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 $\pm 1$.

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.