Masataka Kanki(Kansai University, Osaka, Japan)

Collaborators:
Ryo Kamiya, Tetsuji Tokihiro, Takafumi Mase (Univ. of Tokyo),
Naoto Okubo (Aoyama Gakuin Univ., Kanagawa, Japan)
SIDE13＠Hakata(博多), Fukuoka(福岡), Japan(日本)Nov. 201８ What is an integrable fully-discrete systems?
 Examples of “Definitions”: Sufficient number of conserved quantities,

existence of a Lax representation, multi-dimensional consistencies,…

⇒We want to find easy detectors (criteria) for integrability.

Pioneering works:
 Singularity confinement --- Discrete analogue of the Painlevé test Algebraic entropy --- Complexity of iterated mappings(cf.) Talk and poster on the dynamical degree by T. Mase and R. Willox.
Note: Other “detectors” include the Backlund transformation, the Devronproperty, growth of error-intervals in numerical computation, topologicalentropy, de-autonomisation, and so on.
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[Our initial aim] Construct an integrability “detector” which is universally
applicable to fully-discrete systems.
 We have introduced the coprimeness (CP) condition as an algebraic re-interpretation to the singularity confinement test.

 CP condition has been found to be an useful integrability detector.

[Recent study] Many non-integrable (but interesting) coprimenesspreserving equations (CP eqs) have been discovered.
[Today’s talk] We introduce several non-integrable coprimenesspreserving extensions to the known integrable dynamical systems:e.g.) The Somos-4 sequence, and the 2dim discrete Toda lattice and its
higher dimensional generalisation, higher dimensional HV equation.
(cf.) Integrable CP eqs: Poster on linearizable equations by R. Kamiya.
SIDE13November 20183 Let us consider the discrete system on the integer lattice L.

()

xh =
Φ xh −v1 , xh −v2 , , xh −vN , h ∈ L
 Here Φ is a rational function. Shifts vi ∈ L are independent on  ≥ 0※Independentness is needed for the system to be an evolution equation.
 Set of initial variables 𝐻𝐻0 is chosen for the evolution to be well-defined.

[e.g.]

xn +3 xn +1 + xn2+ 2, H0=
xn + 4=
Somos-4 sequencexn
Discrete KdV equation
wn ,m SIDE13wn1,m11  wn1,m1 ( wn1,m  wn ,m1 )x0 , x1 , x2 , x3 } , L {=, H 0  w0,m , wn ,0 | m, n  0  , L   2November 20184

[Grammaticos, Ramani, Papageorgiou 1991]
 The equation passes the SC test, if the singularities (indeterminantpoints arising from div/0) are resolved after finite iterations and theinformation on the initial data are recovered.
x 1[e.g.] One of the QRT mappings xn1  n 2 passes the SC test.
xn1 xnx0  a  x1  0 
introduce t ⇓aat

t →0⇓01 t2at


indet

 impossibleat (1  t  at 2 )1  O(t )(1  t ) 2a (1  t  at 2 ) 20a 1
recovers information of a

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 Algebraic entropy（AE） [Bellon-Viallet 1999][Def.] Algebraic entropy Λ=log λ（λ≥ 1）of Φ is defined as:log d n,nn  log   lim

d n =deg(    )  deg( n )(λ is called the dynamical degree)n[Working definition]
 Λ=0

⇒

the system is integrable Λ>0⇒

the system is non-integrable[Note on multi-dimensional systems] AE is not defined because of more thanone evolution directions.
[Tentative def.] “Polynomial growth in every direction”=“integrable”
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Coprimeness Property (CP)[Def.] Discrete Dynamical system F satisfies the coprimeness property (CP) if
the following condition is satisfied for any pair xh , xk :
 There exists a constant D>0 independent of h,k such that

dist(h, k )  D  ( xh , xk )  1

Here, dist(,) is a metric on the lattice L.

[Note 1] 𝑥𝑥∗ is a rational function of the initial variables.

[Note 2] Two Laurent polynomials F/M,G/N (M,N:monomials) are coprimein the sense of Laurent polynomials if (F,G)=1.
[Note 3] Two rational functions F/H, G/K are coprime if(F,G)=(F,K)=(H,G)=(H,K)=1 in the sense of Laurent polynomials.
※Monomial common factors are allowed.

