Timeline for Find the order of a class of finite matrices over finite fields
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 23, 2021 at 12:30 | comment | added | Roland Bacher | @Balazs Pozsgay : I could not find your e-mail so easily. Try [email protected] (this should work, my University had recently a namechange and the administrative people at the top considered it a great idea to disable the old adresses) | |
| Dec 23, 2021 at 10:22 | comment | added | Balázs Pozsgay | Dear Roland, I tried to contact you via email, but the address I found was not good. Could you please contact me somehow? (my address can be found easily by search) | |
| Dec 17, 2021 at 14:22 | comment | added | Roland Bacher | Divisibility of $P_{ab}$ by $P_a$ is obviously true (independently of the conjectural truth of the recursion relations for $Q_n$) by the Fourier analysis argument made by Denis Serre (easy proof: Repeat coordinates $(v_1,\ldots, v_{2a})$ of an eigenvector for $M_a$ in order to get an eigenvector of the same eigenvalue for $M_{ka}$. This factorization holds thus over ℤ[𝑥]. Irreducible polynomials over ℤ decay further over primes according to the ramification/intertia stuff of classical algebraic number theory. | |
| Dec 11, 2021 at 4:18 | comment | added | Balázs Pozsgay | Thanks, interesting! For the characteristic polynomial did you work here above the real numbers? I don't see the appearance of the field characteristic. But if the divisibility holds with integer coefficient polynomials, then I understand that it also holds over the finite field case. In any case, if we prove this relation and the divisibility, it could lead to a proof of at least the case $L=2\cdot p^m$. That is already useful. | |
| Dec 10, 2021 at 20:22 | comment | added | Roland Bacher | I have changed my notations to yours and I have added a small experimental observation on the involved characteristic polynomials. | |
| Dec 10, 2021 at 20:20 | history | edited | Roland Bacher | CC BY-SA 4.0 |
changed the notation in order to keep the notation $L$ for the size of all matrices used by the OP. Added an experimental observation.
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| Dec 10, 2021 at 13:12 | comment | added | Balázs Pozsgay | Thank you, this is useful! But there are indeed many problems from a practical point of view. For example, would you expect that the factorization of the char.pol. over the finite field can be done analytically, for every $L$? I don't expect this to happen.... (I guess my $L$ is your $n$) But at least now I know how the large primes can enter the game... Perhaps the nice result for $L=2p^m$ could be proven analytically, without a case by case computation... | |
| Dec 10, 2021 at 13:09 | vote | accept | Balázs Pozsgay | ||
| Dec 10, 2021 at 10:40 | history | answered | Roland Bacher | CC BY-SA 4.0 |