Linear relations for Laurent polynomials and lattice equations

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. We consider a family of nonlinear recurrences with the Laurent property, which were derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalize the Heideman-Hogan recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. The latter are obtained 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 6-point stencil. We introduce another linearizable lattice equation on the same stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6-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.


Introduction
There continues to be a great deal of interest in nonlinear recurrences of the form x n+m x n = P (x n+1 , . . . x m+n−1 ), for a polynomial P , with the surprising property that all of the iterates are Laurent polynomials in the initial data with integer coefficients, that is to say for all n [16]. This Laurent property is a central feature of the generators in cluster algebras, a novel class of commutative algebras introduced by Fomin and Zelevinsky [10], which are defined by recursive relations of the same form as (1) but with the restriction that P should be a binomial expression of a specific kind. The same authors also considered a more general set of sufficient conditions which ensure that the above recurrence has the Laurent property, without requiring P to be a binomial [11]. More recently, this led to the introduction of the broader framework of Laurent phenomenon algebras [26].
Cluster algebras are the focus of much activity due to their connections with diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmüller theory and dimer models [8,13,19]. The structure of a typical cluster algebra may be very complicated, due to the complexity of the recursive process, called mutation, that produces the generators. However, there are certain subclasses of cluster algebras that are associated with discrete integrable systems of some kind, and often these are the examples that are of most interest in applications to other areas. Beyond the finite type cluster algebras, which have a taxonomy that coincides with the Cartan-Killing classification of semisimple Lie algebras and finite root systems [12], and are associated with purely periodic dynamics, the next interesting subclass corresponds to discrete dynamical systems that admit linearization, in the sense that the variables satisfy linear recurrence relations with constant coefficients. The simplest example is the recurrence which arises from mutations of the Kronecker quiver (an orientation of the affine A 1 diagram), for which the iterates satisfy the linear relation is a first integral (independent of n). Linearizability was found for dynamics of cluster variables obtained from affine Dynkin quivers of type A in [15], for types A and D via frieze patterns in [2], and in general for all affine types ADE in [25]. It has been further conjectured (and proved in certain cases) that linearizability holds for sequences of cluster variables obtained from mutation sequences obtained from box products X Y of a finite type Dynkin quiver X and an affine Dynkin quiver Y [29]. A previously known example is provided by Q-systems [6], which arise from the Bethe ansatz for quantum integrable models, and correspond to taking X = A n for n arbitrary and Y = A (1) 1 , so (2) is included when n = 1. The case of a product of a pair of affine quivers X, Y is not linearizable, but is conjectured to be associated with systems that are integrable in the Liouville-Arnold sense [17].
In work by one of us with Fordy [14], concerning cluster algebras obtained from quivers that are mutation-periodic with period 1, in the sense of [15], we showed a further property of the affine type A recurrences, specified by a pair of coprime positive integers p, q as namely that the iterates satisfy additional linear relations with periodic coefficients, of the form x n+2q − J n x n+q + x n = 0, J n+p = J n , x n+2p − K n x n+p + x n = 0, K n+q = K n .
Furthermore, another family of linearizable recurrences from period 1 quivers was found, of the form x n x n+2k = x n+p x n+q + x n+k , p + q = 2k, which includes Dana Scott's recurrence [16] x n x n+4 = x n+1 x n+3 + x n+2 , and these also admit linear relations with periodic coefficients, given by x n+3q − J n+k x n+2q + J n x n+q − x n = 0, J n+p = J n , x n+3p − K n+k x n+2p + K n x n+p − x n = 0, K n+q = K n .
Analogous linear relations with periodic coefficients for affine quivers of type D and E, and associated Liouville integrable systems, appear in [28].
In this paper we are concerned with linearizable recurrences that exhibit the Laurent property but go beyond the setting of cluster algebras. To begin with we will consider the family of recurrences with a fixed parameter a and positive integers k and l. These recurrences were named the "Little Pi" family in [1], where they were shown to be generated by period 1 seeds in the setting of Laurent phenomenon (LP) algebras. They extend the Heideman-Hogan recurrences [20], corresponding to the case l = 1, for which detailed features of the linearization were proved in [22]. Thus our first aim here is to generalize the results of the latter work, and resolve some open conjectures from [30]. In particular, for (8) we obtain the constant coefficient linear relation when 2k and l are coprime, and a counterpart relation if gcd(2k, l) = 2. (All other cases can be reduced to one of these.) In addition to the first integrals (K or A, B, C) that appear as coefficients, we derive periodic quantities and associated linear relations with periodic coefficients. Ordinary difference equations can arise as reductions of two-dimensional lattice equations. For instance, the affine type A recurrences (3) are obtained from the 4-point equation for (s, t) being coordinates on Z 2 (or more generally, on a quadrilateral lattice), which is the relation for a frieze pattern [2]. To obtain (3), one should take the (p, −q) travelling wave reduction u s,t = x n , n = ps + qt, corresponding to a wave moving on the lattice with constant velocity −q/p ∈ Q. Similarly, it was noted in [23] that the 5-point lattice equation which is the relation for a 2-frieze [27], reduces to (5) by substituting in (12) with the replacement p → p − k, q → k, to obtain the (p − k, −k) travelling wave reduction. The authors of [23] also considered the Little Pi family (8) as a reduction of the 6-point lattice equation u s+1,t+2 u s,t = u s+1,t u s,t+2 + a(u s,t+1 + u s+1,t+1 ).
