Linear degree growth in lattice equations

We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear growth of these equations. We then use these recurrences to search for lattice equations that have linear growth and hence are linearizable.


1.
Introduction. Discrete integrable systems have been a topic of many studies for the last three decades, where integrable refers to possession of one or more signature properties that imply 'low complexity' of the dynamics. Complexity of a rational map or a lattice equation can be measured through the so-called algebraic entropy. It has been used as an integrability detector [3,7,14,26,27,29]. If a discrete map or lattice equation has vanishing entropy, i.e. degrees of iterations of the map (or equation) in terms of initial variables grow sub-exponentially, then this can be taken as a definition of being integrable and heuristically it will be accompanied by other signature properties of integrability. A type of sub-exponential growth is polynomial growth. It is noted that all known discrete maps or lattice equations with sub-exponential degree growth have, in fact, polynomial growth [4,5,20].
Algebraic entropy of a map or a lattice equation is often calculated as follows. One can introduce projective coordinates for each variable and write a map or an equation as a rule in these coordinates. By iteratively looking at the degrees of the projective coordinates at each vertex, expressed as functions of the initial conditions after canceling common factors, we obtain the degree sequence of the rule. The next step is to infer a generating function for this sequence and hence extrapolate to calculate algebraic entropy for the rule. It seems that the underlying reason for many integrable maps and lattice equations to have vanishing entropy has not been focused on until recently [14,15,23,20,29]. In [23] we were able to prove polynomial degree growth of a large class of lattice equations (autonomous and nonautonomous) subject to a conjecture. The key ingredient for our approach was to find a recursive formula for the greatest common factor of the coordinate functions and then derive a linear recurrence relation for the actual degrees. The results for lattice equations could then be applied to mappings obtained as reductions of the lattice equations (with some exceptions). In general, many known integrable lattice equations were shown to share the same universal linear recurrence for the actual degrees (see eq. (10) below) which led to quadratic growth.
We note that there is a sub-class of integrable lattice equations which are linearizable, i.e. equations can be brought to linear equations or systems after some rational transformations cf. [2,16,17,18,19,22,24]). One quick test for linearization is using the algebraic entropy test. Linear growth of degrees of an equation indicates that this equation can be linearized. One then can use the symmetry approach given in [17] to transform it to a linear equation or a system of linear equations. Similarly to [23], the first question that arises here is what are the recurrence relations for the actual degrees of some known linearizable lattice equations. If there is such a recurrence, is it related to the recurrence that we have found previously? On the other hand, starting from the recurrence relation that shows quadratic growth, can we find some specialization of it that implies linear growth? Can we then search for candidate equations that satisfy these relations?
In this paper, we try to answer some of these questions. This paper is organized as follows. In section 2, we briefly set up all the notations that will be used for the paper. In section 3, we present some recurrence relations for the actual degrees to have linear growth, i.e. (i)H, (ii)H and (iii)H of Theorem 3.1. In section 4, we show that the recursive formulas for the greatest common factor at each vertex of the discrete Liouville equation [2], Ramani-Joshi-Grammaticos-Tamizhmani equation [22] and the (3, −1) reduction of pKdV [14] give us these degree recurrence relations. We then use the associated linear equations of the lattice equations mentioned in section 4 to prove linear growth of the original equations in section 5. A search for examples of equations on quad-graphs that satisfy those relations is carried out in section 6. This is done by constructing recursive relations for the common factors based on the linear recurrences given in section 3. The Ramani-Joshi-Grammaticos-Tamizhmani equation is obtained though this search. A nice property of algebraic entropy is that it is invariant under birational transformation but need not be under non-rational transformations, a point highlighted in the Appendix.

