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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.00001 |
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| _version_ | 1866917465328451584 |
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| author | Seshadri, Bharath Krishna Sridharan, Shrihari |
| author_facet | Seshadri, Bharath Krishna Sridharan, Shrihari |
| contents | In this manuscript, we study a special class of correspondences on $\mathbb{P}^{1} \times \mathbb{P}^{1}$ given by a polynomial relation, say $P(z, w)$. We focus on what we call restrictive polynomial correspondence and characterise that it can be written as $P (z, w) = g_{1}(w) h_{1}(z) + \cdots + g_ρ(w) h_ρ(z)$, for some appropriate $ρ\in \mathbb{Z}_{+}$, where $g_{r}$ and $h_{r}$ are polynomials. In particular, when $ρ= 2$, we say $P$ is irreducible and observe that the equation $P(z, w) = 0$ can be rewritten as $R(z) = S(w)$, where $R$ and $S$ are rational maps of appropriate degree. Further, we also define an operation that, with the exception of degenerate cases, constructs a new irreducible restrictive polynomial correspondence from any two given irreducible restrictive polynomial correspondences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A classification of restrictive polynomial correspondences Seshadri, Bharath Krishna Sridharan, Shrihari General Mathematics 12D10, 30C15, 32A08 In this manuscript, we study a special class of correspondences on $\mathbb{P}^{1} \times \mathbb{P}^{1}$ given by a polynomial relation, say $P(z, w)$. We focus on what we call restrictive polynomial correspondence and characterise that it can be written as $P (z, w) = g_{1}(w) h_{1}(z) + \cdots + g_ρ(w) h_ρ(z)$, for some appropriate $ρ\in \mathbb{Z}_{+}$, where $g_{r}$ and $h_{r}$ are polynomials. In particular, when $ρ= 2$, we say $P$ is irreducible and observe that the equation $P(z, w) = 0$ can be rewritten as $R(z) = S(w)$, where $R$ and $S$ are rational maps of appropriate degree. Further, we also define an operation that, with the exception of degenerate cases, constructs a new irreducible restrictive polynomial correspondence from any two given irreducible restrictive polynomial correspondences. |
| title | A classification of restrictive polynomial correspondences |
| topic | General Mathematics 12D10, 30C15, 32A08 |
| url | https://arxiv.org/abs/2503.00001 |