(Note that, compared with [23], we have switched the order of the independent variables and introduced the parameter a.) By obtaining linear relations for the above lattice equation, they deduced linear recurrences with periodic coefficients for its (l, −k) travelling wave reduction (8) (cf. Proposition 3.5 and Corollary 3.7 below). In addition, they proved the Laurent property for the lattice equation (14), in the sense that for the initial value problem defined by I = {u s,0 , u s,1 , u 0,t : s, t ∈ N}, the iterates in the positive quadrant in Z 2 are Laurent polynomials in the elements of this set. In this paper we introduce a new 6-point lattice equation, given by and prove the Laurent property for both this and (14) with a much broader set of initial values than just I. We further show that (15) is linearizable, and this feature (as well as the Laurent property) extends to the family of (l, −k) travelling wave reductions Our original motivation for introducing (15) was the fact that, when k = 1, the reduction (16) is the total difference of where the arbitrary parameter b is an integration constant. The latter family of recurrences was referred to as the "Extreme polynomial" in [1], where it was obtained from another set of period 1 seeds in LP algebras, and for b = 0 it was independently found in [30], where it was also shown to be linearizable and have the Laurent property (see [21] for further details). However, the recurrences (16) lie beyond the setting of LP algebras. All of the lattice equations described above fit into the framework of partial differential, differential-difference and partial difference equations described by Demskoi and Tran [5], who considered the family of determinantal equations where |M | = det(M ) is the determinant of an N × N matrix M of Casorati type, with entries specified by M = (u s+i−1,t+j−1 ) 1≤i,j≤N (up to shifts of indices) in the lattice case, or with appropriate modifications to Wronskian type entries in the case of partial differential/differential-difference equations. Equations of the form (18) are connected to 2D Toda lattices with appropriate boundary conditions, as well as Liouville's equation, and they are said to be Darboux integrable, meaning that they admit complete sets of first integrals that do not depend on one or the other of the independent variables s, t. The SL 2 frieze relation (11) is already in the form (18) with N = 2, and has the consequence that the corresponding 3 × 3 determinant vanishes, i.e. u s,t u s,t+1 u s,t+2 u s+1,t u s+1,t+1 u s+1,t+2 u s+2,t u s+2,t+1 u s+2,t+2 = 0, which follows by applying the Dodgson condensation algorithm [7], based on the Desnanot-Jacobi identity for matrix minors [3] (this is also referred to as Sylvester's identity in [5]), that is in which a superscript i (subscript j) on a minor denotes that the ith row (jth column) is deleted. Using similar methods to [14], the right/left null vectors of the 3 × 3 matrix yield the linear relations where the coefficients J = J(t), K = K(s) are first integrals of (11) in the s, t directions respectively, and the two linear relations in (20) reduce to those in (4) after imposing the travelling wave reduction (12). Similarly, applying Dodgson condensation with the 2-frieze relation (13) yields which is the relation for an SL 3 frieze on each of the sublattices obtained by restricting s + t to have odd/even parity, and can be put in the standard form (18) by a linear change of coordinates. A further application of (19) shows that the corresponding 4 × 4 determinant vanishes for the 2-frieze relation, while in [23] it is shown that there is also a constant 3 × 3 determinant and a vanishing 4 × 4 determinant associated with (14), and in the sequel we prove an analogous result for the new lattice equation (15).
In the next section we give a very brief introduction to LP algebras, and explain how nonlinear recurrences of the form (1) can arise in that setting, giving full details for the particular case of the Little Pi family (8). Section 3 is devoted to an independent derivation of the linear recurrences with periodic coefficients found for the Little Pi family in [23], which we then use in Section 4 to derive a constant coefficient relation of order 6kl, of the form (9) or (10), for each pair of coprime positive integers k, l. Section 5 is concerned with the new 6-point lattice equation (15), including the proof of linearization both for the lattice equation and all its travelling wave reductions (16). Finally, in Section 6 we show that the new lattice equation has the Laurent property for suitable band sets of initial values in Z 2 , of the kind described by van der Kamp in [24], and we use this to infer the Laurent property for its reductions (16). We finish by applying the same approach to show the Laurent property for the lattice equation (14) with band sets of initial values.

