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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2402.16206 |
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| _version_ | 1866918202557071360 |
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| author | Mase, Takafumi |
| author_facet | Mase, Takafumi |
| contents | The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. Here we propose a method to rigorously compute the exact degree of each iterate for lattice equations. Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (recurrence relations, typically from ordinary difference equations) from the singularity structure, forms the basis of our idea. The strategy is to extend this method to lattice equations. First, we illustrate, without rigorous details, how to calculate degrees for lattice equations using the lattice version of Halburd's method, and outline the issues that must be resolved to make the method rigorous. We then provide a framework to ensure that all calculations are accurate and rigorous. We further address how to detect the singularity structure in lattice equations. Our method is not only accurate and rigorous but can also be easily applied to a wide range of lattice equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact calculation of degrees for lattice equations: a singularity approach Mase, Takafumi Exactly Solvable and Integrable Systems Mathematical Physics 39A36, 39A14, 37K10 The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. Here we propose a method to rigorously compute the exact degree of each iterate for lattice equations. Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (recurrence relations, typically from ordinary difference equations) from the singularity structure, forms the basis of our idea. The strategy is to extend this method to lattice equations. First, we illustrate, without rigorous details, how to calculate degrees for lattice equations using the lattice version of Halburd's method, and outline the issues that must be resolved to make the method rigorous. We then provide a framework to ensure that all calculations are accurate and rigorous. We further address how to detect the singularity structure in lattice equations. Our method is not only accurate and rigorous but can also be easily applied to a wide range of lattice equations. |
| title | Exact calculation of degrees for lattice equations: a singularity approach |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 39A36, 39A14, 37K10 |
| url | https://arxiv.org/abs/2402.16206 |