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Consider a univariate polynomial of degree $n$ with real coefficients. Are there general equalities/inequalities on its coefficients, so that it has precisely $n$ real roots?

For example for the case of $n=2,3$ the necessary and sufficient inequality is given by the discriminant of the polynomial. If the discriminant is non-negative, the polynomial has only real roots. The case $n=4$ can also be handled since formulas for the roots exist. How about for larger $n$?

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  • $\begingroup$ You've seen Sturm sequences? $\endgroup$ – J. M. is a poor mathematician May 30 '13 at 15:26
  • $\begingroup$ @J.M.: Yes, i have. An idea is to compute the Sturm sequences for a general polynomial, i.e. as a function of its coefficients, take a sufficiently long interval $[a,b]$ and then require that $v(b)-v(a)=n$ where $n$ is the degree of the polynomial, and $v(c)$ is the number of sign changes of the Sturm sequence at $c$. Is that what you have in mind? $\endgroup$ – Manos May 30 '13 at 18:02
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    $\begingroup$ Something like that, yes. $\endgroup$ – J. M. is a poor mathematician May 30 '13 at 18:04
  • $\begingroup$ There is this problem: Sturm sequences count only distinct roots. So if i enforce $v(b)-v(a)=n$, then i am excluding all polynomials with repeated real roots. I have not been able to find other results. However, what i am interested in, is just enforcing all roots to be real, i really don't care about their location, and i have not been able to identify something useful towards this end in the literature. $\endgroup$ – Manos May 30 '13 at 18:12
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    $\begingroup$ If memory serves, there is a modification of that route where there is a clear signal when your polynomial has in fact repeated roots, and that if it runs to completion, then it is guaranteed that your polynomial's roots are all real. Equivalently, you can build a symmetric tridiagonal matrix whose characteristic polynomial is your given polynomial with that route, only if all the polynomial's roots are real. I will have to look up the necessary refs... $\endgroup$ – J. M. is a poor mathematician May 30 '13 at 18:18

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