Timeline for Which sets of roots of unity give a polynomial with nonnegative coefficients?
Current License: CC BY-SA 3.0
9 events
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| May 6, 2019 at 14:22 | comment | added | Joshua P. Swanson | "I believe this is the only way to obtain such a polynomial that is a cyclotomic polynomial": yes, you're correct. The easiest way to see this is plugging in x=1: it's easy to show that $\Phi_m(1) = \begin{cases}0 & m=1 \\ p & m=p^k \\ 1 & \text{otherwise}\end{cases}.$ This essentially says there must be cancellation outside of the prime-power case. | |
| Dec 20, 2018 at 9:41 | history | edited | Wolfgang |
added tag
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| Aug 22, 2015 at 0:43 | history | edited | Louis Deaett | CC BY-SA 3.0 |
I fixed a typo in one equation, and made the title a bit more succinct.
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| Aug 21, 2015 at 20:02 | comment | added | Louis Deaett | @Josep: The reference you gave is extremely interesting. I was totally unaware of Suffridge's result. | |
| Aug 21, 2015 at 19:49 | history | edited | Louis Deaett | CC BY-SA 3.0 |
I changed the format of the citation and also added some additional info regarding the cyclotomic case.
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| Aug 19, 2015 at 14:44 | comment | added | user75485 | In fact the Suffridge polynomials cover the more general case when $1\not\in S,$ S self-conjugate and the arguments of the roots in S are separated by the same angle, except for a pair. | |
| Aug 18, 2015 at 20:42 | comment | added | user75485 | In the wedge case you can compute explicitly the coefficients and see they are positive. It's a particular case of Suffridge's extremal polynomials- you have the expression deduced in Sheil-Small, Complex polynomials, p. 251-252. | |
| Aug 15, 2015 at 1:51 | comment | added | Suvrit | Maybe, the following related result of Kellog is helpful: Let $A$ be a complex $n\times n$ matrix. If all its elementary symmetric functions are positive (so that the characteristic polynomial has alternating signs), then the spectrum of $A$ lies in the set $\{z : |\text{arg}z| \le \pi - \pi/n\}$.... | |
| Aug 14, 2015 at 14:23 | history | asked | Louis Deaett | CC BY-SA 3.0 |