I want to find the relations between coefficients of even order polynomial such that the polynomial has all roots as complex numbers. Please reply if anyone has gone through this.
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.
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