Decidability of polynomial equations over function fields in positive characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915652588011520 |
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| author | Daans, Nicolas |
| author_facet | Daans, Nicolas |
| contents | Let $K$ be a field of positive characteristic with no algebraically closed subfield. Let $F$ be a function field over $K$ and $t \in F$ transcendental over $K$. Refining a result of Eisentr{ä}ger and Shlapentokh, we show that there is no algorithm which, on input a polynomial $f \in \mathbb{Z}[t][X_1, \ldots, X_n]$, determines whether $f$ has a zero in $F^n$. To this end, we revisit and partially extend several recent results from the literature on existential definability in function fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decidability of polynomial equations over function fields in positive characteristic Daans, Nicolas Number Theory Logic Primary: 12L05. Secondary: 03B25, 11R58, 11U05, 12F20 Let $K$ be a field of positive characteristic with no algebraically closed subfield. Let $F$ be a function field over $K$ and $t \in F$ transcendental over $K$. Refining a result of Eisentr{ä}ger and Shlapentokh, we show that there is no algorithm which, on input a polynomial $f \in \mathbb{Z}[t][X_1, \ldots, X_n]$, determines whether $f$ has a zero in $F^n$. To this end, we revisit and partially extend several recent results from the literature on existential definability in function fields. |
| title | Decidability of polynomial equations over function fields in positive characteristic |
| topic | Number Theory Logic Primary: 12L05. Secondary: 03B25, 11R58, 11U05, 12F20 |
| url | https://arxiv.org/abs/2509.02290 |