Laurent phenomenon algebras and recurrence relations
There are various situations where birational transformations of the form (1) arise with more than two monomials on the right-hand side [18,31], and Laurent phenomenon (LP) algebras provide a general framework for such situations which goes beyond the setting of cluster algebras [26]. Like cluster algebras, LP algebras are constructed from collections of objects called clusters: for an LP algebra of rank m, a cluster is a set of m independent quantities called cluster variables. A seed in an LP algebra consists of a cluster together with m polynomials in the cluster variables, called exchange polynomials. There is a process called mutation which allows new seeds to be produced, using the exchange polynomials. Certain conditions must be imposed on the exchange polynomials which ensure that the Laurent property is preserved under arbirary sequences of mutations, in the sense that all of the cluster variables so obtained are Laurent polynomials in the m cluster variables from the initial seed. There is also a concept of periodic seeds [1], analogous to the concept for cluster variables that was introduced in [15], which allows recurrence relations to be generated by particular sequences of mutations.

Construction of an LP algebra
• P i does not contain the variable x i . Definition 2.2. For each seed (x, P) there is a mutation µ k for each k ∈ {1, . . . , m}, producing a new seed µ k ((x, P)) = (x ′ , P ′ ). The process of mutation is defined in the following steps: 1 , . . . , x ±1 m ] to be the unique polynomials such that •P j = P j 1≤i≤n,i =j x a i i for each j and a i ∈ Z ≤0 for each i; and this polynomial is not divisible by P j in this ring.

The new cluster is
x ′ k 4. For each j, remove all common factors withP k | x j ←0 from G j in the unique factorization domain Z[x 1 , . . . , x k , . . . , x j , . . . , x m ], with the hats denoting omitted variables. Denote the polynomials obtained in this way by H j .

The new exchange polynomials are
x m ] such that P ′ j is not divisible by any Laurent monomial in this ring.

The new seed is
Definition 2.3. Two seeds are said to be mutation equivalent if one can be obtained from the other via a finite sequence of mutations. For a choice of initial seed (x, P), the LP algebra A = A(x, P) is the subalgebra of Q(x 1 , . . . , x m ) generated by all cluster variables in seeds that are mutation equivalent to the initial seed. Evidently this does not depend on the choice of initial seed.
The somewhat convoluted construction of LP mutation ensures that the proof of the Laurent property of cluster algebras, via the caterpillar lemma in [10], is still valid in the more general LP case.

Recurrence relations from period 1 seeds
Following [1], we now show how the notion of periodic seeds for cluster algebras, introduced in [15], may be generalized to LP algebras, in the special case where the period is 1. Periodic seeds may be used to show that the iterates of certain recurrence relations correspond to mutations in an LP algebra, hence satisfying the conditions of Theorem 2.4. This proves the Laurent property for these recurrences. In the definition above, we considered unordered seeds, but when we consider recurrence relations it is helpful to fix an ordering.
. . , P m ) be a seed (where the cluster variables and exchange polynomials are ordered according to their subscript), let be the seed obtained from it by applying the mutation µ 1 and then reordering the variables with a cyclic permutation ρ, and define where the shift operator S increases the subscripts on each of the x i appearing by one.
The importance of the above definition is due to the following result, which is Corollary 2.5 in [1], but for completeness we sketch the proof here.
Proof. Since P ′ 2 = SP 1 we have P ′ 2 = P (x 3 , . . . x m+1 ). Applying the mutation µ 1 to the period 1 seed (x, P) gives and setting x m+1 = x ′ 1 agrees with the first iteration of the proposed recurrence (22). After applying the cyclic permutation ρ to reorder the variables and exchange polynomials, the new seed is with new exchange polynomials given by (21). Now applying the mutation µ 2 gives a new cluster variable which is defined to be x m+2 , and produces the new seed Continuing to apply consecutive mutations µ 3 , µ 4 , and so on, one can see that this will give precisely the iterates of (22). Since these iterates are given by compositions of mutations they belong to the LP algebra A generated by the seed (x, P), hence are Laurent polynomials in the initial cluster variables by Theorem 2.4.

Little Pi from a period 1 seed
The Little Pi recurrences (8) are included in many examples of the form (22) found in [1] that can be shown to have the Laurent property by describing them in terms of successive mutations of a period 1 seed, as in Proposition 2.6. In order to apply this result they construct the "intermediate polynomials", that is to say, the other exchange polynomials that appear in the period 1 seed, such that the shifting conditions (21) hold. For Little Pi, this construction is split in to four cases, which we list below.
For convenience, we slightly change the notation compared with the above discussion, where we followed [11] in labelling a cluster of size m with indices from 1 to m. To be consistent with [1], below we label the initial cluster variables x i and exchange polynomials P i with indices 0 ≤ i ≤ m − 1. Note that the inclusion of the coefficient a in (8) means that the Laurent property takes the form but in fact in the next section we will take a → 1. (More details of the Laurent phenomenon over a ring of coefficients are provided in [11].) Only the polynomials P j for j ∈ J := {0, k, 2k, l, k + l} are given here. To find the intermediate polynomial P i for any i, take the largest j ∈ J with j ≤ i and shift P j up by i − j, so that P i = S i−j P j . Note that in all cases we have P 0 = P .

