Integrability of Discrete Equations Modulo a Prime

We apply the ''almost good reduction'' (AGR) criterion, which has been introduced in our previous works, to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion...

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Дата:	2013
Автор:	Kanki, M.
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Опубліковано:	Інститут математики НАН України 2013
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Integrability of Discrete Equations Modulo a Prime / M. Kanki // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 18 назв. — англ.

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spelling	irk-123456789-1493512019-02-22T01:24:10Z Integrability of Discrete Equations Modulo a Prime Kanki, M. We apply the ''almost good reduction'' (AGR) criterion, which has been introduced in our previous works, to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlevé III and IV equations have AGR. We next prove that the Hietarinta-Viallet equation, a non-integrable chaotic system also has AGR. 2013 Article Integrability of Discrete Equations Modulo a Prime / M. Kanki // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 34M55; 37P25 DOI: http://dx.doi.org/10.3842/SIGMA.2013.056 http://dspace.nbuv.gov.ua/handle/123456789/149351 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We apply the ''almost good reduction'' (AGR) criterion, which has been introduced in our previous works, to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlevé III and IV equations have AGR. We next prove that the Hietarinta-Viallet equation, a non-integrable chaotic system also has AGR.
format	Article
author	Kanki, M.
spellingShingle	Kanki, M. Integrability of Discrete Equations Modulo a Prime Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kanki, M.
author_sort	Kanki, M.
title	Integrability of Discrete Equations Modulo a Prime
title_short	Integrability of Discrete Equations Modulo a Prime
title_full	Integrability of Discrete Equations Modulo a Prime
title_fullStr	Integrability of Discrete Equations Modulo a Prime
title_full_unstemmed	Integrability of Discrete Equations Modulo a Prime
title_sort	integrability of discrete equations modulo a prime
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149351
citation_txt	Integrability of Discrete Equations Modulo a Prime / M. Kanki // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 18 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kankim integrabilityofdiscreteequationsmoduloaprime
first_indexed	2025-07-12T21:54:37Z
last_indexed	2025-07-12T21:54:37Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 056, 8 pages Integrability of Discrete Equations Modulo a Prime Masataka KANKI Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan E-mail: kanki@ms.u-tokyo.ac.jp Received April 24, 2013, in final form September 05, 2013; Published online September 08, 2013 http://dx.doi.org/10.3842/SIGMA.2013.056 Abstract. We apply the “almost good reduction” (AGR) criterion, which has been intro- duced in our previous works, to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlevé III and IV equations have AGR. We next prove that the Hietarinta–Viallet equation, a non-integrable chaotic system also has AGR. Key words: integrability test; good reduction; discrete Painlevé equation; finite field 2010 Mathematics Subject Classification: 37K10; 34M55; 37P25 1 Introduction The purpose of this paper is to define the nonlinear discrete integrable equations over finite fields and to investigate how to formulate the integrability of them over finite fields, with the help of the theory of arithmetic dynamics. In the theory