2.
Setting. In this section, we give a procedure to compute degrees of lattice equations defined on a square. This procedure was presented in detail in [23]. We consider a multi-affine equation (i.e. linear in each variable) on the quad-graph where in the second function, we highlight the shorthand we sometimes use: u = u l,m and subscripts 1, 2 denote the shifts in the l and m directions, respectively. One can solve uniquely for each vertex variable of this equation. Suppose that we solve for the top right vertex variable u 12 . By introducing projective coordinates u = x/z etc., we obtain the rule at the top right vertex in projective form: , The functions f (1) and f (2) are homogeneous of degree 3 in their variables, with one term from each vertex. Initial or boundary values in the literature are typically given either on the boundary of the first quadrant 'corner initial values': or on the (1, −1) 'staircase': They are given by blue paths in the following figure. The staircase boundary condition leads to an evolution of degrees on parallel staircases, the degrees being a function of one index labelling the staircase, a case that has been largely studied [7,12,13,27]. The corner initial values lead to the degree being a function of the 2 coordinates (l, m) on the lattice and is a more general situation. Our results for it should apply (via a linear transformation) to other wedge-shaped boundaries [23,26]. We choose initial values for x and z as polynomials in a variable w of the same degree. Using the rule for the top right vertex u 12 , one can find all the points on the right hand side of these boundaries as polynomials in w.
In [23] we conjectured a recursive formula for gcd l,m which led to a recurrence relation for the actual degreesd l,m . For all integrable lattice equations considered in that paper, the recurrence relation is d l−1,m−1 +d l,m+1 +d l+1,m =d l+1,m+1 +d l−1,m +d l,m−1 .
As indicated in Figure 2 the recurrence relation (10) guarantees that the sum of the degrees on the blue vertices and on the red vertices are the same.
3. Conditions for linear growth. In this section, we show that specializations of equation (10) imply linear degree growth.
The equation (10) is equivalent to any of the rearranged equations below: where the right-hand side in each case is the shift of the left-hand side in the diagonal, vertical and horizontal directions, respectively. These equations are illustrated in Figure 3 where the ± denote the coefficients on either side of each equation and again the weighted sum of the blue vertices equals to the weighted sum of the red vertices (where 'weight' refers to the sign of the coefficient). Figure 3. Equations equivalent to equation (10) Therefore, we have Proposition 1. (10) is equivalent to any of the following equivalent state-

Proposition 1. Equation
where k is a function that only depends on l − m, l ,m, respectively.
Proof. We prove that equation (10) is equivalent to (i). We denote the right-hand side of (11) as k l,m . We have k l−1,m−1 = k l,m . It implies that k l,m = k l−m,0 if l ≥ m or k l,m = k 0,m−l if l < m. This means that k l,m depends only on l − m i.e. k l,m = k(l − m). On the other hand, if k l,m = k(l − m), we have k l−1,m−1 = k l,m . This is exactly equation (11) which is equivalent to equation (10). Similarly, one can prove that equation (10) is equivalent to (ii) and (10) is equivalent to (iii). Hence (i), (ii), (iii) are all equivalent themselves.
It has been proved in [23, Theorem 9] thatd l,m has quadratic growth along the diagonals with unit slope when corner initial values are affine in w. Therefore, in general equations (i), (ii) and (iii) of Proposition 1 give such quadratic growth. However, there are special cases that give us linear growth. For example, the (1, −1) staircase version, or the I 2 initial boundary condition (3) of Proposition 1 (10), (i), where k n = k n+2 for the second equation and k n = k n+1 for the third equation.
For the last two equations if k n = 0 thend n has quadratic growth asd n+1 −d n is linear. However, if k n = 0, it is easy to see thatd n grows linearly as the characteristic equations for the last two equations are (λ − 1) 2 = 0 and (λ − 1) 2 (λ + 1) = 0.
On the other hand, with corner boundary conditions (2) in Proposition 1 (i) or slight variants of them to reflect the geometries in (ii) and (iii) (see Figure 3), we have the following Theorem for the homogeneous cases of (i), (ii) and (iii). Proof. 1. For the homogeneous equation (i), we haved l,m =d l0,m +d l,m0 −d l0,m0 . Therefore, ifd l0,m andd l,m0 are linear functions in m and l respectively, thend l,m grows linearly.
Thus we havē It shows that for each fixed j,d l,m0+j grows linearly along the horizontal direction. 3. For the third statement, we just need to swap l and m and then it becomes the second statement.
Recall that the (q, −p) reduction of a lattice equation, where q, p are positive co-prime integers, gives us an ordinary difference equation of order (p + q) for the variable V n := u l,m , where n = lp + mq + 1 cf. [21,23].
Each of the characteristic equations has 1 as a double root and other roots which are distinct roots of unity. 4. Gcd recursions and linear growth of some linearizable equations. Referring back to the method given in [23], we will provide in this section recursive formulas for the gcd of some examples of linearizable equations previously identified in the literature. As a result of these recurrences, the actual degrees of these equations satisfy the homogeneous equation (i)H, and a reduction of (ii)H from Theorem 3.1. We note that these recurrences have been checked using Maple on lattice squares of size given below. Some of these recurrences will be proved rigorously in section 5.