Linear relations with periodic coefficients for Little Pi
Henceforth we shall work with the Little Pi family of recurrences in the form which is obtained from (8) after rescaling x n → ax n . These generalize the family found by Heideman and Hogan [20], which is the case l = 1. In order to find linear relations, we begin by showing that the 3 × 3 matrix has a non-zero periodic determinant. For convenience we set and note the following two identities which are a consequence of (23): has period k.
Proof. First observe that (23) can be rewritten as so using Dodgson condensation, as given by (19) with N = 3, we may write Upon scaling the first column by x n+3k+l , we see that the 2 × 2 determinant δ ′ n+k in (27) satisfies Then we can use (25) and (26) on the left column to obtain and by the same token, but instead manipulating the right column in (27), we have Shifting up n → n + k and comparing with (28) we arrive at so these ratios are periodic with period k, which is the required result.
Lemma 3.2. For each n the determinant δ n = |Ψ n | is non-zero, considered as an element of Q(x 0 , x 1 , . . . , x 2k+l−1 ), the ambient field of fractions in the initial data for (23).
Proof. Without assuming the Laurent property, a priori the iterates of (23) are rational functions of the initial data with rational numbers as coefficients, and the same is true for the determinant δ n . Let us consider the case of substituting real positive initial values x n > 0 for n = 0, . . . , 2k + l − 1. It follows by induction that x n > 0 for all n ∈ Z, hence also z n > 0 for all n. If δ n vanishes for some n then δ ′ n+k vanishes, by (27), but then z n+k z n+2k+l + z n+2k z n+k+l = 0 by (29), which is a contradiction. Hence δ n is a non-zero rational function.
We now consider the corresponding 4 × 4 matrix and use Dodgson condensation once more, with N = 4 in (19), to calculate and then by periodicity of δ n we have the Given that |Ψ n | = 0, we obtain linear relations with periodic coefficients by considering the right and left kernels (i.e. the kernel ofΨ n and that of its transpose).
Remark 3.4. The kernel ofΨ n is one-dimensional, since if it were of dimension greater than one then Ψ n we would have a non-trivial kernel, contradicting Lemma 3.2.
Proposition 3.5. The iterates of (23) satisfy the linear relations x n+3l + γ n x n+2l + β n x n+l + α n x n = 0, with periodic coefficients: K n and K n have period l, α n has period k, and β n and γ n have period 2k.
n , 1) T be in the kernel ofΨ n . (We are justified in scaling the last entry to 1 due to Lemma 3.2.) From the first three rows of we get the matrix equation and by Cramer's rule The last 3 rows of (32) give The equations (33) and (34) imply that K n and K n both have period l. Now set Ψ T n (α n , β n , γ n , 1) T = 0, and analogous arguments to the preceding ones give and the result that α n is k-periodic, and β n and γ n are 2k-periodic.
We can derive further relations between the coefficients in (30) and (31) by using (33) and (34), as well as the corresponding equations for the left kernel ofΨ n . Lemma 3.6. The periodic coefficients in (30) are related to one another by K Proof. From the first 2 rows of (32) we have Then solving for K (2) n and K The sequence of z j satisfy the same matrix equation (33) as x j , obtained by replacing each x j → z j in (33), due to the l-periodicity of K (2) and K (3) , so Assuming that K and the left-hand side above is periodic with period l so the right-hand side should be too, i.e. z n+3k z n+5k = z n+3k+l z n+5k+l ⇐⇒ z n+3k z n+3k+l z n+5k z n+5k+l = 0, and this determinant is δ ′ n+3k from (27), but by the proof of Lemma 3.2 this cannot be identically zero, which gives a contradiction. Hence K Remark 3.8. The latter results were previously obtained via a different method, using the travelling wave reduction of (8), in [23] (see Corollary 3.2 and Proposition 3.3 therein).
We close this section by proving some conjectures made for l = 1 in [30], and extending them to arbitrary l. Proposition 3.9. The periodic coefficients of the linear relation (31) satisfy the following set of identities: Proof. From the left kernel analogue of (34) we have so we can express β n and γ n as Upon shifting β n → β n+k we can equate the bracketed terms above as Now if we write the z j in terms of the x i and replace the x n+k+4l that appears as Since the kernel ofΨ n is one-dimensional we can scale and equate coefficients in (41) and an appropriate shift of (35) to get three equations, namely α n = β n + γ n+k − 1, γ n+k α n+l = β n+l + β n+k+l − α n+l γ n , α n+l = β n+l + γ n+k+l − 1.
where the third of these is simply a shift of the first, and these rearrange to give (37) and (38). The identity (39) follows from (36) and the fact that δ n has period k.

Linear relations with constant coefficients
In this section we derive the linearization of the Little Pi family (23), in the form of linear relations with constant coefficients, which were not previously considered in [23]. The key is to use monodromy arguments, similar to those employed in [14] in the case of the cluster algebra recurrences (3) and (5). We start by defining the sequences of matrices which vary with overall periods l and 2k, respectively, and allow the linear relations (30) and (31) to be rewritten in matrix form as where as before Ψ n is given by (24). The point of this is that if we define the pair of monodromy matrices M n := L n L n+2k L n+4k · · · L n+2k(l−1) ,M n :=L n+(2k−1)l · · ·L n+2lLn+lLn (43) then right multiplication by M n will shift Ψ n by 2k upwards l times, that is and left multiplication byM n will shift Ψ n by l upwards 2k times, so that Remark 4.1. If d := gcd(k, l) > 1 then the recurrence (23) splits into d copies of itself, so without loss of generality we can take d = 1. Then with d = 1, if l is odd then gcd(2k, l) = 1 and lcm(2k, l) = 2kl, while if l is even then gcd(2k, l) = 2 and lcm(2k, l) = kl, and we need to deal with these two different cases separately.

