arXiv subjects

Takafumi Mase

Publications and source records attributed to Takafumi Mase.

At least 19 recordsLinked to original sources

arXiv2025

Exact calculation of degrees for lattice equations: a singularity approach

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.

nlin.SI↗

arXiv2024

Deautonomisation by singularity confinement and degree growth

In this paper we give an explanation of a number of observations relating to degree growth of birational mappings of the plane and their deautonomisation by singularity confinement. These observations are of a link between two a priori unrelated notions: firstly the dynamical degree of the mapping and secondly the evolution of parameters required for its singularity structure to remain unchanged under a sufficiently general deautonomisation. We explain this correspondence for a large class of birational mappings of the plane via the spaces of initial conditions for their deautonomised versions. We show that even for non-integrable mappings in this class, the surfaces forming these spaces have effective anticanonical divisors and one can define a period map parametrising them, similar to that in the theory of rational surfaces associated with discrete Painlevé equations. This provides a bridge between the evolution of coefficients in the deautonomised mapping and the induced dynamics on the Picard lattice which encode the dynamical degree.

nlin.SI↗

arXiv2024

A study on the domain independence of the Laurent property, the irreducibility and the coprimeness in lattice equations

We study the Laurent property, the irreducibility and the coprimeness for lattice equations (partial difference equations), mainly focusing on how the choice of initial value problem (the choice of domain) affects these properties. We show that these properties do not depend on the choice of domain as long as they are considered together. In other words, these properties are inherent to a difference equation. Applying our result, we discuss the reductions of lattice equations. We show that any reduction of a Laurent system, even if the lattices have torsion elements, preserves the Laurent property.

math-ph↗

arXiv2024

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the following simple root systems: $A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to addition of a point in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that mutation equivalent to the $q$-Painlevé III quiver found by Okubo. The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

nlin.SI↗

arXiv2024

Singularities and growth of higher order discrete equations

We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are integrable through spectral methods, typically belonging to the QRT family, to a variety of linearisable ones. We show that by judiciously choosing these linearisable mappings, it is possible to obtain higher order mappings that exhibit the maximal degree growth compatible with integrability, i.e. for which the degree grows as a polynomial of order equal to the order of the mapping. In all the cases we analysed, we found that maximal degree growth was associated with the existence of an unconfining singularity pattern. Several cases with submaximal growth but which still possess unconfining singularity patterns are also presented. In many cases the exact degrees of the iterates of the mappings were obtained by applying a method due to Halburd, based on the preimages of specific values that appear in the singularity patterns of the mapping, but we also present some examples where such a calculation appears to be impossible.

nlin.SI↗

arXiv2020

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

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 analogue 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 Appendix we review the coprimeness-preserving discrete KdV like equation whichis a base equation for our main system and prove the properties such as the coprimeness.

nlin.SI↗

arXiv2020

Discrete Hamiltonians of discrete Painlevé equations

We express discrete Painlevé equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painlevé equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.

math-ph↗

arXiv2019

Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type

We introduce a family of extensions of the Hietarinta-Viallet equation to a multi-term recurrence relation via a reduction from the coprimeness-preserving extension to the discrete KdV equation. The recurrence satisfies the irreducibility and the coprimeness property although it is nonintegrable in terms of an exponential degree growth. We derive the algebraic entropy of the recurrence by an elementary method of calculating the degree growth. The result includes the entropy of the original Hietarinta-Viallet equation.

math-ph↗

arXiv2019

Singularity confinement as an integrability criterion

In this paper we present a rigorous method for deciding whether a birational three point mapping that has the singularity confinement property is integrable or not, based only on the structure of its (confined) singularity patterns. We also explain how the exact value of the dynamical degree for such a mapping may be deduced from the singularity patterns.

math-ph↗

arXiv2018

Detecting discrete integrability: the singularity approach

We describe the various types of singularities that can arise for second order rational mappings and we discuss the historical and present-day, practical, role the singularity confinement property plays as an integrability detector. In particular, we show how singularity analysis can be used to calculate explicitly the dynamical degree for such mappings.

math-ph↗

arXiv2018

On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

In this article we investigate the coprimeness properties of one and two-dimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.

math-ph↗

arXiv2018

Singularity patterns and dynamical degrees

We explain on a selection of mappings how the method introduced by Halburd and our simplified variant thereof, the so-called express method, can be used to calculate the dynamical degree of second-order rational mappings from nothing more than their singularity structure.

math-ph↗

arXiv2018

Toda type equations over multi-dimensional lattices

We introduce a class of recursions defined over the $d$-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, their singularities exhibit a nature similar to that of integrable systems in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.

math-ph↗

arXiv2018

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

Coprimeness property was introduced to study the singularity structure of discrete dynamical systems. In this paper we shall extend the coprimeness property and the Laurent property to further investigate discrete equations with complicated pattern of singularities. As examples we study extensions to the Somos-$4$ recurrence and the two-dimensional discrete Toda equation. By considering their non-autonomous polynomial forms, we prove that their tau function analogues possess the extended Laurent property with respect to their initial variables and some extra factors related to the non-autonomous terms. Using this Laurent property, we prove that these equations satisfy the extended coprimeness property. This coprimeness property reflects the singularities that trivially arise from the equations.

math-ph↗

arXiv2018

A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence

We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somos-like recurrences, and introduce one of its two dimensional extensions. The two dimensional lattice equation we present is linearizable in both directions, and has the Laurent and the coprimeness properties. Moreover, its reduction produces a generalized family of the Heideman-Hogan recurrence. Higher order examples of two dimensional linearizable lattice equations related to the Dana-Scott recurrence are also discussed.

math-ph↗

arXiv2017

Calculating the algebraic entropy of mappings with unconfined singularities

We present a method for calculating the dynamical degree of a mapping with unconfined singularities. It is based on a method introduced by Halburd for the computation of the growth of the iterates of a rational mapping with confined singularities. In particular, we show through several examples how simple calculations, based on the singularity patterns of the mapping, allow one to obtain the exact value of the dynamical degree for nonintegrable mappings that do not possess the singularity confinement property. We also study linearisable mappings with unconfined singularities to show that in this case our method indeed yields zero algebraic entropy.

math-ph↗

arXiv2017

Integrable mappings and the notion of anticonfinement

We examine the notion of anticonfinement and the role it has to play in the singularity analysis of discrete systems. A singularity is said to be anticonfined if singular values continue to arise indefinitely for the forward and backward iterations of a mapping, with only a finite number of iterates taking regular values in between. We show through several concrete examples that the behaviour of some anticonfined singularities is strongly related to the integrability properties of the discrete mappings in which they arise, and we explain how to use this information to decide on the integrability or non-integrability of the mapping.

math-ph↗

arXiv2017

Studies on spaces of initial conditions for nonautonomous mappings of the plane

We study nonautonomous mappings of the plane by means of spaces of initial conditions. First we introduce the notion of a space of initial conditions for nonautonomous systems and we study the basic properties of general equations that have spaces of initial conditions. Then, we consider the minimization of spaces of initial conditions for nonautonomous systems and we show that if a nonautonomous mapping of the plane with a space of initial conditions, and unbounded degree growth, has zero algebraic entropy, then it must be one of the discrete Painlevé equations in the Sakai classification.

math-ph↗