Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions

Technical Summary: Counting Rational Maps on $\mathbb{P}^1$ with Prescribed Local Conditions

Problem Statement
This paper investigates distribution questions for rational maps on the projective line $\mathbb{P}^1$ over $\mathbb{Q}$ within the framework of arithmetic dynamics. Drawing an analogy to the arithmetic statistics of elliptic curves, the authors address counting problems for rational maps $\phi$ of a fixed degree $d \ge 2$ with prescribed reduction properties. Specifically, the work focuses on determining the density of rational maps that possess a "globally minimal resultant."

A rational map $\phi \in \mathbb{Q}(z)$ is represented by a pair of coprime homogeneous polynomials $F, G \in \mathbb{Z}[X, Y]$ of degree $d$. The resultant $\text{Res}(\phi) = \text{Res}(F, G)$ is a global invariant. The map is said to have a globally minimal resultant if $\text{Res}(\phi)$ equals the minimal resultant $R_\phi$, defined as the product of local minimal resultants over all non-archimedean places. The central question is whether a positive proportion of rational maps (ordered by height or box density) satisfy this minimality condition, analogous to results concerning the density of global minimal Weierstrass models for elliptic curves (e.g., Cremona and Sadek [CS23]).

Methodology
The authors employ a combination of arithmetic geometry, sieve theory, and explicit computational algebra to establish lower bounds on the densities of specific subsets of rational maps.

1. Density Frameworks: The paper utilizes two primary density measures:

   • Height Density: Based on the absolute multiplicative Weil height $H(\phi)$ of the point $P(\phi) \in \mathbb{P}^{2d+1}(\mathbb{Q})$ associated with the coefficients of the map.
   • Weak Box Densities: Following Poonen's work [Poo03], these densities consider the distribution of coefficients within boxes in affine space $\mathbb{Z}^{2d+2}$, allowing for independent limits on coordinate bounds.

2. Local Conditions and Sieving: The authors define subsets of rational maps via local conditions at primes. A map is included in a set if its reduction modulo $p^k$ satisfies specific criteria (e.g., good reduction or $p^k$-free resultant).

   • They establish that if a set of maps satisfies an "admissibility condition" (specifically, that the density of maps failing the condition at large primes vanishes), the global density is the product of local densities (Proposition 3.4).
   • This relies on a sieve-theoretic argument to show that the density of maps with a common factor (non-minimal) or high valuation of the resultant is negligible.

3. Algebraic Geometry of Resultants:

   • The resultant polynomial $r$ of two generic degree $d$ polynomials is analyzed. The authors utilize the fact that $r$ is absolutely irreducible (Proposition 4.11).
   • To bound the number of points where the resultant vanishes modulo $p^k$, they analyze the singular locus defined by $r=0$ and its partial derivatives $s=0$.
   • Using results from Warning, Schwartz-Zippel, and Hensel's Lemma, they derive upper bounds for the number of solutions to these systems over finite fields and their lifts to $\mathbb{Z}/p^k\mathbb{Z}$.

4. Explicit Computation for $d=2$: For degree 2, the authors perform explicit primary decomposition of the ideal generated by the resultant and its partial derivatives using the software Macaulay2. This allows for precise counting of points in $(\mathbb{Z}/p\mathbb{Z})^6$ and $(\mathbb{Z}/p^2\mathbb{Z})^6$ to compute exact local densities for small primes and rigorous bounds for larger primes.

Key Contributions and Results

• Theorem A (Height Density): For odd integers $d \ge 3$, the set of rational maps with $2d$-power free resultant and good reduction at a finite set of primes $\Sigma$ has a positive height density. Consequently, a positive proportion of such maps have globally minimal resultants.
• Theorem B (Weak Box Density): For any integer $d \ge 2$, the set of rational maps with $d$-power free resultant and good reduction at a finite set of primes $\Sigma$ has a positive weak box density $\mu_n$. This implies a positive proportion of maps have globally minimal resultants under this density measure.
• Theorem C (Explicit Bound for $d=2$): For degree 2 rational maps, the authors provide an explicit lower bound. They demonstrate that more than 32.7% of degree 2 rational maps over $\mathbb{Q}$ possess a squarefree (and hence minimal) resultant with respect to weak box densities.

Significance and Claims
The paper positions itself as a continuation of the analogy program between arithmetic dynamics on $\mathbb{P}^1$ and the Diophantine geometry of elliptic curves. While previous statistical results in arithmetic dynamics often relied on stochastic processes involving random composition of maps, this work addresses the distribution of static rational maps with prescribed arithmetic properties.

The authors claim that their results provide the first proofs that a positive proportion of rational maps have globally minimal resultants, analogous to the density results for elliptic curves by Cremona and Sadek. The explicit computation for $d=2$ serves as a concrete illustration of the theoretical framework, yielding a specific, non-trivial lower bound (32.7%).

The paper concludes modestly, noting that while the local conditions used provide positive lower bounds, they do not yet describe the entire collection of maps with minimal resultants. The question of whether a positive proportion of rational maps have globally minimal resultants with respect to the standard box density (as opposed to weak box densities) is identified as a topic for future work.