The case gcd(2k, l) = 1
Here lcm(2k, l) = 2kl, so l is odd, and with the monodromy matrices M n andM n defined as in (43) above, we see that due to the l-periodicity of L n and the cyclic property of the trace, the quantity K := tr(M n ) has period l, and similarly tr(M n ) has period 2k. Now from (44) and (45) we have so K has period gcd(2k, l) = 1, hence is a first integral for (23), independent of n. The same argument applies to the quantityK := tr (M −1 n ) = tr (M −1 n ), which we will now show is equal to K. Proof. The result holds in the case l = 1, since we have K = tr(M n ) = tr(L n ) = K n+k , K = tr(L −1 n ) = K n , and K n has period 1, hence K =K. (Another proof for l = 1 is given in [22].) Rewriting K as tr(M n ), and similarly forK = tr(M −1 n ), and (setting n = 0 without loss of generality) this implies an algebraic relation between the 5k quantities α 0 , . . . , α k−1 , β 0 , . . . , β 2k−1 , γ 0 , . . . , γ 2k−1 , namely that must hold as a consequence of the relations in Proposition 3.9 for l = 1. Due to periodicity there are 2k independent relations of the form (37), as well as k independent relations of the form (38), together with (39), but in fact there are only 3k independent relations in total, so these equations define an affine variety of dimension 2k. Now for odd l > 1 note that for K =K to holds if and only if tr (L σ(2k−1)Lσ(2k−2) · · ·L σ(0) ) = tr (L −1 where σ is the permutation of the indices 0, 1, . . . , 2k − 1 defined by σ(i) = il mod 2k, which satisfies the properties With all indices read mod 2k (or mod k in the case of α j ), it follows from these properties that σ acts by permuting the coordinates α j , β j , γ j in the identities for l = 1 in Proposition 3.9, so that the identities for each odd l are just and similarly for (39). In other words, the identities for odd l > 1 are just permutations of those for l = 1, so the algebraic relation (47) holds as an immediate consequence of the relation (46) when l = 1.

Remark 4.3.
There is an implicit assumption in the above proof, namely that when l = 1 the map from the initial values x 0 , x 1 , . . . , x 2k for (23) to the variety defined by the relations in Proposition 3.9 is surjective, which ensures that the identity tr(M n ) = tr (M −1 n ) must be an algebraic consequence of these relations. In particular, it is enough to check that for l = 1 there is collection of 2k independent 2k-periodic functions of the initial data (e.g. either of the sets β 0 , . . . , β 2k−1 or γ 0 , . . . , γ 2k−1 should be functionally independent). While this is a straightforward but laborious task for any given k, we do not know of a simple verification that is valid for all k. However, for any l, a direct algebraic proof of Proposition 4.2 is provided by the argument used to prove Theorem 4.5 in [21], which we will revisit in the proof of Theorem 5.4 below.

The case gcd(2k, l) = 2
In this case lcm(2k, l) = kl, with l even and k odd. With the same definition (43) for M n , each matrix in the product appears twice in the same order relative to its neighbours, so M n is a perfect square, and we can define the square root M * n = M 1/2 n by the same product with half as many factors, and similarly forM * n =M 1/2 n . The total shift for Ψ n is now kl instead of 2kl, so we have Again tr (M * n ) has period 2k and tr (M * n ) has period l, but now this implies that K n := tr (M * n ) = tr (M * n ) has period gcd(2k, l) = 2, and similarly forK n := tr ((M * n ) −1 ). The analogue of Proposition 4.2 requires more work in this case. We begin with Now due to (39) we have α n = −1 and using (37) we get β n = −γ n+1 , hence K n =K n+1 , and the result holds in this case. This implies an algebraic identity between the entries of the monodromy matrix M * n and its shift, namely that tr (L 0 L 2 · · · L l−2 ) = tr (L −1 l−1 L −1 l−3 · · · L −1 1 ) (where we set n = 0 without loss of generality), which is just a tautology in terms of the l quantities K 0 , K 1 , . . . , K l−1 that appear in the entries. Similarly to the argument in the proof of Proposition 4.2, we have that for k > 1 the required identity of traces for M * n and (M * n+1 ) −1 just corresponds to a permutation σ of the indices of the quantities K j , given by σ(j) = 2jk mod l, so the relation K n =K n+1 holds for all k.
Proposition 4.6. The iterates of (23) satisfy a linear relation with period 2 coefficients, given by Proof. This follows by the Cayley-Hamilton theorem, as in the proof of Theorem 4.4, but with different traces appearing.
Theorem 4.7. When gcd(2k, l) = 2, the iterates of (23) satisfy the constant coefficient relation Proof. Let S be the shift operatorm such that S(f n ) = f n+1 for any function of n. Then applying the operator S 3kl − K n+1 S 2kl + K n S kl − 1 to equation (50) gives the required result.