Liouville equation.
In this section, we study growth of degrees of the discrete Liouville equation. The discrete Liouville equation is given as follows The discrete Liouville equation was first introduced by Adler and Startsev [2]. It is known that this equation is Darboux integrable and linearizable. Therefore, it should have linear growth. In fact, this equation is equivalent to Equation 22 in [13] which has been checked to have linear growth with staircase initial values I 2 of (3) of Figure 1 (right). We now consider corner initial values I 1 of (2) of Figure 1 (left).

DINH T. TRAN AND JOHN A. G. ROBERTS
where parameters r, s, t are free functions in l, m. In terms of projective coordinates, we have x l+1,m+1 = r l,m s l,m+1 x l,m x l,m+1 z l+1,m + r l,m t l,m+1 x l,m z l,m+1 z l+1,m − r l,m+1 s l,m x l,m x l+1,m z l,m+1 − r l,m+1 t l,m x l+1,m z l,m z l,m+1 Using the same method described in section 4.1, one finds that gcd 2,1 = x 1,0 and gcd 1,2 = s 0,1 x 0,1 + t 0,1 z 0,1 . Therefore, we can build the recurrence for l, m ≥ 2. If l = 0 or m = 0 or (l, m) = (1, 1), we take G l,m = 1. If l = 1, m > 1 we take G l,m = G l,m−1 (s l−1,m−1 x l−1,m−1 + t l−1,m−1 z l−1,m−1 ). If l > 1, m = 1, we take G l,m = G l−1,m x l−1,m−1 . We have checked with random integers for r, s, t at each edge and random initial values as polynomials of degree 1 in w that G l,m = gcd l,m (up to a constant factor) for l, m ≤ 12. Assuming that this holds in general for all l, m > 1, the analysis of the previous subsection shows that equation (28) implies equation (24), or (i)H for the degrees. Moreover, it is easy to see thatd l,m = l+m+1 for l, m ≥ 1. Hence,d l,m grows linearly.
Therefore for n ≥ 5, we have It suggests that we should try the following recursive formula for n > 4 and G n = gcd n for n ≤ 8.
We now take initial values as random polynomials of degree 1 in w. We have checked for n ≤ 40 that G n = gcd n (up to a constant factor). Thus, we conjecture that gcd n = G n . Taking degrees of both sides of (33), we obtain This implies thatd which is equation (18) with (q, p) = (3, −1), i.e. a reduction of (ii)H of Theorem 3.1. The characteristic equation for this linear equation is This equation has the following roots: 1 (double root), −1, i, −i. This meansd n grows linearly for n ≥ 5. On the other hand, we know thatd i = 1 for 1 ≤ i ≤ 4, andd i = 2i − 7 for i = 5 ≤ i ≤ 8. It is easy to prove thatd n = 2n − 7 for n ≥ 5. It again confirms that the sequenced n has linear growth. We remark that a direct proof of this also follows from the linearized version of (30) presented in [14].

