# Is this a possible strengthening of the Lehmer conjecture?

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 $$\pm 1$$.