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

1. Problem Statement

The paper addresses a fundamental difficulty in arithmetic geometry: the computation of the essential minimum ($\text{ess}(h)$) of height functions on algebraic varieties.

• Context: The height of an algebraic point measures its arithmetic complexity. The essential minimum is the infimum of the limit inferior of heights over all "generic" sequences of algebraic points (sequences that eventually escape any proper Zariski closed subset).
• The Challenge: While two classical methods exist to bound this value—one providing lower bounds (via polynomials) and one providing upper bounds (via measures)—they have historically been treated as separate techniques. For many important height functions (e.g., the Zhang–Zagier height or the Faltings height), these bounds do not coincide, leaving a "gap" where the true essential minimum lies. No practical algorithm exists to approximate these values to arbitrary precision.
• Goal: The authors aim to prove that these two bounding methods are dual to each other in the sense of linear programming and that strong duality holds. This would imply the lower and upper bounds converge to the exact same value, effectively "closing the gap."

2. Methodology

The authors employ a synthesis of Potential Theory, Arakelov Geometry, and Linear Programming (LP) Duality.

A. Height Functions and Green Functions

The study focuses on height functions $h_g$ on $\mathbb{A}^1(\mathbb{Q}) = \mathbb{Q}$ defined by a Green function $g: \mathbb{C} \to \mathbb{R}$.

• $g$ is continuous, invariant under complex conjugation, and has logarithmic growth at infinity ($g(z) = \log|z| + o(\log|z|)$).
• The height of an algebraic number $\alpha$ is defined as the average of $g$ over its Galois orbit, adjusted by the leading coefficient of its minimal polynomial.

B. The Two Bounding Methods

1. Lower Bound (Smyth's Method):

   • Based on the product formula and properties of polynomials with integer coefficients.
   • Defined as $D(g) = \sup \left( \inf_{z \in \mathbb{C}} \left( g(z) - \sum a_i \log|P_i(z)| \right) \right)$, where $P_i \in \mathbb{Z}[x]$ and $a_i \geq 0$.
   • This represents a maximization problem over linear combinations of logarithmic potentials.

2. Upper Bound (Measure Theory):

   • Based on the equidistribution of Galois orbits.
   • Defined as $P(g) = \inf \left( \int g \, d\mu \right)$, where the infimum is taken over a specific set of probability measures $\mu$.
   • The set of admissible measures, denoted $P^{\mathbb{Z}}_{\log}(\mathbb{C})$, consists of limits of Galois orbits of algebraic integers. A key characterization (Theorem B) states that a measure $\mu$ belongs to this set if and only if it is conjugation-invariant and $\int \log|Q| \, d\mu \geq 0$ for all non-zero $Q \in \mathbb{Z}[x]$.

C. Linear Programming Formulation

The authors reformulate these problems as a primal-dual pair in infinite-dimensional linear programming:

• Primal Problem ($P(g)$): Minimize $\int g \, d\mu$ subject to linear constraints $\int \log|Q| \, d\mu \geq 0$.
• Dual Problem ($D(g)$): Maximize the lower bound derived from the constraints.
• Weak Duality: It is known that $D(g) \leq P(g)$.
• Strong Duality: The core technical challenge is proving $D(g) = P(g)$.

D. Technical Tools

To prove strong duality in this infinite-dimensional setting, the authors introduce:

• Sweetened Truncation: A refined method for truncating measures to compact sets while preserving the crucial inequality $\int \log|Q| \, d\mu \geq 0$. This overcomes issues with standard truncation where the constraint might be violated.
• Log-Weak Convergence: A topology on the space of measures with logarithmic growth ($P_{\log}(\mathbb{C})$) that corresponds to the $L^1_{\text{loc}}$ convergence of their potential functions. This topology is shown to be the correct setting for potential theory on non-compact supports.
• Density Results: Proving that the set of measures generated by specific rational maps (pullbacks of the Haar measure on the unit circle) is dense in $P^{\mathbb{Z}}_{\log}(\mathbb{C})$.

3. Key Contributions and Results

Theorem D (Strong Duality)

The central result is Theorem 6.5:
$$D(g) = \text{ess}(h_g) = P(g)$$
This equality proves that the gap between the best possible lower bound (Smyth's method) and the best possible upper bound (measure optimization) is zero. The essential minimum is exactly the optimal value of this linear programming problem.

Theorem A (Realization by Algebraic Integers)

The authors prove that the essential minimum is not just an infimum but is attained (or approached arbitrarily closely) by a generic sequence of algebraic integers.

• Specifically, there exists a sequence of distinct algebraic integers $(\alpha_n)$ such that $h_g(\alpha_n)$ decreases monotonically to $\text{ess}(h_g)$.
• Furthermore, the set of height values of algebraic integers is dense in the interval $[\text{ess}(h_g), \infty)$.

Theorem E (Computability)

If the Green function $g$ is computable (meaning it can be approximated algorithmically and its asymptotics are effectively controlled), then the essential minimum $\text{ess}(h_g)$ is a computable real number.

• This is achieved by reducing the optimization problems to countable subsets (using rational coefficients and specific polynomial families), allowing for the construction of an algorithm that produces converging sequences of lower and upper bounds.

Application to Faltings Height

The paper applies these results to the Faltings height ($h_F$) of elliptic curves.

• It confirms that the essential minimum of the Faltings height can be realized by a sequence of elliptic curves with good reduction everywhere (specifically, those with integral $j$-invariants).
• It provides a theoretical framework to compute the essential minimum of the Faltings height, a value previously only known to lie within a narrow numerical interval.

4. Significance

1. Unification of Methods: The paper unifies two disparate areas of number theory (Diophantine approximation via polynomials and equidistribution via potential theory) under the rigorous framework of linear programming duality.
2. Resolution of the "Gap": It resolves the long-standing issue of the gap between lower and upper bounds for essential minima, proving they are identical.
3. Computability: It establishes that these fundamental arithmetic invariants are computable. While the proposed algorithm is theoretical and not yet practical for high-precision numerical computation, it proves that an effective procedure exists, opening the door for future algorithmic improvements.
4. New Characterization of Measures: The characterization of the set of limit measures of Galois orbits ($P^{\mathbb{Z}}_{\log}(\mathbb{C})$) via the condition $\int \log|Q| \, d\mu \geq 0$ extends previous results (Smith, Orloski-Sardari) to non-compactly supported measures, providing a robust tool for studying the distribution of algebraic integers.
5. Counterexamples and Nuance: The paper also highlights subtleties, such as the fact that the infimum in the dual problem is not always attained by a single measure (Theorem 6.4 example), and that the set of equilibrium measures of capacity-one compact sets is not dense in the space of all relevant measures.

In summary, this work provides a powerful theoretical bridge between arithmetic geometry and optimization, transforming the problem of finding essential minima from a heuristic search into a well-defined, solvable linear programming problem with guaranteed convergence properties.