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When a system satisfies the coprimeness condition it belong to
the coprimeness-preserving class (CP class).

xn + 4 xn xn +3 xn +1 + xn2+ 2[e.g.] Somos-4 recurrence =(D. Gale 1991, Integer sequence No. A006720) belongs to the CPclass.
Somos-4 is transformed into a nonlinear form (one of the QRTmappings)
un  1un1 un1un2
via the transformation

SIDE13un xn3 xn1.
2xn2November 20188

[Prop.] [JPA2014] For a system un1 un  1
un1un2(un , u=1 if | n − m |≥ 3.
m)※In the Def. of coprimeness we takedist (un , um )  n  m , D  3.

[Experiment] For initial values (u0,u1)=(s,t )1  t st (1  t  st 2 ) (1  t )(1  2t    s 2t 3 ),,s, t , 2 ,st(1  t ) 2s (1  t  st 2 ) 2(1  t  st 2 )(1  3t    s 3t 4 ) st (1  2t    s 2t 3 )(1  s  4t    s 4t 6 ),,
3 4 2t (1  2t    s 2t 3 ) 2(1  s  3t    s t )SIDE13

November 20189 We can define a coprimeness property over the initial variables &several “additional” functions.

R Let I be the set of initial variables, and let=[ I ], E ⊂ R − {0}.

 The system has the extended Laurent property if every iterate is in

R  f −1 | f ∈ E  =: R  E −1  The system has the extended coprimeness property if every pair ofiterates is coprime in the quotient field of R  E −1  .

※If E={monomials of R}, we have the original Laurent and coprimenessproperties.
※If E=R-{0}, R  E −1  is a field, and the coprimeness is trivially satisfied forany equation.
SIDE13November 201810 Generalized Somos-4 [Fomin-Zelevinsky 2002]

xn + 4 xn =xnm+3 xnl +1 + xnk+ 21, 2, 4,8,)( GCD(k , l , m) =※ We call it the CP-Somos-4.
[Prop.] It has irreducible Laurentness and the coprimeness. [JMP2015]
 Nonlinear form of the CP-Somos-4:( un+ 4 − 1)( un − 1) = unm+3unl +1
kunk+ 2
( un+ 2 − 1)

 Correspondence:

un + 2 =xn + 4 xn
xnk+ 2 (Different from the relation between

Somos-4 and a QRT system)

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November 201811 Nonlinear form

( un+ 4 − 1)( un − 1) = unm+3unl +1  †kku
1−u( n+2 )n+2

is constructed from the reduction of 2-dim discrete Toda.
 By un + 2 =xn + 4 xn
it corresponds with CP-Somos-4 non-trivially:
xnk+ 2[Sketch]xn + 4 xn − xnk+ 2
Fn + 4 F n† ⇔ Fn,1==mlkxn + 3 xn +1
Fn + 2
 𝐹𝐹𝑛𝑛 is explicitly written by F0 , F1 , F2 , F3 .

mlk

Fn ⋅ xn + 3 xn +1 + xn + 2 includes the CP Non-autonomous system xn + 4 xn =Somos-4.

 It is well-defined over the initial variables { F0 , F1 , F2 , F3 , x4 , x5 , x6 , x7 } .
SIDE13November 201812 [Key Lemma] Each iterate 𝑥𝑥𝑛𝑛 (non-auto system) has the extended

Laurent property as E ′−1  :=   g −1 | g ∈ E ′
xn ∈ =E′{x , x , x , x , x x − x , x x − x , x x − x , x x − x }4567k
2

4 05 1k
36 2k
47 3k
5

Moreover, it is irreducible and pairwise coprime.
 [Thm.] Each iterate 𝑢𝑢𝑛𝑛 has the extended coprimeness property asIf n ≠ n′ mod 2 or | n − n′ |≥ 5, then, (un , un′ )=1 in Q :Q = (Quotient field of   E −1  ),=E{u , u , u , u , ( u − 1) , ( u − 1) , ( u − 1) , ( u − 1)}.
2

3452

34k
2

5(Sketch) Use the key Lemma and u2 − 1 =x4 x0 k− x .
x2

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[Thm.][JMP2017] Let us studyk1k2

l1l2

t1,n,m1t1,n1,m  t ,n1,m  t ,n,m1   t ,n,m  t ,n1,m1 

GCD(k1 , k2 , l1 , l2 ) = 1, 2, 4,8,

Each iterate is an irreducible Laurent polynomial int ,n ,m     0,n ,m  , 1,n ,m  

Moreover, two distinct iterates are pairwise coprime.
※Original 2dim-dToda if superscripts are all 1.

