of real roots greater than a given value of x. To get what you want, use a sufficiently large negative x, i.e. look at the highest order term in each of the polynomials in the sequence (there will be N+1).