On the geometry of a 4-dimensional extension of a $q$-Painlevé I equation with symmetry type $A_1^{(1)}$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911024125313024 |
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| author | Stokes, Alexander Takenawa, Tomoyuki Carstea, Adrian Stefan |
| author_facet | Stokes, Alexander Takenawa, Tomoyuki Carstea, Adrian Stefan |
| contents | We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlevé I equation with symmetry of type $A_1^{(1)}$. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from $({\mathbb P}^1)^{\times 4}$ by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group $\widetilde{W}(A_1^{(1)})\times \widetilde{W}(A_1^{(1)})$ by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of $q$-Painlevé equations, one of which is a deautonomisation of the original autonomous integrable map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the geometry of a 4-dimensional extension of a $q$-Painlevé I equation with symmetry type $A_1^{(1)}$ Stokes, Alexander Takenawa, Tomoyuki Carstea, Adrian Stefan Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry Dynamical Systems We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlevé I equation with symmetry of type $A_1^{(1)}$. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from $({\mathbb P}^1)^{\times 4}$ by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group $\widetilde{W}(A_1^{(1)})\times \widetilde{W}(A_1^{(1)})$ by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of $q$-Painlevé equations, one of which is a deautonomisation of the original autonomous integrable map. |
| title | On the geometry of a 4-dimensional extension of a $q$-Painlevé I equation with symmetry type $A_1^{(1)}$ |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2506.21092 |