Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$
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| Format: | Preprint |
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2025
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| _version_ | 1866915213839695872 |
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| author | Fischler, Stéphane Rivoal, Tanguy |
| author_facet | Fischler, Stéphane Rivoal, Tanguy |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_20345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$ Fischler, Stéphane Rivoal, Tanguy Number Theory 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. |
| title | Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.20345 |