Theory of Heat Equations for Sigma Functions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2017
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| Acceso en línea: | |
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| _version_ | 1866929636875698176 |
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| author | Eilbeck, J. C. Gibbons, J. Ônishi, Y. Yasuda, S. |
| author_facet | Eilbeck, J. C. Gibbons, J. Ônishi, Y. Yasuda, S. |
| contents | We consider the heat equations satisfied by the sigma function associated with a planar curve, extending and developing earlier pioneering work of Buchstaber and Leykin. These heat equations lead to useful {\em linear} recursive relations for the coefficients of power series expansion of the sigma function. In particular we exhibit explicit results for curves of genus 3, and give a new constructive proof of an explicit expression for the main matrix in the theory for {\em any} hyperelliptic curve. We also state and prove a new explicit formula for the eigenvalues of the linear operators associated with this matrix, as well as other practical formulae. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1711_08395 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Theory of Heat Equations for Sigma Functions Eilbeck, J. C. Gibbons, J. Ônishi, Y. Yasuda, S. Exactly Solvable and Integrable Systems Algebraic Geometry Number Theory 37K20, 11G05, 14H45 We consider the heat equations satisfied by the sigma function associated with a planar curve, extending and developing earlier pioneering work of Buchstaber and Leykin. These heat equations lead to useful {\em linear} recursive relations for the coefficients of power series expansion of the sigma function. In particular we exhibit explicit results for curves of genus 3, and give a new constructive proof of an explicit expression for the main matrix in the theory for {\em any} hyperelliptic curve. We also state and prove a new explicit formula for the eigenvalues of the linear operators associated with this matrix, as well as other practical formulae. |
| title | Theory of Heat Equations for Sigma Functions |
| topic | Exactly Solvable and Integrable Systems Algebraic Geometry Number Theory 37K20, 11G05, 14H45 |
| url | https://arxiv.org/abs/1711.08395 |