On the two-dimensional Jacobian conjecture: Magnus' formula revisited, IV
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909278055432192 |
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| author | Lee, Kyungyong Li, Li |
| author_facet | Lee, Kyungyong Li, Li |
| contents | Let $(F,G)$ be a Jacobian pair with $d=w\text{-deg}(F)$ and $e=w\text{-deg}(G)$ for some direction $w$. A generalized Magnus' formula approximates $G$ as $\sum_{γ\ge 0} c_γF^{\frac{e-γ}{d}}$ for some complex numbers $c_γ$. We develop an approach to the two-dimensional Jacobian conjecture, aiming to minimize the use of terms corresponding to $γ>0$. As an initial step in this approach, we define and study the inner polynomials of $F$ and $G$. The main result of this paper shows that the northeastern vertex of the Newton polygon of each inner polynomial is located within a specific region. As applications of this result, we introduce several conjectures and prove some of them for special cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_01279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the two-dimensional Jacobian conjecture: Magnus' formula revisited, IV Lee, Kyungyong Li, Li Algebraic Geometry Commutative Algebra Primary: 14R15 Secondary: 13F20, 14M25, 11P21 Let $(F,G)$ be a Jacobian pair with $d=w\text{-deg}(F)$ and $e=w\text{-deg}(G)$ for some direction $w$. A generalized Magnus' formula approximates $G$ as $\sum_{γ\ge 0} c_γF^{\frac{e-γ}{d}}$ for some complex numbers $c_γ$. We develop an approach to the two-dimensional Jacobian conjecture, aiming to minimize the use of terms corresponding to $γ>0$. As an initial step in this approach, we define and study the inner polynomials of $F$ and $G$. The main result of this paper shows that the northeastern vertex of the Newton polygon of each inner polynomial is located within a specific region. As applications of this result, we introduce several conjectures and prove some of them for special cases. |
| title | On the two-dimensional Jacobian conjecture: Magnus' formula revisited, IV |
| topic | Algebraic Geometry Commutative Algebra Primary: 14R15 Secondary: 13F20, 14M25, 11P21 |
| url | https://arxiv.org/abs/2408.01279 |