Superintegrability of Little Pi
The birational map defined by the Little Pi recurrence (23) is measure-preserving, in the sense that where Ω is the volume form We can use this to show that the map ϕ is maximally superintegrable, in the sense that it admits an (anti-) invariant Poisson structure, and the number of independent first integrals is one less than the dimension of the phase space.
In the case gcd(2k, l) = 1, it appears that the l-periodic quantities K 0 , . . . , K l−1 are independent of one another, hence any cyclically symmetric functions of these quantities are first integrals: so this provides l independent first integrals for (23). Similarly, subject to the relations in Proposition 3.9 one can take 2k independent 2k-periodic quantities, and cyclically symmetric functions of these provide 2k independent first integrals. However, in total this should give exactly 2k + l − 1 independent first integrals I 1 , I 2 , . . . , I 2k+l−1 , since the identity tr (M n ) = tr (M n ) gives a relation between these two sets of cyclically symmetric functions. Then by a result from [4], taking all but one of these first integrals together with the covolume form V = x 0 · · · x 2k+l−1 ∂ ∂x 0 ∧ · · · ∧ ∂ ∂x 2k+l−1 (i.e. the (2k + l)-multivector field that contracts with Ω to give 1) yields a Poisson bracket defined by { f, g } = V (df, dg, dI 1 , dI 2 , . . . , dI 2k+l−2 ), and this bracket is invariant/anti-invariant under the action of ϕ, according to the parity of l, that is for any pair of functions f, g on the (2k + l)-dimensional phase space. By construction, the first integrals I 1 , I 2 , . . . , I 2k+l−2 are Casimirs for this bracket, but the additional first integral I 2k+l−1 is not, so it defines a non-trivial Hamiltonian vector field.
As an example, we take the simplest case k = l = 1, when ϕ is defined by The quantity K 0 = K is a first integral, which can be written as by Theorem 4.4, and then rewritten as a function of the initial values x 0 , x 1 , x 2 by using (51). In fact, by Theorem 1.2 in [22], the explicit expression is where and Also, by Proposition 3.9 we have and then from (31) we can find two independent 2-periodic quantities by solving for β 0 , β 1 in terms of the x j from the pair of linear equations Then the two symmetric functions provide two independent first integrals, but they are related to K by Finally, contracting the covolume form with the one-form dI 1 gives the Poisson bracket commutes with the map, and its level sets are curves defined by I 1 = const, I 2 = const.

Linearization and reductions of a 6-point lattice equation
In this section we consider the new 6-point lattice equation (15), which can be rewritten as an equality of two 2 × 2 determinants, in the form u s,t+1 u s,t+2 u s+1,t+1 u s+1,t+2 + a = u s,t + a u s,t+1 u s+1,t u s+1,t+1 , or in the form of a conservation law, as ∆ s au s,t+1 = ∆ t u s,t u s,t+1 u s+1,t u s+1,t+1 .
By imposing the constraint u s,t = u s+k,t−l for integers k, l, one obtains the (l, −k) travelling wave reduction u s,t = x n n = ls + kt, which produces the family of recurrences (16). Upon making use of the conservation law (52), we can write the reduction as and as both sides are a total difference this can be integrated to give where b is an integration constant. In other words, b is a first integral for (16). The particular case k = 1, given by (17), is the "Extreme polynomial" family found from period 1 seeds in LP algebras in [1], whose linearization was studied in [21] for b = 0. For k > 1, the recurrences (54) are not of the type (1) that can arise from periodic seeds in LP algebras, and none of the recurrences (16) are of this type. Nevertheless, these recurrences turn out to have the Laurent property for any k, as a consequence of the fact that the lattice equation (15) has the Laurent property.

Linearization of new lattice equation
Similarly to the results on linearization of the lattice equation (14) obtained in [23], the iterates of the new 6-point equation (15), or equivalently (52), satisfy two types of linear relation, with coefficients that are independent of one or the other of the lattice variables s, t.
Proof. Dividing both sides of (15) by u s,t+1 u s+1,t+1 , we immediately find a quantity which is invariant under shifts in the s direction, since the equation becomes the total difference The definition of J rearranges to give an inhomogeneous linear relation for u s,t , that is u s,t+2 − J(t)u s,t+1 + u s,t + a = 0, and by applying the difference operator ∆ t to this we are led to the homogeneous relation (55). Since the coefficients of the latter are fixed under shifting s, we can write down four shifts of the relation in the form of a matrix linear system, that is     u s,t u s,t+1 u s,t+2 u s,t+3 u s+1,t u s+1,t+1 u s+1,t+2 u s+1,t+3 u s+2,t u s+2,t+1 u s+2,t+2 u s+2,t+3 u s+3,t u s+3,t+1 u s+3,t+2 u s+3,t+3 Thus we see that the 4 × 4 matrix above has determinant zero, so we may take a vector (C, B, A, 1) in the left kernel, and then it is apparent that the entries of this vector are invariant under shifting t. This kernel gives the second linear relation (56).
In the course of the proof, we observed the following result, which is also proved for (14) in [23].

