On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915350678863872 |
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| author | Lelis, Jean Miranda, Bruno De Paula Moreira, Carlos Gustavo |
| author_facet | Lelis, Jean Miranda, Bruno De Paula Moreira, Carlos Gustavo |
| contents | In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of $\overline{\mathbb{Q}}^m$ contained in the unit polydisc $Δ(0;1)$, closed under complex conjugation and containing the zero vector, is the exceptional set of uncountably many transcendental analytic functions in several variables with integer coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients Lelis, Jean Miranda, Bruno De Paula Moreira, Carlos Gustavo Number Theory Primary 11J81, Secondary 32A15 In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of $\overline{\mathbb{Q}}^m$ contained in the unit polydisc $Δ(0;1)$, closed under complex conjugation and containing the zero vector, is the exceptional set of uncountably many transcendental analytic functions in several variables with integer coefficients. |
| title | On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients |
| topic | Number Theory Primary 11J81, Secondary 32A15 |
| url | https://arxiv.org/abs/2506.15914 |