On the geometry of a 4-dimensional extension of a $q$-Painlev\'e I equation with symmetry type $A_1^{(1)}$

Alexander Stokes; Tomoyuki Takenawa; Adrian Stefan Carstea

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arXiv · 2506.21092

On the geometry of a 4-dimensional extension of a $q$-Painlev\'e I equation with symmetry type $A_1^{(1)}$

Alexander Stokes·Tomoyuki Takenawa·Adrian Stefan Carstea

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Abstract

We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlev\'e 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 $q$-Painlev\'e equations, one of which is a deautonomisation of the original autonomous integrable map.

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Alexander Stokes, Tomoyuki Takenawa, Adrian Stefan Carstea. 2025-06-26. On the geometry of a 4-dimensional extension of a $q$-Painlev\'e I equation with symmetry type $A_1^{(1)}$. https://arxiv.org/abs/2506.21092

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