Here is another possible refinement of the Lehmer conjecture.

For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained in the field $\mathbb{Q}(\alpha)$. The discriminant of a minimal equation of $\alpha$ over a number field $F$ is an ideal of $O_F$ that we denote by $\Delta_F(\alpha)$. This is just the discriminant of the finite $O_F$-algebra $O_F[\alpha]$.

Let us write $d := [\mathbb{Q}(\alpha):\mathbb{Q}]$ and $d^{\mathrm{rel}} := [\mathbb{Q}(\alpha):C_{\alpha}] = [\mathbb{Q}^{\mathrm{ab}}(\alpha) : \mathbb{Q}^{\mathrm{ab}}]$, the absolute and the relative-over-cyclotomic degrees. The Amoroso-Dvornicich-Zannier relative Lehmer problem asks whether $h(\alpha) \gg 1 / d^{\mathrm{rel}}$ whenever $\alpha$ is not a root of unity. (This has been proved up to an $\epsilon$ in the exponent.) The Mahler discriminant inequality (an argument with the Vandermonde determinant and the Hadamard volume inequality) shows $$ |\Delta_{\mathbb{Q}}(\alpha)|^{1/d} \leq d M(\alpha)^2 $$ and $$ |N_{C_{\alpha}/\mathbb{Q}}\Delta_{C_{\alpha}}(\alpha)|^{1/d} \leq d^{\mathrm{rel}} M(\alpha)^2, $$ where $M(\alpha) := \exp(d h(\alpha))$ is the Mahler measure, and we define $$ \delta := \frac{d}{|\Delta_{\mathbb{Q}}(\alpha)|^{1/d} } \in [M(\alpha)^{-2},d], \quad \delta^{\mathrm{rel}} := \frac{d^{\mathrm{rel}}}{|N_{C_{\alpha}/\mathbb{Q}}\Delta_{C_{\alpha}}(\alpha)|^{1/d} } \in [M(\alpha)^{-2},d^{\mathrm{rel}}]. $$ Matveev's enhancement of Dobrowolski's theorem states $$ \log{M(\alpha)} \geq \Big( \frac{\log{\log{\max(\delta,3)}}}{\log{\max(\delta,3)}} \Big)^3 $$ (for all but finitely many algebraic numbers $\alpha$), just provided that the mild conditions $\mathbb{Q}(\alpha^p) = \mathbb{Q}(\alpha)$ hold for all primes $p < (\log{\delta})^2$. From this it is easy to see that the Lehmer conjecture holds true for all $\alpha$ having $\delta(\alpha)$ bounded from above: the case of large discriminants. Just note that, by a simple application of the Vahlen-Capelli theorem, $[\mathbb{Q}(\alpha) : \mathbb{Q}(\alpha^p)] > 1$ for a given prime $p$ entails the existence of an algebraic number $\eta \in \overline{\mathbb{Q}}^{\times}$ from the smaller field $\mathbb{Q}(\alpha^p)$, not a root of unity if $\alpha$ isn't, having a strictly lower degree $[\mathbb{Q}(\alpha^p):\mathbb{Q}]=d / [\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^p)]$ and with $M(\eta) = M(\alpha)$ and $\delta(\eta) \ll_p \delta(\alpha)$.

This last observation becomes a little more interesting once one realizes computationally that, among the algebraic integers of a Mahler measure $M(\alpha) < 2$ and a growing degree bound $D$, Matveev's defect $\delta(\alpha)$ appears to typically be bounded, say bounded with asymptotic probability $1$ as $D \to \infty$. But it may certainly go to infinity, as $\log\delta(\zeta_{N}) = \log{\log{d}} - \log{\log{\log{d}}} + O(1)$ when $N$ is a primorial level (the product of the first $k$ primes for some $k$), by an easy application of the prime number theorem.

But the argument in Matveev's paper yields a relative version of his result in which, crudely speaking, the left-hand side $\log{M(\alpha)}$ of the above lower bound is replaced by $$ \frac{\log{M(\alpha)}}{[C_{\alpha}:\mathbb{Q}]} = d^{\mathrm{rel}} h(\alpha), $$ the discriminant defect $\delta$ is replaced by its relative version $\delta^{\mathrm{rel}}$, and the exponent ``$3$'' on the right hand side is weakened to the slightly worse exponent $4$. By the same argument, the relative Lehmer conjecture (and a fortiori the original Lehmer conjecture) is true for all $\alpha$ having $\delta^{\mathrm{rel}}$ bounded above.

Thus I wanted to ask whether or not the latter could be anticipated as a possible refinement of the Lehmer conjecture:

Question. Is there any possibility to fancy an absolute constant upper bound on the relative discriminant defect $\delta^{\mathrm{rel}}$ of an algebraic number?

It is quite a strong statement, but I fail to see any way of refuting it, even heuristically. Of course, for the relative Lehmer problem, it is enough to restrict attention to the $\alpha$ with $h(\alpha) < 1 / d^{\mathrm{rel}}$. I am already interested in just a construction of any $\alpha$ having $\delta^{\mathrm{rel}}(\alpha) \to \infty$. The above cyclotomic examples are now suddenly rendered irrelevant in this relativized question. A special case is whether there is any hope to expect a $\log{|\Delta_{\mathbb{Q}}(\alpha)|} = d\log{d} - O(d)$ asymptotic on the logarithmic discriminant of a degree-$d$ algebraic number $\alpha$ with $M(\alpha) < 2$ for which the field $\mathbb{Q}(\alpha)$ does not include any roots of unity besides $\pm 1$.


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