※ Initial variables are all the points on the n-m planes
with t=0,1.

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The equation we study (shifted as 1+V* = U* for a technical reason):U l1 1 1U l2 1 1
t ,n − ,m−t ,n + ,m+(U t +1,n ,m − 1)(U t −1,n ,m − 1)
22
2
2=.
k1
k2
k1
k2
U 1 1U 1 1 t ,n + ,m−t ,n − ,m+2
222
 U t , n + 1 , m − 1 − 1  U t , n − 1 , m + 1 − 1
2
222

 

Related to the ”tau-function” form:
k1
k2
l1l2





F
, t1,n,m t1,n,mt ,n1/2,m1/2
t ,n1/2,m1/2
t1,n ,mt ,n1/2,m1/2
t ,n1/2,m1/2

k1k2
 F
FFFtnmtn
mtn
mtn1/2,m1/2 .

1,,1,,,1/2,1/2,
Correspondence:

τ t +1,n ,mτ t −1,n ,mU t ,n,m = k.
kτ t ,n +1/2,m −1/2τ t ,n −1/2,m +1/2
1

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Main result on Cp-2dim-dToda (nonlinear form)ーExtended Coprimeness Propertyー [Key Lemma] The tau-function form has the irreducible

Laurent and the coprimeness properties in S := [ E −1 ].

Here,τ 2,n ,m ,τ 3,n ,m , (τ 2,n ,mτ 0,n ,m − τ 1,k1n +1/2,m −1/2τ 1,k2n −1/2,m +1/2 ), E=
 .
k1
k2−ττ
τ
τ()
 3,n ,m 1,n ,m 2,n +1/2,m −1/2 2,n −1/2,m +1/2
n ,m [Thm.] [JPA2018-2]

Two iterates U t ,n ,m , U t ′,n′,m′ of the nonlinear form is in thequotient field of T := [G ] and is pairwise coprime on T if
the following condition is satisfied:| t − t ′ |> 2 or (n ', m ') ≠ (n, m),(n − 1, m + 1),(n + 1, m − 1).

=G {U 0,n ,m , U1,n ,m , (U 0,n ,m − 1) , (U1,n ,m − 1)} .
Here,n,mNovember 2018
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（１）(a+b)-dimensional “discrete Toda lattice”(non-integrable except for a=b=1, k=m=1)

τ t +1,nτ t −1,na∏ (τi= 1) (τkit ,n+ei

t ,n-eia +b) + ∏ (τmin = ( n1 , n2 , , na +b ) , ei : unit vectori= a +1) (τ )kit ,n+ei

mit .n-ei（２）Higher dimensional analogue of the Hietarinta-Vialletequation (also related to CP-discrete KdV eq. when d=1)


aibi

xt +1,n =− xt −1,n + ∑ +kimi
 x

i =1x
t ,n-ei t ,n+eid(SIDE13) ()(ki , mi ∈ 2)

November 201817 [Thm.] [JPA-Special issue 2018, arXiv:1803.06774] The equation

τ t +1,nτ t −1,na∏ (τi= 1) (τkit ,n+eit ,n-eia +b) + ∏ (τmii= a +1) (τ )kit ,n+ei

mit .n-ei

has the Laurent property. Moreover, if GCD ( k=1, , a + b )i , li | i
is a non-negative power of 2, it satisfies the irreducibility and thecoprimeness.

{

Here, the initial variables are τ 0,n ,τ 1,n | n ∈ 
the positive direction of t.

a +b} , and it evolves towards[Thm. 2] Its evolution is expressed as a mutation of the LP algebra.
(Details are omitted. Mainly done by N. Okubo.)
SIDE13November 201818abii
xt +1,n =− xt −1,n + ∑ +kimi
 [Thm.] Let us study
 x
i =1xn+en-e,,ttii
d) (()(ki , mi ∈ 2)with a certain condition on the indices k, m. (details are omitted here)()

Its two iterates xt ,n , xt ′,n′ are coprime in  { x0,m , x1,m }m∈Z ,{ai , bi } ,dIf | t − t ′ |> 2 or dist(n,n′)>2.
 Sketch: Use the Laurent property and the irreducibility of the “tau-function” form.

