Timeline for On Lehmer's conjecture for a given degree
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2021 at 11:23 | comment | added | Gerry Myerson | No, it's complete through degree $44$. Look through Mossinghoff's site, he links to the papers where the work is done. | |
| Jan 3, 2021 at 10:20 | comment | added | Mare | @GerryMyerson Thanks. How do they know that the list is complete for degree at most 44? Maybe complete here just means that it is the complete list of polynomials that have been found by humans until now for those degrees. | |
| Jan 3, 2021 at 6:12 | comment | added | Gerry Myerson | Verger-Gaugry's second attempt: arxiv.org/abs/1911.10590 | |
| Jan 3, 2021 at 6:06 | comment | added | Gerry Myerson | The mjm webpages have not been updated since 2011. I don't know whether there has been any progress since then. Jean-Louis Verger-Gaugry claims to have settled the problem in "A proof of the conjecture of Lehmer and of the conjecture of Schinzel-Zassenhaus," arxiv.org/abs/1709.03771 a paper of $164$ pages. I am not qualified to offer an opinion on this work. It is discussed at mathoverflow.net/questions/286640/… | |
| Jan 3, 2021 at 5:53 | comment | added | Gerry Myerson | One of the items at the site @Felipe gives is "The complete list of irreducible, noncyclotomic polynomials having degree at most $44$ and Mahler measure below $1.3$." So we can say $m_n$ is known for $n\le44$, and that Lehmer's polynomial beats the competition at least up to that degree. By the way, Lehmer didn't conjecture that the measure he found was the minimum, as he felt he didn't have enough evidence to justify a conjecture. And of course he didn't use the term, Mahler measure. Mahler's work was still $30$ years in the future when Lehmer wrote his paper. | |
| Jan 2, 2021 at 21:51 | comment | added | Mare | Ah I think I got it: We can assume that the polynomial is self-reciprocal and when the degree is odd then f(-1)=0 and thus -1 is a zero and f is not irreducible. Thus we can assume the degree is even. | |
| Jan 2, 2021 at 21:34 | comment | added | Mare | @FelipeVoloch Thanks, Im new to this problem and I do not know much. This links is very interesting. Probably a stupid question: Why do the lists in this link contain only polynomials of even degree? | |
| Jan 2, 2021 at 20:06 | comment | added | Felipe Voloch | cecm.sfu.ca/~mjm/Lehmer | |
| Jan 2, 2021 at 18:41 | comment | added | Stanley Yao Xiao | The record for degree dependent bounds is due to Dobrowolski, with the constant improved by Voutier in 1996. Have you looked at those papers? | |
| Jan 2, 2021 at 17:46 | history | asked | Mare | CC BY-SA 4.0 |