Lower bounds for heights in cyclotomic extensions

Abstract

We show that the height of a nonzero algebraic number α that lies in an abelian extension of the rationals and is not a root of unity must satisfy h(α) > 0.155097.

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

“…This has the same form as Theorem 1, although the constant in Theorem 1 is a little bit better than ours (and our estimate is trivial for m = 2). On the other hand, it is interesting that such an elementary argument produces a lower bound that agrees with the best known lower bounds [3,4,6,10] up to an additional O(1/m 2 ).…”

“…The next theorem is our main result for number fields. As we will see, it generalizes [3,4,6,10] (Theorem 1), albeit with worse constants. Later we will prove an elliptic curve version of this theorem and its consequences.…”

“…Of course, the bound that we obtain is not sharp. However, our goal is not to get sharp bounds in this particular case, where other authors [3,4,6,10] have used intricate techniques to obtain better bounds than we could obtain even if we took more care. Instead, we aim to show how to obtain nontrivial bounds that, among other things, imply that Lehmer's conjecture is true for an interesting class of polynomials that is larger than the class considered in [3,4,6,10].…”

“…We mention that an earlier paper [4] does the case of non-reciprocal polynomials, and subsequent papers [6,10] give improved values for C m , although asymptotically they all have the form C m = log(m/2) + O(1/m 2 ). We also note the papers [17,18] which give various generalizations of Theorem 1, including weakening the congruence condition (2), working over number fields, and considering heights of points and subspaces in projective space.…”

On a Theorem of Garza Regarding Algebraic Numbers With Real Conjugates

“…The above argument can be extended to arbitrary algebraic numbers α by using some basic algebraic number theory and properties of the absolute height (cf. [3,4] for the case of Schinzel's result).…”