Dear Pranav,

Take a look at

http://pmpu.ru/vf4/detse/discrime

Subsection "Reality of zeros".

WBR, Alexei

Dear Pranav,

Hadamard's theorem can be helpful in your question.

Check Appendix B in the following link:

https://www.researchgate.net/publication/272429520_On_the_Transverse_Low_Frequency_Vibrations_of_a_Traveling_String_With_Boundary_Damping

Best regards,

Nick

Article On the Transverse, Low Frequency Vibrations of a Traveling S...

There is a simple sufficient condition, given by Descartes' rule of signs (see the link to Wikipedia):

If p(x) is a polynomial with real coefficients, then the number of positive (real) roots of p is at most the number of change of signs between the monomials of p(x), when ordered from greater to smaller degree. Similarly, the number of negative (real) roots of p is at most the number of sign changes between the monomials of p(-x).

So, if p(x) has only positive monomials of even degree, and has nonzero independent term, then p has only complex (not real) roots. For example p(x)=x^6+2x^2+1 satisfies these criteria.

https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs

Thanks George.. It really looks great. Will go through it in detail. Also, your previous link about discussion including Hermite form of polynomial was very interesting.