Proof of linear growth of certain lattice equations via their corresponding linearizations. It is clear that a linear lattice equation has linear growth.
However, it is not trivial to prove linear growth of some linearizable equations. In this section, we provide rigorous proofs of linear growth of some equations given in section 4 by proving their gcd recurrences. Our method is based on their associated linear equations and the existence of rational transformations that transform them to these linear equations. Non-rational linearizing transformations need not preserve linear growth properties -see Appendix for some examples.
Proof. By iterating the rule, it is easy to see that u l,m is a linear combinations of initial values given on the first quadrant, i.e.
This gives us the following linear equation for v v l+1,m+1 + r l,m v l+1,m = C l (s l,m v l,m+1 + t l,m v l,m ), where C l = v l+1,1 +r l,0 v l+1,0 s l,0 v l,1 +t l,0 v l,0 . We note that this is a complicated version of (36). We take general corner boundary initial values for the original RJGT equation (25) , i.e. taking {x i,0 , z i,0 , x 0,j , z 0,j } i,j≥0 as initial values. This helps to build initial values for equation (45) as follows. For l ≥ 0, m ≥ 1, we take v l,m = X l,m Z l,m , X l,0 = z l,0 , Z l,0 = 1, We notice that some of the initial values of (45) lie on the line y = 1 corresponding to m = 1. This is because equation (45) becomes a trivial equation when m = 1. This boundary condition also defines the parameter C l = (x l+1,0 + r l,0 z l+1,0 )/(s l,0 x l,0 + t l,0 z l,0 ). We want to prove (28) using the linear equation (45).
Moreover, we note that the degree of C i in each k l,m can not exceed 1, i.e. k l,m is affine linear in C i . In particular, the total degree of variables C 0 , . . . , C l−1 in k l,m 0,j is l and in k l,m i,1 is l − i. This can be done by induction on N = l + m. Since v i,1 = x i,0 , using formulas for C i and v 0,j = X 0,j /Z 0,j given above, we havē Z l,m = z 0,1 . . . z 0,m−1 (s 0,0 x 0,0 + t 0,0 z 0,0 ) . . . (s l−1,0 x l−1,0 + t l−1,0 z l−1,0 ), which is also a common divisor for all the terms in (46) andZ l,m is the denominator of v l,m after cancelling some common factors of X l,m and Z l,m . This implies that deg(X l,m ) = l + m (by adding the degree of v i,1 to degZ l,m ). Thus, we have It implies that degx l,m = degz l,m ≤ l + m. If we takex l,m =X l,m+1 then z l,m =X l,m z 0,m =x l,m−1 z 0,m . We also know that for the RJGT equation (25) In this section we prove linear growth of the discrete Liouville equation (20). This equation is linearized by the three-point transformation given by cf. [2] where v l,m satisfies the linear equation This equation implies that v l,m = v l,0 + v 0,m − v 0,0 .
Initial values for Liouville equation are given on the first quadrant which are {x l,0 , z l,0 , x 0,m , z 0,m } l,m≥0 . We want to express v 0,j and v i,0 in terms of these initial values and v 0,0 , v 1,0 , v 0,1 . We denote K = v 0,1 − v 0,0 and On the horizontal axis we have This yields Therefore, we obtain on iterating Similarly, for the vertical axis we have By choosing K = v 0,0 and v 0,0 = 1 and using the fomulas for u 0,0 , we obtain .
We also have It is easy to check thatx l,m andz l,m do not have any common factors asz l,m does not vanish at any zero ofx l,m . This implies that deg u l,m = 2(l + m).
If k = 1, this equation is actually the RJGT equation (25) we considered above, which is known to be linearizable [22,17].
We can use the similar argument for relation (iii)H. Thus, we can take respectively for (ii)H and (iii)H: We assume that this common factor B 2,1 first appears at vertex (2, 1). In addition, we try to mimic the behaviour that we got with the (3, −1) reduction of H 1 ; that is we assume that B 2 2,1 |A 2,2 . We follow the steps that were given above to search for equations, however in this case we do not obtain any non-degenerate equations.
We can do similarly by swapping l with m and we do not obtain any non-degenerate equations either.
7. Conclusion. In this paper, based on the recurrence relation (10) found previously for the degrees of many integrable lattice equations [23], we derived some linear recurrences (i)H, (ii)H and (ii)H of Theorem 3.1 that are special cases of (10) and imply linear degree growth. We then used these recurrences to build recursive formulas for the gcds. Thus, we were able to search for examples of linearizable equations with certain gcd patterns, for example the RJGT equation and discrete Burgers equation. A symmetry method given in [17] was used to linearize the RJGT equation. Moreover, we have proved linear degree growth of certain equations such as the Liouville equation, RJGT equation and lattice equations linearizable via Möbius transformations (for example see [17]) by using their associated linear equations. We have also noted in the Appendix that there are linearizable equations with exponential growth [9,19]. This is because the transformations to bring these equations to linear equations are not rational.
We have also noticed that some other linearizable equations such as Equation 15 in [12] where p 3 = 0 and another form of Liouville equation given in [16] satisfy the recurrence (60) and hence satisfy (i)H. Furthermore, we have not found a lattice equation whose gcd relations give us the homogeneous equations (ii)H or (iii)H of Theorem 3.1 directly. Moreover, it is known that there are also other linearizable equations (for example see [6,10]) that we have not studied in this paper. It would be interesting to study their gcd recurrences in the future.