Linear relations for travelling wave reductions
Upon applying the reduction (53), the linear relations (55) and (56) for the lattice equation (15) reduce to linear recurrences with periodic coefficients for (16).
Proposition 5.3. The iterates of the equation (16) for the (l, −k) travelling reduction of (15) satisfy linear recurrence relations with periodic coefficients, given by x n+3l + A n x n+2l + B n x n+l + C n x n = 0, where the coefficient J n is periodic with period l, and A n , B n , C n are periodic with period k.
Proof. The travelling wave reduction (53) applied to (15) gives the nonlinear recurrence (16), and under this reduction the quantity J(t) defined by (57) becomes which is periodic with period l, while the coefficients A(s), B(s) and C(s) in (56) become k-periodic quantities, denoted A n , B n , C n . The linear relations (60) and (61) can also be constructed directly from the observation that J n defined by (62) has period l. (See [21] for details in the case k = 1.) Making appropriate adjustments compared with the case of Little Pi, we redefine and set so that we have the linear matrix equations As before, we can make the assumption gcd(k, l) = 1, since otherwise (16) splits into several copies of lower dimension.
Theorem 5.4. When gcd(k, l) = 1, the iterates of (16) satisfy the constant coefficient linear relation where K is the trace of the monodromy matrix, which is a first integral, as well as the linear relation Proof. From the matrix linear relations (63) we have where the second monodromy matrix is M n :=L n+(k−1)lLn+(k−2)l · · ·L n+lLn .
Then since the entries of L n andL n have periods l and k respectively, it follows that K = tr (M n ) = tr (M n ) has period gcd(k, l) = 1, by assumption. By an analogous permutation argument to the one used in the proof of Proposition 4.2 and in the proof of Proposition 4.5, we have that tr (M n ) = tr (M −1 n ), and then the relation (64) follows by applying the Cayley-Hamilton theorem to M n , just as in the proof of Theorem 4.4. However, we can get a stronger result by considering (62), and defining the matrices which are related by the inhomogeneous equation Then paraphrasing the steps of the proof of Theorem 4.5 in [21], we introduce the 2 × 2 monodromy matrix M * n = L * n L * n+k · · · L * n+k(l−1) , and find a matrix equation of the form with κ = tr (M * n ), where (like those of L * n ) the entries of the matrixC * n are periodic with period l. The top leftmost entry of (66) gives the equation for some l-periodic quantityJ n , and if we apply the operator S kl − 1 then we obtain (64) together with the relation tr (M n ) = K = tr (M * n ) + 1; this also gives an independent proof that tr (M n ) = tr (M −1 n ). However, we can instead apply the operator S l − 1 to (67), giving the homogeneous linear relation (65), which is of lower order than (64) when k > 1.
Remark 5.5. For k = 1, the quantity κ = tr (M * n ) is given by the explicit formula and the formula for k > 1 is obtained by a permutation of indices. The term of each distinct homogeneous degree in the above expression corresponds to a first integral of the dressing chain for one-dimensional Schrödinger operators (see [21] and references). With minor modifications, the preceding argument shows that the trace of the 3 × 3 monodromy matrix M n in (43), defined by a product over matrices L n as in (42), is related via (68) to the trace of a 2 × 2 monodromy matrix M * n expressed in terms of l-periodic entries J n = K n − 1, and this gives an independent proof of Proposition 4.2, deriving tr(M n ) = tr (M −1 n ) as an equality between cyclically symmetric functions of K 0 , K 1 , . . . , K l−1 .
Remark 5.6. The birational map ϕ in dimension 2k+l defined by (16) is measure-preserving, with the volume form There should be l independent l-periodic quantities J n , which appear as coefficients in (60), but it is unclear how many of the k-periodic quantities appearing in (61) should be independent, so the question of superintegrability of (16) remains open.

Laurent property for linearizable lattice equations
In this section we discuss the Laurent property for both of the 6-point lattice equations (14) and (15). Following van der Kamp [24], we construct a family of bands of initial values, as well as some special sets, that give well-defined solutions on the whole Z 2 lattice. Given suitable conditions on the initial values, linear relations with coefficients fixed in one lattice direction can be used to prove the Laurent property. We show that the band sets of initial values satisfy the necessary criteria. for all s, t ∈ Z 2 .
In the equation (15) there is a single coefficient a, so we have A = {a}, and we write L[a] for the ring of Laurent polynomials associated with an initial value set I.

Construction of band sets of initial values
In [24] an algorithm is given which finds (in almost all cases) an unique solution to a lattice equation on an arbitrary stencil, given a band of initial values I. We apply this to the 6-point domino-shaped stencil that (15) is defined on.
First we define the lines L 1 and L 2 , each with positive rational gradient, such that L 2 = L 1 + (1, −2). For a given pair of lines L 1 , L 2 related in this way, the associated band set of initial values I = I(L 1 , L 2 ) consists of all the lattice points lying between these two lines, including the points on L 1 but not those on L 2 . An example with gradient 1/3 is shown in Figure 1, where the points in the band set I are coloured yellow. We will consider each of the points on L 1 to be the top left corner of a 6-point domino, hence L 2 will pass through the points diagonally opposite. By the results of [24], taking initial values between these lines and on L 1 (but not on L 2 ) allows us to find an unique solution of (15) for each choice of gradient. The first step is to calculate the values on L 2 , drawn in blue, using the yellow initial values. We then shift the lines perpendicularly to their direction until they pass through another point of the domino, corresponding to the dashed lines in the figure. The new L 2 will pass through the next points to be calculated, drawn in red. This process is continued until we fill the whole lattice below L 1 . We can also shift the lines in the opposite direction to fill the whole lattice above L 1 .

