Given a polynomial equation $x^n+a\_{n-1}x^{n-1}+\cdots+a\_1x+a\_0=0$, where $n$ is even and all the coefficients $a\_i$ are real, what is the best way to determine whether it has a real root or not?


I know [Sturm's theorem](http://en.wikipedia.org/wiki/Sturm%27s_theorem), but I am wondering if it's possible to determine this by the sign of some form of resolvent (*e.g.*, [discriminant](http://en.wikipedia.org/wiki/Discriminant) works for $n=2$, but not 4 or beyond)?


Please point me to some reference if this has already been studied. Thanks for your time!




---


I apologize for not searching hard enough - I just realized there is a similar question discussed [here](https://mathoverflow.net/questions/9431/when-does-a-real-polynomial-have-a-pair-of-complex-conjugate-roots) before, and has already got lots of useful comments. My hope is to find a function of these real coefficients and whether the polynomial has real root or not is determined by its sign.



Criteria to determine whether a real-coefficient polynomial has real root?
There is indeed an easy way to check if a univariate poly with real coefficients has a real root, without computing the roots.


Note that the answer for odd degree polynomials is always yes. For an even degree polynomial $p(x)$ do the following:


1. Compute the Hermite form of the polynomial. This is a symmetric matrix defined e.g. [here](http://stellar.mit.edu/S/course/6/sp10/6.256/courseMaterial/topics/topic2/lectureNotes/lecture-05/lecture-05.pdf): (on page 4, near the bottom denoted by $H\_1(p)$). The entries of this matrix can be filled using [Newton identities](http://en.wikipedia.org/wiki/Newton%27s_identities).
2. The number of real roots of $p(x)$ is equal to the signature of the Hermite matrix $H\_1(p)$, i.e., the number of positive eigenvalues minus the number of negative eigenvalues.
3. Since the Hermite form is symmetric, its characteristic poly has
only real roots. Therefore, we can apply [Descartes' rule of signs](http://en.wikipedia.org/wiki/Descartes%27_rule_of_signs) to its char.
polynomial, which would give us exactly the number of positive and negative
eigenvalues of the Hermite form.


This process gives you the exact number of real roots of $p(x)$ without computing the roots, and in particular you can see if the polynomial has a real root.


Hope this helps.


-Amirali Ahmadi


