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Real roots criterion

  1. Jul 23, 2007 #1
    given a Polynomial or a trigonometric Polynomial

    [tex] K(z)= \sum_{n=0}^{N}a_{n}x^{n} [/tex] and

    [tex] H(x)= \sum_{n=0}^{N}b_{n}e^{inx} [/tex]

    is there a criterion to decide or to see if K(z) or H(x) have ONLY real roots
  2. jcsd
  3. Jul 23, 2007 #2


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    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).
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