# Energy integrals over local fields and global height bounds

@article{Fili2013EnergyIO, title={Energy integrals over local fields and global height bounds}, author={Paul Fili and Clayton Petsche}, journal={arXiv: Number Theory}, year={2013} }

We solve an energy minimization problem for local fields. As an application of these results, we improve on lower bounds set by Bombieri and Zannier for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In the case of fields with mixed archimedean and non-archimedean splitting conditions, we are able to combine our bounds with similar bounds at the archimedean places for totally real fields.

## 8 Citations

Quantitative height bounds under splitting conditions

- MathematicsTransactions of the American Mathematical Society
- 2019

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions, such as being…

Energy integrals and small points for the Arakelov height

- Mathematics
- 2015

We study small points for the Arakelov height on the projective line. First, we identify the smallest positive value taken by the Arakelov height, and we characterize all cases of equality. Next we…

Constructing totally p-adic numbers of small height

- MathematicsInternational Journal of Number Theory
- 2021

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime…

Small totally p-adic algebraic numbers

- MathematicsInternational Journal of Number Theory
- 2018

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 [Formula: see…

A dynamical construction of small totally p-adic algebraic numbers

- MathematicsJournal of Number Theory
- 2019

Log-Coulomb gases in the projective line of a $p$-field

- Mathematics
- 2021

This article extends recent results on log-Coulomb gases in a p-field K (i.e., a nonarchimedean local field) to those in its projective line P(K), where the latter is endowed with the PGL2invariant…

A Branching Random Walk on the Positive Half-Line Associated with Incompressible Navier-Stokes Equation and Euler ’ s DiLogarithmic Function

- Mathematics
- 2013

This paper treats two branching Markov chains (BMC) on the positive half-line that arise naturally in the analysis of three-dimensional incompressible Navier-Stokes equations in terms of the…

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