Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation—investigation into their extended Laurent properties

Kamiya, Ryo; Kanki, Masataka; Mase, Takafumi; Tokihiro, Tetsuji

doi:10.1088/1751-8121/aad074

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…The role of the Laurent property in discrete integrable systems is rather intriguing: on the one hand, most recurrences with the Laurent property are not integrable; and on the other hand, most discrete integrable systems do not possess the Laurent property, when written in their standard coordinates. However, the Laurent property is an essential feature of discrete Hirota equations for tau functions [38,44], and, in a suitable algebro-geometric setting, there is an expectation that all discrete integrable systems should admit 'Laurentification' [31,32], that is, a lift to a new set of coordinates (tau functions or their analogues) in which the Laurent property does hold. For a recent review of cluster algebras in the context of discrete integrable systems, see [29].…”

Section: Discussionmentioning

confidence: 99%

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Linear relations for Laurent polynomials and lattice equations

Hone

Pallister

2020

Nonlinearity

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A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmüller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky’s theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed). Here we consider a family of nonlinear recurrences with the Laurent property, referred to as ‘Little Pi’, which was derived by Alman et al via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the Heideman–Hogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter from linear relations with periodic coefficients, which were found recently by Kamiya et al from travelling wave reductions of a linearizable lattice equation on a six-point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable. We also introduce another linearizable lattice equation on the same six-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the six-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.

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Section: Discussionmentioning

confidence: 99%

“…Lemma 3.6. The periodic coefficients in (38) are related to one another by K (2) n+k = −K (3) n . Proof.…”

Section: Lemma 32 For Each N the Determinant δmentioning

confidence: 99%

Linear relations for Laurent polynomials and lattice equations

Hone

Pallister

2020

Nonlinearity

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Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice

Kamiya

Kanki

Mase

et al. 2021

Journal of Mathematical Physics

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We introduce an equation defined on a multi-dimensional lattice, which can be considered as an extension to the coprimeness-preserving discrete KdV like equation in our previous paper. The equation is also interpreted as a higher-dimensional analog of the Hietarinta–Viallet equation, which is famous for its singularity confining property while having an exponential degree growth. As the main theorem, we prove the Laurent and the irreducibility properties of the equation in its “tau-function” form. From the theorem, the coprimeness of the equation follows. In Appendixes A–D, we review the coprimeness-preserving discrete KdV like equation, which is a base equation for our main system, and prove the properties such as the coprimeness.

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Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation—investigation into their extended Laurent properties

Cited by 2 publications

References 15 publications

Linear relations for Laurent polynomials and lattice equations

Linear relations for Laurent polynomials and lattice equations

Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice

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