Timeline for When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Feb 12, 2017 at 21:50 | comment | added | Federico Poloni | I think there is still some work to be done to get a full solution (for instance, to remove denominators and reduce to the polynomial case). It is quite late now in my time zone and I don't have the time to write a complete answer, but if you know how to put everything together please go on and write it yourself -- I will happily leave the imaginary internet points to you. :) | |
| Feb 12, 2017 at 21:43 | comment | added | Max Alekseyev | @FedericoPoloni: Thanks for the nice observation. It reduces this new question to the old one. Please post your comments as an answer, and I'll accept it (to keep this question for the reference). | |
| Feb 12, 2017 at 21:34 | comment | added | Federico Poloni | Another observation derived from similar problems over $\mathbb{C}$ is that $f(\exp(i\theta)+\exp(-i\theta)) = g(\exp(i\theta))g(\exp(-i\theta)) = |g(\exp(i\theta)|^2 \geq 0$, so a necessary condition is that $f(x)\geq 0$ for each $x\in[-2,2]$. | |
| Feb 12, 2017 at 21:28 | comment | added | Federico Poloni | I suspect the two questions may be related via the following substitution (Cayley transform): if $t = \frac{1+x}{1-x}$, then $\frac12 (t+t^{-1}) = \frac{1+x^2}{1-x^2}$. | |
| Feb 12, 2017 at 21:12 | history | asked | Max Alekseyev | CC BY-SA 3.0 |