of arithmetic dynamics, we are interested in how the properties of the mappings change as we change the set on which the mappings are defined [15]. In particular, the system over the field of p-adic integers and its reduction modulo a prime to the finite field attracts much attention. We have another interest in the dynamical systems over finite fields in terms of cellular automata, of which the underlying set consists of a finite number of elements and the mapping is given by recurrence formulae [18]. Let us first explain the problems we encounter and review some of the previous results. In the case of linear discrete equations, there is no problem in defining the equations over finite fields just by changing the field on which the equations are defined to finite fields. This approach can also be valid for equations such as bilinear equations. In this direction, the discrete KP and KdV equations and their soliton solutions over finite fields have been investigated [2]. However, in order to deal with nonlinear discrete equations, we frequently pass through division by zero modulo prime and indeterminacies (e.g. 0/0, ∞ ± ∞), which prevent us from obtaining the evolution. One of the methods to tackle this difficulty is to restrict the domain of definition so that we do not need to treat indeterminacies. For example we can terminate the evolution of the equation when it hits these points. Integrability of the discrete equations over finite fields has been investigated in terms of the lengths of the periodic and terminating orbits [12, 13]. The graph structures of the discrete Toda equation whose dependent variables are limited to non-zero values have been studied [16]. We take another approach in this paper: we extend the space of initial conditions instead of restricting it. We have two main ways of extension. The first method is to apply Sakai’s theory for discrete Painlevé equations [14] over the finite field. According to this theory we can construct the birational mapping over the extended space of initial conditions by blowing up at each singular point. The space of initial conditions for the discrete Painlevé II equation has been established over Fp and the bijection between a finite number of points has been constructed [8]. kanki@ms.u-tokyo.ac.jp http://dx.doi.org/10.3842/SIGMA.2013.056 2 M. Kanki The second approach uses the field of p-adic numbers. We study this method in detail from here on. We define the discrete integrable equations over the field of p-adic numbers Qp, and then define them over Fp, so that they are compatible with the reduction modulo prime from those over Qp. Rational mappings are said to have good reduction if, roughly speaking, the reduction and the evolution of the system commute. One of the typical examples with good reduction is the fractional linear transformation related to the projective linear group PGL2. Recently, birational mappings over finite fields have been investigated in terms of integrability [12]. In the previous papers, we defined the generalized notion of good reduction so that it could be applied to wider class of integrable mappings. We called this notion “almost good reduction” (AGR), and proved that discrete and q-discrete Painlevé II equations have AGR [7, 8]. Our conjecture was that AGR is also satisfied for other discrete Painlevé equations and that AGR is closely related to the integrability of dynamical systems over finite fields. In this paper, we prove that several types of q-discrete analogues of the Painlevé equations [11] have AGR for an appropriate domain, thereby verifying the conjecture. We also study the application of AGR to a chaotic system – Hietarinta–Viallet equation [5] – and conclude that AGR can be seen as an arithmetic analogue of the singularity confinement test [3]. 