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Arithmetic properties of multiplier polynomials for certain polynomial maps

Yuya Murakami, Kaoru Sano, Kohei Takehira

TL;DR

This work establishes integrality and monicness for multiplier polynomials across several 1-parameter polynomial families, notably $f_c(z)=z^d+c$ and $f_c(z)=z^{d+1}+cz$, using Newton polygons and resultant techniques to connect arithmetic properties with dynamical parameters. It derives uniform height bounds for parabolic parameters and provides concrete determinations of quadratic parabolic parameters for $z^2+c$, including conditional counts under irreducibility conjectures. The authors extend the framework to non-unicritical families and discuss the limits of uniform integrality by analyzing $z^{d+2}+cz^2$, highlighting both the power and the boundaries of the current methods. Overall, the paper advances the arithmetic of dynamical moduli by giving explicit integrality, monicness, and height-bounding results, and by detailing both unconditional and conditional classifications of parabolic parameters. The methods blend algebraic techniques with Newton polygons to study roots and multipliers without relying on analytic at-infinity arguments.

Abstract

We investigate the arithmetic properties of the multiplier polynomials for certain $1$-parameter families of polynomials. In particular, we prove integrality theorems of multiplier polynomials for $z^d+c$, $(z-c)z^d + c$ and $z^{d+1}+cz$. As a corollary, we obtain the uniform upper bound of the naive height of parabolic parameters of unicritical polynomials. Moreover, we determined the quadratic parabolic parameters for $z^2 + c$. We also conditionally list parabolic parameters for $z^2 + c$ of fixed degrees.

Arithmetic properties of multiplier polynomials for certain polynomial maps

TL;DR

This work establishes integrality and monicness for multiplier polynomials across several 1-parameter polynomial families, notably and , using Newton polygons and resultant techniques to connect arithmetic properties with dynamical parameters. It derives uniform height bounds for parabolic parameters and provides concrete determinations of quadratic parabolic parameters for , including conditional counts under irreducibility conjectures. The authors extend the framework to non-unicritical families and discuss the limits of uniform integrality by analyzing , highlighting both the power and the boundaries of the current methods. Overall, the paper advances the arithmetic of dynamical moduli by giving explicit integrality, monicness, and height-bounding results, and by detailing both unconditional and conditional classifications of parabolic parameters. The methods blend algebraic techniques with Newton polygons to study roots and multipliers without relying on analytic at-infinity arguments.

Abstract

We investigate the arithmetic properties of the multiplier polynomials for certain -parameter families of polynomials. In particular, we prove integrality theorems of multiplier polynomials for , and . As a corollary, we obtain the uniform upper bound of the naive height of parabolic parameters of unicritical polynomials. Moreover, we determined the quadratic parabolic parameters for . We also conditionally list parabolic parameters for of fixed degrees.
Paper Structure (29 sections, 46 theorems, 114 equations, 8 figures, 5 tables)

This paper contains 29 sections, 46 theorems, 114 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental facts on Number fields
  4. Fundamental facts on Complete valuation fields
  5. Fundamental facts on dynamics
  6. Multiplier polynomials for $z^d+c$
  7. Integrality
  8. Valuations of roots
  9. Monicness
  10. Integrality of parameters with periodic points of fixed multiplier
  11. Multiplier polynomials for $z^{d+1}+ cz$
  12. Replacement of polynomials
  13. Integrality
  14. Valuations of roots
  15. Monicness
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $f_c(z) = z^d + c$ for an integer $d\geq 2$. Then for all integers $n, m$ with $m | n$ and $m < n$, there is a polynomial $\Psi_{n,m}(t) \in \mathbb{Z}[t]$ such that Moreover, the polynomial $(-1)^{e_{n,m}}\Psi_{n,m}(t)$ is monic, where $d_m = \sum_{k|m} d^k \mu\left(\frac{m}{k}\right)$, $\varphi(n)$ is Euler's totient function, and

Figures (8)

  • Figure 1: Parabolic parameters of degree at most 2.
  • Figure 2: The Newton polygons of $f_c^{\circ k}(z)$ and $f_c^{\circ k}(z) - z$.
  • Figure 3: The Newton polygons of $\mathop{\mathrm{Res}}\nolimits_z(f_c^{\circ k}(z)-z, x - (f_c^{\circ m})'(z))$ and $\delta_m(x)$.
  • Figure 4: The Newton polygon of $\tilde{f}(z)$.
  • Figure 5:
  • ...and 3 more figures

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2: $\risingdotseq$ Hug21
  • Remark 1.3
  • Theorem 1.4
  • Conjecture 1.5: Morton-Silverman
  • Theorem 1.6
  • Conjecture 1.7: The irreducibility conjecture
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 77 more