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Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$

Stéphane Fischler, Tanguy Rivoal

TL;DR

The paper investigates zeros of $E$-functions and exponential polynomials over $\overline{\mathbb Q}$, proposing a gcd-based and minimal-polynomial analogue framework to understand multiplicities and common zeros. It proves a special case of Jossen's conjecture showing that if $f=g^m$ and $L(g)/g$ is entire then $L(g)/g$ must be a polynomial with algebraic coefficients, and establishes a constructive decomposition for globally bounded $E$-functions with equal-multiplicity zeros, including denominators and holonomicity of the root. Under Schanuel's conjecture, it derives strong gcd-type results for common zeros of exponential polynomials, provides irreducibility results for certain exponential-polynomial objects related to Bessel functions, and outlines a conjectural factorization theory for $E$-functions, together with a generalized notion of minimal polynomials for zeros such as those of $\pi$ and $J_\alpha$. The work connects zero structure to differential/algebraic properties, yielding deep implications for special values like $\pi$, logarithms of algebraic numbers, and zeros of Bessel functions, and sets a program for broader factorization and zero-analysis of $E$-functions.

Abstract

Zeros of Bessel functions $J_α$ play an important role in physics. They are a motivation for studying zeros of exponential polynomials defined over $\overline{\mathbb{Q}}$, and more generally of $E$-functions. In this paper we partially characterize $E$-functions with zeros of the same multiplicity, and prove a special case of a conjecture of Jossen on entire quotients of $E$-functions, related to Ritt's theorem and Shapiro's conjecture on exponential polynomials. We also deduce from Schanuel's conjecture many results on zeros of exponential polynomials over $\overline{\mathbb{Q}}$, including $π$, logarithms of algebraic numbers, and zeros of $J_α$ when $2α$ is an odd integer. For the latter we define (if $α\neq\pm1/2$) an analogue of the minimal polynomial and Galois conjugates of algebraic numbers. At last, we study conjectural generalizations to factorization and zeros of $E$-functions.

Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$

TL;DR

The paper investigates zeros of -functions and exponential polynomials over , proposing a gcd-based and minimal-polynomial analogue framework to understand multiplicities and common zeros. It proves a special case of Jossen's conjecture showing that if and is entire then must be a polynomial with algebraic coefficients, and establishes a constructive decomposition for globally bounded -functions with equal-multiplicity zeros, including denominators and holonomicity of the root. Under Schanuel's conjecture, it derives strong gcd-type results for common zeros of exponential polynomials, provides irreducibility results for certain exponential-polynomial objects related to Bessel functions, and outlines a conjectural factorization theory for -functions, together with a generalized notion of minimal polynomials for zeros such as those of and . The work connects zero structure to differential/algebraic properties, yielding deep implications for special values like , logarithms of algebraic numbers, and zeros of Bessel functions, and sets a program for broader factorization and zero-analysis of -functions.

Abstract

Zeros of Bessel functions play an important role in physics. They are a motivation for studying zeros of exponential polynomials defined over , and more generally of -functions. In this paper we partially characterize -functions with zeros of the same multiplicity, and prove a special case of a conjecture of Jossen on entire quotients of -functions, related to Ritt's theorem and Shapiro's conjecture on exponential polynomials. We also deduce from Schanuel's conjecture many results on zeros of exponential polynomials over , including , logarithms of algebraic numbers, and zeros of when is an odd integer. For the latter we define (if ) an analogue of the minimal polynomial and Galois conjugates of algebraic numbers. At last, we study conjectural generalizations to factorization and zeros of -functions.

Paper Structure

This paper contains 9 sections, 21 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. A special case of Jossen's conjecture
  3. Proof of Theorem \ref{['theo:1intro']}
  4. Existence of $g$ entire of order $\le 1$ such that $f=g^m$ and $g(0)=1$
  5. Denominators of the Taylor coefficients of $g$ at the origin
  6. Holonomicity of $g$
  7. Zeros of exponential polynomials over $\overline{\mathbb Q}$
  8. Factorization of $E$-functions
  9. Zeros of $E$-functions

Key Result

Theorem 1.2

Let $g$ be an entire function such that $g^m$ is an $E$-function for some $m\geq 1$. Let $L\in\overline{\mathbb Q}(x)[d/dx]$ be a differential operator such that ${L(g)}/g$ is an entire function. Then ${L(g)}/g\in\overline{\mathbb Q}[x]$.

Theorems & Definitions (44)

  • Conjecture 1.1: Jossen
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Conjecture 1.6: Schanuel
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Conjecture 1.10
  • ...and 34 more