2 Reduction modulo a prime Let p be a prime number. Each non-zero rational number x ∈ Q× has a unique representation x = pvp(x) uv where vp(x), u, v ∈ Z and u and v are coprime integers neither of which is divisible by p. The p-adic norm \|x\|p is defined as \|x\|p := p−vp(x) (\|0\|p := 0). The field of p-adic numbers Qp is defined as a completion of Q with respect to the p-adic norm. The ring of p-adic integers is defined as Zp := {x ∈ Qp \| \|x\|p ≤ 1}. The ring Zp has the unique maximal ideal p = pZp = {x ∈ Zp \| \|x\|p < 1}. We define the reduction of x modulo p by Zp 3 x 7→ (x mod p) ∈ Zp/p ∼= Fp, and denote (x mod p) as x̃. The above mapping defines a reduction of p-adic integers to the (simple) finite field. Note that if we limit x to be a (rational) integer, then x̃ is nothing but x modulo p. The reduction is naturally generalised to Qp: Qp 3 x 7→ { x̃, x ∈ Zp, ∞, x ∈ Qp \ Zp ∈ PFp. We also denote the right hand side by x̃. Here the space PFp denotes the projective line P1(Fp) defined over the finite field Fp. As a set, we have PFp = Fp ∪ {∞}. Next we define the reduction of maps of the plane. Let φ be a rational map of the plane Q2 p given by two rational functions defined over (x, y) ∈ Q2 p: φ(x, y) = (f(x, y), g(x, y)), where f, g ∈ Qp(x, y) are rational functions. By multiplying numerators and denominators of f and g by suitable powers of p, the coefficients of both f and g can be taken in Zp and that at least one of the coefficients is in Z×p . From here on we assume this “minimal” form for rational Integrability of Discrete Equations Modulo a Prime 3 functions. If neither of the denominator of the minimal form of f or that of g modulo p is zero, then φ̃ is defined as the system whose coefficients are reduced to Fp: φ̃(x, y) = ( f̃(x, y), g̃(x, y) ) ∈ (Fp(x, y))2, where (x, y) ∈ Q2 p. The map φ is said to have good reduction (modulo p) on the domain D ⊆ Z2 p, if we have φ̃(x, y) = φ̃(x̃, ỹ) for any (x, y) ∈ D [15]. Although the good reduction is useful in arithmetic dynamical systems, the discrete Painlevé equations (expressed as a dynamical system) do not have good reduction since they frequently pass through singularities after reducing the equations modulo a prime. Also, the discrete Painlevé equations are non-autonomous mappings, and we need a generaliza- tion of good reduction to a non-autonomous mapping. Therefore, we have modified the good reduction so that it can be applied to wider class of systems, in particular to the discrete Painlevé equations. Definition 2.1 ([7]). A (non-autonomous) rational map of the plane φn has almost good re- duction (AGR) modulo p on the domain D(n) ⊆ Z2 p ∩ φ−1n (Q2 p) if for any p = (x, y) ∈ D(n) and any time step n, there exists a positive integer mp;n such that ˜φ mp;n n (x, y) = φ̃ mp;n n (x̃, ỹ). Here, the iteration φmn is defined as φmn := (φn+m−1 ◦ φn+m−2 ◦ · · · ◦ φn) \|D(n) for m > 0. If the domain D(n) does not depend on n, we just denote it by D. Note that, in particular, if we can take mp;n = 1 for all points p ∈ D(n) and all n, then the mapping φn has good reduction. Therefore AGR is weaker than good reduction. The following simple mapping Ψγ illustrates how almost good reduction works. Let us define Ψγ : xn+1 = axn + 1 xγnyn , yn+1 = xn, (1) where \|a\|p ≤ 1 and γ ∈ Z≥0 are parameters. Note that we omitted the cases of \|a\|p > 1, since we have vp(a) < 0 in this case, and we have to deal with a mapping such as φ : xn+1 = xn p + 1 xγnyn = xn + p pxγnyn , yn+1 = xn, whose reduction of coefficients φ̃ is not well-defined (note that 1̃/p = ∞). The map (1) is known to be integrable if and only if γ = 0, 1, 2. In these cases the map is a symmetric QRT mapping [9]. We have proved in our previous work that the following proposition holds. Proposition 2.2 ([7]). The rational mapping (1) has almost good reduction modulo p on the domain D if and only if γ = 0, 1, 2. Here D := Z2 p ∩ Ψ−1γ (Q2 p). If γ = 1, 2 then D = { (x, y) ∈ Z2 p \|x 6= 0, y 6= 0 } . If γ = 0 then D = { (x, y) ∈ Z2 p \| y 6= 0 } . We have also proved that the discrete and q-discrete Painlevé II equations also have almost good reduction [7, 8]. From these observations we have conjectured that, for the map of the plane Φn defined over the field of p-adic numbers, having almost good reduction on the domain D(n) = Z2 p∩Φ−1n (Q2 p) is equivalent to passing the singularity confinement test [3]. In this article, we support this conjecture by presenting further applications of the almost good reduction 4 M. Kanki principle to other integrable equations such as several types of q-discrete Painlevé equations and a chaotic equation. Note that in this paper we only deal with simple finite fields Fp. To study the equations over a general finite field Fpm (m > 1), we need to use the field extension of Qp. Since a field extension L of finite degree m over Qp is a simple extension, there exists an element α ∈ L such that L = Qp(α). The reduction from L to the extension of the finite field Fp(α) is defined as L 3 m−1∑ i=0 xiα i 7→  m−1∑ i=0 x̃iα i, ∀ i xi ∈ Zp, ∞, ∃ i xi ∈ Qp \ Zp. From Fp(α) ∼= Fpm , we obtain a system over a general finite field. Since the p-adic norm of Qp is extended to L, propositions in this paper are still valid for L, with slight modifications (e.g. D becomes ( Z⊕mp )2 ∩Ψ−1γ (L2) in Proposition 2.2). 3 q-difference analogue of Painlevé equations over a finite field In this section we prove that the q-discrete analogues of Painlevé III and IV equations have almost good reduction. These two equations are indeed integrable in the sense that they pass the singularity confinement test [3]. Note that, although passing singularity confinement test is not equivalent to the integrability of the given discrete equation, singularity confinement can be seen as a discrete analogue of the Painlevé property of the continuous Painlevé equations. In the geometrical setting, these two equations have rational surfaces of initial conditions on which the equations are birational [14]. From here on we occasionally write a ≡ b for a, b ∈ Zp, to indicate that ã = b̃ ∈ Fp. 3.1 q-discrete Painlevé III equation The q-discrete analogue of Painlevé III equation has the following form xn+1xn−1 = ab(xn − cqn)(xn − dqn) (xn − a)(xn − b) , where a, b, c, d and q are parameters [11]. It is convenient to rewrite it as the following coupled system form Φn : xn+1 = ab ( xn − cqn )( xn − dqn ) yn(xn − a)(xn − b) , yn+1 = xn. (2) Proposition 3.1. Suppose that a, b, c, d, q are distinct parameters with \|a\|p = \|b\|p = \|c\|p = \|d\|p = 1, and we also suppose that a+ b 6≡ (c+ d)q3 and a 6≡ b, then the mapping (2) has almost good reduction modulo p on the domain D := { (x, y) ∈ Z2 p \|x 6= a, b, y 6= 0 } . Proof. Let (xn+1, yn+1) = Φn(xn, yn). In the case when x̃n 6= ã, b̃ and ỹn 6= 0, we have x̃n+1 = ãb̃ ( x̃n − c̃q̃n )( x̃n − d̃q̃n ) ỹn(x̃n − ã)(x̃n − b̃) , ỹn+1 = x̃n. since the reduction modulo pZp is a ring homomorphism. Hence clearly ˜Φn(xn, yn) = Φ̃n(x̃n, ỹn). Next we examine other cases. We investigate the iterated maps for every initial condition Integrability of Discrete Equations Modulo a Prime 5 (xn, yn) ∈ D. There are six cases to consider. They are essential since the behaviors around the singular points are involved. From here we sometimes abbreviate ã as a, b̃ as b for simplicity. (i) Let us first consider the case where xn ≡ a and (a− b)(a+ b− cq − dq)ỹn 6≡ b(a− c)(a− d). In this case, Φ̃n(ã, ỹn) is not well-defined. This is because we have x̃n − ã = ã − ã = 0 in the denominator of x̃n+1. We also learn from yn+2 = xn+1, that ỹn+2 is not defined either. Therefore Φ̃2 n(ã, ỹn) is not well-defined. Here Φ̃2 n := ˜Φn+1 ◦ Φn. However, at the third iteration step, Φ̃3 n(ã, ỹn) is well-defined and we have ˜Φ3 n(xn, yn) = Φ̃3 n(x̃n = ã, ỹn) = ( a(b− cq2)(b− dq2)ỹn b(a− c)(a− d)− (a− b)(a+ b− cq − dq)ỹn , b ) . From the assumption of the case (i), the denominator is nonzero and is in F×p . (ii) Next we investigate the case of xn ≡ a and (a− b)(a+ b− cq − dq)ỹn ≡ b(a− c)(a− d) In this case, none of Φ̃i n(ã, ỹn) is well-defined for i = 1, 2, 3, 4. Next we calculate the fifth iteration Φ5 n at ỹn ≡ b(a−c)(a−d) (a−b)(a+b−cq−dq) and simplify the outcome. Then we learn that Φ̃5 n(ã, ỹn) is well-defined and we have ˜Φ5 n(xn, yn) = Φ̃5 n(x̃n = ã, ỹn) = ( b(a− cq4)(a− dq4) (a− b)(a+ b− cq3 − dq3) , a ) . (iii) If x̃n = b̃ and (a − b)(a + b − cq − dq)ỹn 6≡ −a(b − c)(b − d), by an argument similar to that in (i), we have ˜Φ3 n(xn, yn) = Φ̃3 n(x̃n = b̃, ỹn) = ( b(a− cq2)(a− dq2)ỹn a(b− c)(b− d) + (a− b)(a+ b− cq − dq)ỹn , a ) . (iv) If x̃n = b̃ and (a − b)(a + b − cq − dq)ỹn ≡ −a(b − c)(b − d), by an argument similar to that in (ii), we have ˜Φ5 n(xn, yn) = Φ̃5 n(x̃n = b̃, ỹn) = ( − a(b− cq4)(b− dq4) (a− b)(a+ b− cq3 − dq3) , b ) . (v) If ỹn = 0 and x̃n 6= 0, ˜Φ3 n(xn, yn) = Φ̃3 n(x̃n, ỹn = 0) = ( 0, ab x̃n ) . (vi) If ỹn = 0 and x̃n = 0, ˜Φ4 n(xn, yn) = Φ̃4 n(x̃n = 0, ỹn = 0) = (0, 0) . We have now fully investigated the behaviours around singularities and have completed the proof. � 6 M. Kanki 3.2 q-discrete Painlevé IV equation The q-discrete analogue of Painlevé IV equation has the following form (xn+1xn − 1)(xnxn−1 − 1) = aq2n(x2n + 1) + bq2nxn cxn + dqn , where a, b, c, d and q are parameters [10, 11]. It can be rewritten as follows: Φn : xn+1 = τ2(ax2n + bxn + a) + (xnyn − 1)(xn + τ) xn(xnyn − 1)(xn + τ) , yn+1 = xn, (3) where τ = qnτ0. Here we took τ0 = d/c and redefined a, b as ac/d2 → a and bc/d2 → b. Proposition 3.2. Suppose that \|a\|p = \|b\|p = \|q\|p = \|τ0\|p = 1, and we also suppose that aq2τ0 6≡ 1 and aq4τ0 6≡ 1. Then the mapping (3) has almost good reduction modulo p on the domain D(n) := { (x, y) ∈ Z2 p ∣∣x 6= 0, xy 6= 1, x 6= −qnτ0 } . Proof. In the proof we use the abbreviation as ã→ a, b̃→ b, τ̃0 → τ0. By an argument similar to that in Proposition 3.1, we have only to consider the cases at the singular points modulo a prime. (i) If x̃n = 0 and 1 + q2(−1 + aτ0 − bτ20 + qτ20 + τ0yn − aτ20 yn) 6≡ 0, the first and second iterations, Φ̃n(0, ỹn) and Φ̃2 n(0, ỹn) are not well-defined. However, at the third iteration we have ˜Φ3 n(xn, yn) = Φ̃3 n(x̃n = 0, ỹn) = ( −1− q3τ20 − bq4τ20 + aq6τ30 + q2(1 + bτ20 − τ0ỹn + aτ20 ỹn) q2τ0{1 + q2(−1 + aτ0 − bτ20 + qτ20 + τ0ỹn − aτ20 ỹn)} ,−q2τ0 ) . (ii) If x̃n = 0 and 1 + q2(−1 + aτ0 − bτ20 + qτ20 + τ0yn − aτ20 yn) ≡ 0, we iterate the map further from (i) until the reduced map is well-defined at ỹn = 1+q2(−1+aτ0−bτ20+qτ20 ) q2τ0(aτ0−1) . At the fifth iteration, ˜Φ5 n(xn, yn) = Φ̃5 n(x̃n = 0, ỹn) = ( −1 + q2 + aq4τ0 + q7τ20 − bq8τ20 q4τ0(−1 + aq4τ0) , 0 ) . Since we assumed that aq4τ0 6≡ 1, it is well-defined. (iii) If x̃n = −qnτ0 and ỹn 6= −τ−10 , ˜Φ3 n(xn, yn) = Φ̃3 n(x̃n = −qnτ0, ỹn) = ( −1− τ0ỹn + ( q3 − bq4 ) τ20 (1 + τ0ỹn) + q2{1 + bτ20 + τ0ỹn + aτ20 (−τ0 + ỹn)} q2τ0(−1 + aq2τ0)(1 + τ0ỹn) , 0 ) , where we assumed aq2τ0 6≡ 1. (iv) If x̃n = −qnτ0 and ỹn = −τ−10 , ˜Φ5 n(xn, yn) = Φ̃5 n ( x̃n = −qnτ0, ỹn = −τ−10 ) = ( − 1 aq6τ20 ,−aq6τ20 ) . (v) If x̃nỹn = 1, ˜Φ5 n(xn, yn) = Φ̃5 n ( x̃n = 1 ỹn , ỹn ) = ( 1 aq6τ30 ỹn , aq6τ30 ỹn ) . � Integrability of Discrete Equations Modulo a Prime 7 4 Hietarinta–Viallet equation over a finite field The Hietarinta–Viallet equation [5] is the following difference equation: xn+1 + xn−1 = xn + a x2n , (4) with a as a parameter. The equation (4) passes the singularity confinement test [3], which is a notable test for integrability of equations, yet is not integrable in the sense that its algebraic entropy [1] is positive and that the orbits display chaotic behaviour [5, 17]. We prove that the AGR is satisfied for this Hietarinta–Viallet equation. We rewrite (4) as the following coupled system: Φn : xn+1 = xn + a x2n − yn, yn+1 = xn. (5) Proposition 4.1. Suppose that \|a\|p = 1, then the mapping (5) has almost good reduction modulo p on the domain D := { (x, y) ∈ Z2 p \|x 6= 0 } . Proof. If x̃n 6= 0 then, the next step is immediately well-defined: ˜Φn(xn, yn) = Φ̃n(x̃, ỹ). If x̃n = 0, we have to iterate the map four times to obtain ˜Φ4 n(xn, yn) = Φ̃4 n(x̃n = 0, ỹn) = (ỹn, 0). � Therefore we learn that the AGR works similarly to the singularity confinement test in distinguishing the integrable systems from the non-integrable ones. In fact, the AGR can be seen as an arithmetic analogue of the singularity confinement test. 5 Concluding remarks We studied the integrable discrete equations over a finite field by reducing them from the field of p-adic numbers. We considered the “almost good reduction” (AGR), which had been proposed to be closely related to the integrability of discrete dynamical systems over finite fields. We proved that q-discrete Painlevé III and IV equations also have AGR, which has been a conjecture in our previous article. We also treated the Hietarinta–Viallet equation, which is non-integrable yet passes singularity confinement test. We proved that it also has AGR. From these observations, we have concluded that AGR is not a complete integrability test, however, we can safely state that the AGR is an arithmetic dynamical analogue of the singularity confinement method. One of the future problems is to modify AGR so that it can identify the systems which are non- integrable yet pass the singularity confinement test, like the Hietarinta–Viallet equation. Other future problems are listed below: (i) to study the geometric construction of the initial value space by blowups of the projective space of Qp, and its relation to the Sakai’s theory [14], (ii) to study the relation of our methods to the algebraic entropy [1] and its arithmetic analogue [4], (iii) to formulate the properties of the reduction modulo prime of the higher dimensional mappings like those in [6], (iv) to extend our methods to lattice equations with soliton solutions, such as the discrete Korteweg–de Vries equation and the discrete nonlinear Schrödinger equation. Acknowledgements The author wish to thank Professors Jun Mada, K.M. Tamizhmani, Tetsuji Tokihiro and Ralph Willox for insightful discussions and comments. He also thanks the detailed suggestions by the referees. This work is supported by Grant-in-Aid for JSPS Fellows (24-1379). 8 M. Kanki References [1] Bellon M.P., Viallet C.M., Algebraic entropy, Comm. Math. Phys. 204 (1999), 425–437, chao-dyn/9805006. 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Integrability of Discrete Equations Modulo a Prime

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