Timeline for Sum of the coefficients of the characteristic polynomial of periodic matrices
Current License: CC BY-SA 4.0
16 events
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| Sep 4, 2020 at 6:10 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 4, 2020 at 5:50 | vote | accept | Mare | ||
| Sep 4, 2020 at 4:32 | history | became hot network question | |||
| Sep 3, 2020 at 22:32 | comment | added | YCor | If $\Phi(t)$ is a cyclotomic polynomial other than $t\pm 1$ then $\Phi$ has no real root, hence has a constant sign on the reals. Since it's monic, it's then positive on reals, and hence $\ge 1$ on integers. | |
| Sep 3, 2020 at 22:27 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 3, 2020 at 22:20 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 3, 2020 at 22:19 | answer | added | Qiaochu Yuan | timeline score: 10 | |
| Sep 3, 2020 at 22:19 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 3, 2020 at 22:17 | comment | added | Mare | @JoeSilverman That answers Q1 together with the wikipedia link I guess. If you want you can turn it into an answer. | |
| Sep 3, 2020 at 22:14 | comment | added | Joe Silverman | Usually people say that $M$ has finite order, rather than calling it periodic. In any case, the fact that $M^n=I$ means that the eigenvalues are roots of unity, which in turn implies that its characteristic polynomial is a product of cyclotomic polynomials (since the characteristic polynomial is in $\mathbb Z[x]$). Hence as noted earlier, your question is whether cyclotomic polynomials are non-negative when evaluated at 1, it has nothing ot do with matirces. | |
| Sep 3, 2020 at 22:12 | comment | added | Mare | @AbdelmalekAbdesselam I forgot whether this holds in general, but it holds for acyclic quiver algebra and thus your comment answers Q3! | |
| Sep 3, 2020 at 22:08 | comment | added | Abdelmalek Abdesselam | For Q1 one could perhaps show that $p_A$ is a product of cyclotomic polynomials and then the result would follow from en.wikipedia.org/wiki/Cyclotomic_polynomial#Polynomial_values | |
| Sep 3, 2020 at 22:02 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 3, 2020 at 21:25 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 3, 2020 at 20:48 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 3, 2020 at 20:29 | history | asked | Mare | CC BY-SA 4.0 |