Polynomial correspondences expressible as maps of $d$-tuples
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913491636453376 |
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| author | Sridharan, Shrihari G., Subith Tiwari, Atma Ram |
| author_facet | Sridharan, Shrihari G., Subith Tiwari, Atma Ram |
| contents | In this paper, we consider polynomial correspondences $f (x, y)$ in $\mathbb{C}[x, y]$ of degree $d \ge 2$ in both the variables and obtain necessary and sufficient conditions in order that the equation $f (x, y) = 0$ can be expressed as $ϕ(x) = ψ(y)$, where $ϕ$ and $ψ$ are fractional degree $d$ rational maps in the Riemann sphere. In the absence of involutions that played a vital role towards characterising quadratic correspondences ($d = 2$), we employ certain elementary ideas from theory of equations and matrices to achieve our results. We further explore certain symmetry conditions on the matrix of coefficients of correspondences that satisfy the above factorisation. We conclude this short note with a few examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13894 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polynomial correspondences expressible as maps of $d$-tuples Sridharan, Shrihari G., Subith Tiwari, Atma Ram Dynamical Systems Complex Variables In this paper, we consider polynomial correspondences $f (x, y)$ in $\mathbb{C}[x, y]$ of degree $d \ge 2$ in both the variables and obtain necessary and sufficient conditions in order that the equation $f (x, y) = 0$ can be expressed as $ϕ(x) = ψ(y)$, where $ϕ$ and $ψ$ are fractional degree $d$ rational maps in the Riemann sphere. In the absence of involutions that played a vital role towards characterising quadratic correspondences ($d = 2$), we employ certain elementary ideas from theory of equations and matrices to achieve our results. We further explore certain symmetry conditions on the matrix of coefficients of correspondences that satisfy the above factorisation. We conclude this short note with a few examples. |
| title | Polynomial correspondences expressible as maps of $d$-tuples |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2303.13894 |