Timeline for Is the infimum of Salem numbers > 1?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2015 at 15:15 | answer | added | Bobby Grizzard | timeline score: 3 | |
| Nov 23, 2013 at 23:43 | vote | accept | blabler | ||
| Sep 20, 2013 at 23:35 | answer | added | Vesselin Dimitrov | timeline score: 9 | |
| Sep 20, 2013 at 15:21 | answer | added | P.. | timeline score: 4 | |
| Jul 26, 2013 at 15:42 | comment | added | Anthony Quas | In the section of the wiki article on "partial results", the statement that there exists an absolute (but non-explicit) constant $C$ such that for all d, $M(P)>=1+C/(d\log d)$ for all polynomials of degree $d$ is attributed to Blanksby and Montgomery; and Stewart. If it were the case that $M(P)>1+\delta$ for all $P$, then the above result would be trivial: just choose $C=2\log 2\delta$. Then automatically you would have $M(P)>1+C/(d\log d)$ for all polys of degree $d$. | |
| Jul 26, 2013 at 10:29 | comment | added | blabler | @IanAgol: Thank you for these very relevant references! | |
| Jul 26, 2013 at 10:26 | comment | added | blabler | @AnthonyQuas: Perhaps I am missing something but I do not understand how "These bounds would be rendered moot if there were a lower bound independent of the degree (i.e. $\inf T>1$)". Recall that $\inf T > 1$ is only the statement that $\theta > 1 + \delta$ for all Salem numbers $\theta$ and some $\delta > 0$. I don't see how that implies that $M(P) > 1 + \delta$ for all polynomials $P$ with $M(P) > 1$ but hopefully there is some confusion here on my part that you can explain? | |
| Jul 26, 2013 at 7:16 | history | edited | Greg Martin | CC BY-SA 3.0 |
improved title
|
| Jul 26, 2013 at 3:49 | comment | added | Ian Agol | Some progress on the Salem number conjecture: ams.org/mathscinet-getitem?mr=1824892 ams.org/mathscinet-getitem?mr=1953192 ams.org/mathscinet-getitem?mr=2105815 | |
| Jul 25, 2013 at 23:54 | comment | added | Gerry Myerson | Historical note --- Lehmer would remind people that he had not published it as a conjecture, only a question, since he didn't feel he had enough evidence for it to call it a conjecture. | |
| Jul 25, 2013 at 23:15 | comment | added | Igor Rivin | See also mathworld.wolfram.com/SalemConstants.html | |
| Jul 25, 2013 at 23:11 | comment | added | Anthony Quas | Obviously it has not been disproved that $\inf T>1$. Otherwise Lehmer's conjecture would be known to be false. Further, if you look at the (reasonably authoritative wiki page) you will see that there are bounds on the Mahler measure which converge to 0 in the degree. These bounds would be rendered moot if there were a lower bound independent of the degree (i.e. $\inf T > 1$). Hence, assuming that the editors of the wiki page did not fall asleep for the last few years, one can reasonably conclude that it has neither been proved nor disproved that $\inf T>1$. | |
| Jul 25, 2013 at 22:37 | history | edited | blabler | CC BY-SA 3.0 |
added 69 characters in body
|
| Jul 25, 2013 at 22:29 | comment | added | blabler | Yes, but my question is "Has it been proved or disproved that $\inf T > 1$" | |
| Jul 25, 2013 at 22:15 | comment | added | Anthony Quas | From wiki, it's believed that Lehmer's conjecture holds, and hence a fortiori it's believed that $\inf T > 1$. | |
| Jul 25, 2013 at 21:52 | comment | added | blabler | If Lehmer conjecture is true then $\inf T > 1$, but is it true that $\inf T > 1$ implies Lehmer conjecture? | |
| Jul 25, 2013 at 21:46 | comment | added | Anthony Quas | I mean "yes it's still open" according to wiki: "It is widely believed that this example represents the true minimal value: that is, mu=1.176280818... in Lehmer's conjecture." | |
| Jul 25, 2013 at 21:42 | comment | added | blabler | I don't see anything to that effect on Wiki's page... | |
| Jul 25, 2013 at 21:34 | comment | added | Anthony Quas | Yes according to wikipedia en.wikipedia.org/wiki/Lehmer%27s_conjecture | |
| Jul 25, 2013 at 21:29 | comment | added | blabler | I am aware that the title of this question is not the most fortunate one, but I think that it captures the gist of the question rather succintly. | |
| Jul 25, 2013 at 21:27 | history | asked | blabler | CC BY-SA 3.0 |