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Lehmer's conjecture states that the Mahler measure of a monic integer polynomial as at least $u:=1.176280818...$ when it is greater than 1, see for example https://en.wikipedia.org/wiki/Lehmer%27s_conjecture.

Question: For which $n$ is Lehmer's conjecture true/known when one restricts to polynomial of a given fixed degree $n$?

For example it is easy to see that is true for degree $n=2$. Is it for example true for degrees $n=10$, where the first polynomial with measure $u$ was found?

For a fixed $n$, define $m_n$ as the infimum of Mahler measures >1 of polynomials of degree $n$.

Question: For which $n$ is $m_n$ known?

What is $m_{10}$ or $m_{11}$?

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    $\begingroup$ The record for degree dependent bounds is due to Dobrowolski, with the constant improved by Voutier in 1996. Have you looked at those papers? $\endgroup$
    – Stanley Yao Xiao
    Commented Jan 2, 2021 at 18:41
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    $\begingroup$ cecm.sfu.ca/~mjm/Lehmer $\endgroup$
    – Felipe Voloch
    Commented Jan 2, 2021 at 20:06
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    $\begingroup$ 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. $\endgroup$
    – Gerry Myerson
    Commented Jan 3, 2021 at 5:53
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    $\begingroup$ 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/… $\endgroup$
    – Gerry Myerson
    Commented Jan 3, 2021 at 6:06
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    $\begingroup$ Verger-Gaugry's second attempt: arxiv.org/abs/1911.10590 $\endgroup$
    – Gerry Myerson
    Commented Jan 3, 2021 at 6:12

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