Equations involving the modular $j$-function and its derivatives
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908600823185408 |
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| author | Aslanyan, Vahagn Eterović, Sebastian Mantova, Vincenzo |
| author_facet | Aslanyan, Vahagn Eterović, Sebastian Mantova, Vincenzo |
| contents | We show that for any polynomial $F(X,Y_0,Y_1,Y_2) \in \mathbb{C}[X, Y_0, Y_1, Y_2]$, the equation $F(z,j(z),j'(z),j''(z))=0$ has a Zariski dense set of solutions in the hypersurface $F(X,Y_0,Y_1,Y_2)=0$, unless $F$ is in $\mathbb{C}[X]$ or it is divisible by $Y_0$, $Y_0-1728$, or $Y_1$.
Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09974 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equations involving the modular $j$-function and its derivatives Aslanyan, Vahagn Eterović, Sebastian Mantova, Vincenzo Complex Variables Logic Number Theory 11F03, 11F23, 11U09 We show that for any polynomial $F(X,Y_0,Y_1,Y_2) \in \mathbb{C}[X, Y_0, Y_1, Y_2]$, the equation $F(z,j(z),j'(z),j''(z))=0$ has a Zariski dense set of solutions in the hypersurface $F(X,Y_0,Y_1,Y_2)=0$, unless $F$ is in $\mathbb{C}[X]$ or it is divisible by $Y_0$, $Y_0-1728$, or $Y_1$. Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions. |
| title | Equations involving the modular $j$-function and its derivatives |
| topic | Complex Variables Logic Number Theory 11F03, 11F23, 11U09 |
| url | https://arxiv.org/abs/2312.09974 |