Decidability of some complicated structures definable in $\mathbb{C}(t)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918129879220224 |
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| author | Scanlon, Thomas |
| author_facet | Scanlon, Thomas |
| contents | Several properly countable unions of algebraic sets in $\mathbb{C}^n$ are definable in $\mathbb{C}(t)$ including the set CM of $j$-invariants of complex elliptic curves with complex multiplication. It has been suggested that one could prove the undecidability of $\operatorname{Th}(\mathbb{C}(t))$ by showing that the theory of the structure $\mathsf{CM} := (\mathbb{C},+,\cdot,0,1,CM)$ of the field of complex numbers considered with a unary predicate picking out CM is undecidable. We show using an effective version of the André-Oort conjecture that to the contrary $\operatorname{Th}(\mathsf{CM})$ is stable and decidable. We discuss some related structures on the complex numbers definable in $\mathbb{C}(t)$ and how their theories may be connected to the Zilber-Pink conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decidability of some complicated structures definable in $\mathbb{C}(t)$ Scanlon, Thomas Logic Algebraic Geometry Number Theory Several properly countable unions of algebraic sets in $\mathbb{C}^n$ are definable in $\mathbb{C}(t)$ including the set CM of $j$-invariants of complex elliptic curves with complex multiplication. It has been suggested that one could prove the undecidability of $\operatorname{Th}(\mathbb{C}(t))$ by showing that the theory of the structure $\mathsf{CM} := (\mathbb{C},+,\cdot,0,1,CM)$ of the field of complex numbers considered with a unary predicate picking out CM is undecidable. We show using an effective version of the André-Oort conjecture that to the contrary $\operatorname{Th}(\mathsf{CM})$ is stable and decidable. We discuss some related structures on the complex numbers definable in $\mathbb{C}(t)$ and how their theories may be connected to the Zilber-Pink conjectures. |
| title | Decidability of some complicated structures definable in $\mathbb{C}(t)$ |
| topic | Logic Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2508.17485 |