[Lemma]By the transformation

xt ,n =f t +1,n f t −1,nd∏fi =1

kit , n + ei

f t l,in −ei we obtain the tau-function formof the equation w.r.t. f*. Each iterate f* satisfies the irreducibility.
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 One of the reductions of (2):ab

xn + xn − p − q = k + k (k : even).
xn − q xn − p An extension of the Hietarinta-Viallet equation to a multi-termrecurrence.

[Prop.](in prep.) Its dynamical degree is the maximal real root of

λ p + q − k λ p − k λ q + 1 =0.
[Cor.] When p=1, q=2, k=2, we recover the original H-V equation.
※The root of a “characteristic” polynomial related to the singularity
structure gives the dynamical degree.
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[Conclusion]① Defined the coprimeness and the extended coprimeness properties.
② Constructed a coprimeness-preserving non-integrable equation related

to the two dimensional Toda.

③ Constructed its generalisations to equations over a higher-dimensionallattice.

[Future Problems]① Relations of the coprimeness with various notions of entropy.

(e.g.) Diophantine entropy & algebraic entropy (growth of complexity ofan iteration) vs. topological entropy & Lyapunov exponent (growth of asmall perturbation).

② Integrated method of proving the coprimeness.
③ Possible application to coprime sequences of integers in number theory.
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[GrammaticosRamaniPapageorgiou] B. Grammaticos, A. Ramani, V. Papageorgiou: Phys. Rev. Lett. 67 (1991), 1825—1828.
[Bellon-Viallet] Algebraic Entropy, CMP 204 (1999), 425—437.
[Hietarinta-Viallet] Singularity Confinement and Chaos in Discrete Systems, Phys. Rev. Lett. 81 (1998), 325.
[JPA2014] M. Kanki, J. Mada, T. Mase, T. Tokihiro: J. Phys. A: Math. Theor. 47 (2014), 465204 (15pp).
[T] M. Toda: Vibration of a chain with nonlinear interaction, J. Phys. Soc. Jpn. 22, (1967), 431—436.
[M] A. V. Mikhailov: Integrability of a two-dimensional generalization of the Toda chain, JETP Lett. 30, (1979), 414—418.
[LS] A. N. Leznov, M. V. Saveliev: Comm. Math. Phys. 89 (1983), pp59—75.
[Hirota] R. Hirota: J. Phys. Soc. Jpn. 50, (1981), pp3785--3791.
[JMP2017] R. Kamiya, M. Kanki, T. Mase, T. Tokihiro: Coprimeness-preserving non-integrable extension to the two-dimensional
discrete Toda equation, arXiv: 1610.02646, J. Math. Phys. 58 (2017), 012702 (12pp).
[JPA2018-1] R. Kamiya, M. Kanki, T. Mase, T. Tokihiro: Two dimensional lattice equation as extension of Heideman-Hogan
recurrence, arXiv: 1709.8930, J. Phys. A: Math. Theor. 51, (2018), 125203.
[JPA2018-2] R. Kamiya, M. Kanki, T. Mase, T. Tokihiro: Nonlinear forms of co-primeness preserving extensions to the Somos-4
recurrence and the two-dimensional Toda lattice equation –investigation into their extended Laurent properties-,
arXiv:1708.07917, J. Phys. A: Math. Theor. 51, 355202, (2018).
[JPA-Special Issue] R. Kamiya, M. Kanki, T. Mase, N. Okubo, T. Tokihiro: Toda type equations over multi-dimensional lattices, J. Phys.
A: Math. Theor. (Special Issue on fifty years of the Toda lattice) 51, 364002, (2018).
[LP] T. Lam, P. Pylyavskyy: Laurent Phenomenon Algebras, Cambridge J. Math. 4, 121—162 (2016).
[HV-k] M. Kanki, T. Mase, T. Tokihiro: Algebraic entropy of the extended Hietarinta-Viallet equation, J. Phys. A: Math. Theor. 48
355202, (2015).

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