The Laurent property for lattice equation (15)
To prove the Laurent property we will use the linear relation (58), but first we must prove that the coefficients J(t) belong to the ring of Laurent polynomials. Proof. Since J(t) defined by (57) is independent of s we may shift it in the s direction until the index values appears, and then we have Proof. For each s we use induction on t and the relation noting that, by the previous lemma, J(t) is in the Laurent ring. The base case for the induction is given by This proves Laurentness for t >t + 1, and the proof for t <t is similar.
Theorem 6.4. The Laurent property for (15) holds for the band sets of initial values I, as described in Subsection 6.1. Proof. To calculate u s,t we only have to divide by u s+1,t+1 and vice versa, so we know all the values we calculate are Laurent polynomials until we have to divide at one of the blue points in Figure 1, and the corresponding points above L 1 . These we mark in green in Figure 2. We draw L ′ 2 parallel to and below L 2 through the first non-green point and L ′ 1 parallel to and above L 1 through the last green point. Equivalently L ′ 2 = L 2 + (−1, −1), L ′ 1 = L 1 + (−1, 1), hence L ′ 2 = L ′ 1 + (1, −4). Since the minimal distance between L ′ 1 and L ′ 2 is √ 17 > 4 any line that intersects I will intersect at least four elements of L. For Lemma 6.2 we take horizontal lines with height t and see that they intersect at least four green or yellow points, at least one of which will be yellow. Hence J(t) ∈ L for all t. For Theorem 6.3 we take vertical lines for each s and see that these intersect at least two green or yellow points. Hence the conditions of the preceding theorem hold and we have the Laurent property for these initial values.
In the special case where the gradient is 0 it is prescribed in [24] that we should take an extra line of initial values perpendicular to L 1 and L 2 , as shown in Figure 3, and this case also has the Laurent property. However, we note that Laurentness does not hold for all well-posed initial value problems, for example the yellow set shown in Figure 4. In this case one can see from the form of (15) that to calculate the value of u s,t at the blue node we must divide by a polynomial (not a monomial) in the surrounding initial values.
Note that the Laurent property for the reductions (16) is easily seen from (62). In fact, since J n has period l, the only initial variables that can appear in the denominator are x k , x k+1 , . . . , x k+l−1 . In particular, setting each of these to be 1 will give a polynomial sequence in the remaining initial values. So we have

The Laurent property for the lattice Little Pi
The Laurent property for the lattice equation (14) was proved in [23] for points (s, t) in the positive quadrant with the initial value set I = {u s,0 , u s,1 , u 0,t : s, t ∈ N} (note that we switched s and t compared with the original reference). We have drawn the above set I in yellow in Figure 5, extended to include indices s, t in the whole of Z. Again we can define the associated Laurent ring L (without the coefficient a, since we set a → 1 here), and provide a different proof of Laurentness, similar to the proof for (15). From Theorem 2.1 and Proposition 2.6 in [23], respectively, we have that u s,t+6 − β(t + 1)u s,t+4 + β(t)u s,t+2 − u s,t = 0 (69) with β = β(t) (independent of s) being given by β(t) = 1 + u 0,t u 0,t+3 + u 0,t+1 u 0,t+4 + u 0,t+2 u 0,t+5 u 0,t+2 u 0,t+3 Note that in this expression s has been set to zero, but due to the fact that β is s-independent the same formula is valid with each term u 0,j replaced by u s,j for j = t, t+1, . . . , t+5. Similarly to the proof of Theorem 6.4, we colour the values that only require division by elements of I in green in Figure 5. Due to the shape of (14) we end up with more green vertices than we had for (15).
Proposition 6.6. The lattice equation (14) has the Laurent property for the initial values I = {u s,0 , u s,1 , u 0,t : s, t ∈ Z}.
Proof. Again we fix s and use induction on t. The induction starts with the vertical line of six values in L, shown in yellow and green in Figure 5. We can see from (70) that, for this I, β(t) ∈ L for all t, and we increase t by solving (69) for u s,t+6 to obtain Laurent polynomials for all t > 0, while to extend to t < 0 we solve for u s,t instead.
For the band sets of initial values we have to work harder. then β(t) ∈ L.
Proof. In the expression for β(t) in (70), we shift s until us ,t+2 and us ,t+3 appear in the denominator, and the terms in the numerator belong to L by assumption, so the result follows.
Theorem 6.8. For a given initial set I, if the conditions of Lemma 6.7 hold for all t, and if for all s there is at such that {u s,t , u s,t+1 , u s,t+2 , u s,t+3 , u s,t+4 , u s,t+5 } ⊂ L, then equation (14) has the Laurent property. Proof. The proof is the same as for Proposition (6.6).
Theorem 6.9. The equation (14) has the Laurent property if I is a band of initial values.
Proof. Similarly to the proof of Theorem 6.4 we have L ′ 1 = L 1 + (−1, −2), L ′ 2 = L 2 + (1, −2), so L ′ 2 = L ′ 1 + (3, −6) and the minimal distance between them is √ 45 > 6. Hence for any vertical or horizontal line intersecting the lattice we have at least six consecutive values in L, and at least two of these will be neighbours and in I.