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Related question: Totally real number fields with bounded regulators

Given a number field $K$ with degree $n$ and determinant $D$, what is the "best" upper bound for its regulator $R$, if any? I know that there are many studies about lower bounds (for example "Analytic Formula for the Regulator of a Number Field"), but I have not found any reference on upper bounds.

I am also interested (rather then bounding the regulator of any number fields), on the existence of number fields of small determinant, for general $n$ and $K$ (maybe assuming a totally complex/totally real extension).

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    $\begingroup$ Franz Lemmermeyer's answer to the related question you link to actually provides an upper bound for the regulator (R < c(n)*D^(1/2)*log^(n-1)(D)) due to Landau and cites a subsequent improvement due to Remak. Of course both of these results are over 80 years old... $\endgroup$
    – user1073
    Nov 9 '15 at 16:32
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    $\begingroup$ This answers the firs part of the question - but since my German is, let's say, null, I couldn't see any details of the proof in the paper. Also, is it possible to describe how is the constant c(n) increasing with n? $\endgroup$
    – Campello
    Nov 10 '15 at 9:18
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This is an extended version of the comment I made above and a response to the OP's follow-up question.

Franz Lemmermeyer's response to this question provides an upper bound for the regulator of a number field due to Landau.

  • E. Landau, Verallgemeinerung eines Polyaschen Satzes auf algebraische Zahlkörper Gött. Nachr. 1918, 478--488

Landau's bound is $$R<c(n)D^{1/2}\log^{n-1}(D),$$

and was later improved upon by Remak.

  • R. Remak, Elementare Abschätzungen von Fundamentaleinheiten und des Regulators eines algebraischen Zahlkörpers, J. Reine Angew. Math. 167 (1932), 360-378.

Note that Siegel also has related results.

  • C. L. Siegel, Abschätzung von Einheiten, Nachr. Göttingen 9 (1969) 71-86.

All of these results are in german. The best english reference I could find is due to Sands.

  • J. Sands, Generalization of a theorem of Siegel. Acta Arithmetica 58, 47–56 (1991).

To be more precise, Sands considered the problem of obtaining an (effective) upper bound for the regulator $R_\mathcal{O}$ of an order $\mathcal{O}$ contained in the ring of integers $\mathcal{O}_K$ of $K$. Let $h_\mathcal{O}$ be the class number of $\mathcal{O}$, $w_\mathcal{O}$ the order of the torsion subgroup of $\mathcal{O}$ and $r_1$ the number of real places of $K$.

Sand's main theorem is:

Theorem: We have the inequality $$2^{r_1}R_\mathcal{O}h_\mathcal{O}/w_\mathcal{O} < 4\left(\frac{4}{n-1}\right)^{n-1}|d_\mathcal{O}|^{1/2}(\log|d_\mathcal{O}|)^{n-1}(\log\log|d_\mathcal{O}|)^{n/2}.$$

When $\mathcal{O}=\mathcal{O}_K$ (the case you are interested in) Sands notes that one can remove the $(\log\log|d_\mathcal{O}|)^{n/2}$ term.

While I do not have an answer to your second question I merely want to mention the discriminant bounds of Odlyzko (see this paper and this one).

Theorem: Let $\gamma=0.57721\dots$ be the Euler–Mascheroni constant and $K$ be a number field of signature $(r_1,r_2)$ (hence degree $n=r_1+2r_2$) and absolute value of discriminant $D$. Then $$D^{1/n} > (4\pi e^{1+\gamma})^{r_1/n}\left(4\pi e^\gamma\right)^{2r_2/n}-O(n^{-2/3}).$$

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