Small totally p-adic algebraic numbers

Abstract

The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than log(3) /p.

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“…Recently, Pottmeyer [Pot18] has improved upon Petsche's upper bound, and obtained the existence of totally p-adic α such that 0 < h(α) ≤ log p p .…”

A dynamical construction of small totally p-adic algebraic numbers

“…In particular, Q (p) has what Bombieri-Zannier call the Bogomolov property, which means that h(α) ≥ C > 0 for some constant C and all nonzero, non-root-of-unity α ∈ Q (p) . The lower bound in (2) was later improved slightly by Fili-Petsche [12], who showed that lim inf α∈Q (p) h(α) ≥ p log p 2(p 2 −1) , and more significantly by Pottmeyer [16], who showed that lim inf α∈Q (p) h(α) ≥ log(p/2) p+1 and lim inf α∈Q (2) h(α) ≥ log 2 4 . To prove the upper bound in (2), Bombieri-Zannier used a fairly intricate construction which begins with the polynomial (x − 1)(x − 2) .…”

Totally $T$-adic functions of small height

“…A number of authors have studied totally real or totally p-adic algebraic numbers in Q of small height; examples include Bombieri/Zannier [2], Fili [4], Petsche/Stacy [10], Pottmeyer [11], Schinzel [12], and Smyth [13]. By analogy with these works, one is led to ask how small the height h(α) can be for nonconstant α ∈ T q , and whether such a bound can be given which does not depend on the degree d = [F q (T )(α) : F q (T )].…”

“…The first proof is brief and geometric, so it appears in the next part of the introduction. The second proof is analytic and inspired by a result on heights of totally p-adic algebraic numbers by Pottmeyer [11], following an earlier unpublished argument of the first author. Considering the partition of the local field F q ((T )) into the three regions ord 0 < 0, ord 0 = 0, and ord 0 > 0, the proportion of the algebraic conjugates of α lying in each region may be bounded above in terms of the height h(α) and the (projective) T -adic size of the region.…”

“…It is worth contrasting with the situation of small totally p-adic algebraic numbers. Pottmeyer [11] has shown for odd p that h(α) > log(p/2)/(p + 1) whenever α ∈ Q is totally p-adic, nonzero, and not a root of unity. This is currently the best known lower bound, but indeed, as indicated, equality in Pottmeyer's bound is not possible.…”