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Closing the gap around the essential minimum of height functions with linear programming

José Burgos Gil, Ricardo Menares, Binggang Qu, Martín Sombra

TL;DR

This work addresses the fundamental problem of determining the essential minimum ess(h_g) of height functions defined by a Green function g. By formulating two infinite-dimensional linear programs that capture lower Smyth-type bounds and upper limit-distribution bounds, the authors prove a strong duality result: D(g) = P(g) = ess(h_g). This duality, together with a refined log-weak convergence framework and the sweetened truncation technique, extends the Smith-Orloski-Sardari program to non-compact support measures and shows that ess(h_g) is computable when g is computable. A key corollary is that ess(h_g) can be attained by sequences of algebraic integers, and in the Faltings height case, the essential minimum is reachable by elliptic curves with good reduction everywhere, with computations connected to equilibrium measures and capacity-one sets. The paper thus links potential theory, Arakelov geometry, and computability to provide a rigorous, computable characterization of the minimal height landscape for a broad class of heights.

Abstract

For many common height functions, the explicit determination of the essential minimum is an open problem. We consider a classical method to obtain lower bounds that goes back at least to C.J. Smyth, and a method to obtain upper bounds based on the knowledge of the limit distribution of integral points. We use an infinite dimensional linear programming scheme to show that that both methods agree in the limit, by showing that the principle of strong duality holds in our situation. As a corollary we prove that the essential minimum can be attained by sequences of algebraic integers. Recent results by A. Smith and B. Orloski--N. Sardari, furnish a characterization of compactly supported measures that can be approximated by complete sets of conjugates of algebraic integers, in terms of infinitely many nonnegativity conditions. We establish an extension of this characterization to measures with non necessarily compact support. As an application of this result and of strong duality, we show that the essential minimum is a computable real number when the Green function used to define the height is computable. We systematically use potential theory for measures that can integrate functions with logarithmic growth.

Closing the gap around the essential minimum of height functions with linear programming

TL;DR

This work addresses the fundamental problem of determining the essential minimum ess(h_g) of height functions defined by a Green function g. By formulating two infinite-dimensional linear programs that capture lower Smyth-type bounds and upper limit-distribution bounds, the authors prove a strong duality result: D(g) = P(g) = ess(h_g). This duality, together with a refined log-weak convergence framework and the sweetened truncation technique, extends the Smith-Orloski-Sardari program to non-compact support measures and shows that ess(h_g) is computable when g is computable. A key corollary is that ess(h_g) can be attained by sequences of algebraic integers, and in the Faltings height case, the essential minimum is reachable by elliptic curves with good reduction everywhere, with computations connected to equilibrium measures and capacity-one sets. The paper thus links potential theory, Arakelov geometry, and computability to provide a rigorous, computable characterization of the minimal height landscape for a broad class of heights.

Abstract

For many common height functions, the explicit determination of the essential minimum is an open problem. We consider a classical method to obtain lower bounds that goes back at least to C.J. Smyth, and a method to obtain upper bounds based on the knowledge of the limit distribution of integral points. We use an infinite dimensional linear programming scheme to show that that both methods agree in the limit, by showing that the principle of strong duality holds in our situation. As a corollary we prove that the essential minimum can be attained by sequences of algebraic integers. Recent results by A. Smith and B. Orloski--N. Sardari, furnish a characterization of compactly supported measures that can be approximated by complete sets of conjugates of algebraic integers, in terms of infinitely many nonnegativity conditions. We establish an extension of this characterization to measures with non necessarily compact support. As an application of this result and of strong duality, we show that the essential minimum is a computable real number when the Green function used to define the height is computable. We systematically use potential theory for measures that can integrate functions with logarithmic growth.
Paper Structure (29 sections, 43 theorems, 193 equations)

This paper contains 29 sections, 43 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. The essential minimum
  3. Strong duality
  4. Approximation of measures by algebraic integers
  5. Computability of the essential minimum
  6. Methods and organization of the paper
  7. Potential theory and limit distribution of algebraic integers
  8. Potential theory on the complex plane and the Fekete--Szegő theorem
  9. Smith and Orloski--Sardari theorems
  10. $\log$-weak convergence of measures
  11. Subharmonic functions and $L^1_{\mathrm{loc}}$ convergence
  12. The sweetened truncation
  13. Sweetened truncation
  14. Sweetened truncation and asymptotically logarithmic functions
  15. Approximating measures by algebraic integers and by capacity one compact sets
  16. ...and 14 more sections

Key Result

Theorem A

Let $\mathrm{g}\colon \mathbb{C} \longrightarrow \mathbb{R}$ be a Green function. Then, $\mathop{\mathrm{h}}\nolimits_\mathrm{g}(\overline{\mathbb{Z}})$ is dense in $[\mathop{\mathrm{\mathop{\mathrm{ess}}\nolimits(\mathop{\mathrm{h}}\nolimits_\mathrm{g})}}\nolimits,\infty)$. In particular, there ex

Theorems & Definitions (97)

  • Theorem A: Corollary \ref{['ess_attained_by_integer']} and Corollary \ref{['spectrum']}
  • Theorem B: Strong duality, Theorem \ref{['strong_duality_essential_minimum']}
  • Theorem 1.1: Smith, Orloski--Sardari
  • Theorem C: Theorem \ref{['SOS non compact']}
  • Theorem D: Theorem \ref{['negativo']}
  • Theorem E: Theorem \ref{['ess_min_computable']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 87 more