Algebraic entropy of (integrable) lattice equations and their reductions
John A. G. Roberts, Dinh T. Tran
TL;DR
This paper investigates the algebraic entropy of lattice equations, demonstrating polynomial growth for integrable cases and exponential growth for non-integrable ones, using degree growth analysis and conjectures.
Contribution
It introduces a universal method to determine degree growth in lattice equations, linking algebraic entropy to integrability and providing new insights into their complexity.
Findings
Polynomial degree growth in integrable lattice equations
Exponential degree growth in non-integrable equations
A universal linear partial difference equation for degree growth
Abstract
We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler-Bobenko-Suris equations and Viallet's and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain…
| mKdV | |
|---|---|
| sG | |
| E16 | |
| E21 | |
| E22 | |
| E24 | |
| E25 |
| KdV, MX | |
|---|---|
| Tz | |
| mLV |
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Algebraic entropy of (integrable) lattice equations and their reductions
John A. G. Roberts, Dinh T. Tran
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
[email protected], [email protected]
Abstract.
We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler-Bobenko-Suris equations and Viallet’s and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain from this linear partial difference equation an ordinary difference equation for degrees that implies quadratic or linear degree growth. We also study growth of degree of several non-integrable lattice equations. Exponential growth of degrees of these equations, and their mapping reductions, is also proved subject to a conjecture.
1. Introduction
In recent years, there has been growing interest in discrete dynamical systems (e.g. partial or ordinary difference equations) that are integrable – see [10] and references therein. Similar to their continuous time and space counterparts, the term integrable in discrete dynamical systems refers to possession of one or more signature properties (e.g. singularity confinement, Lax pairs, consistency around a cube) that imply ‘low complexity’ of the dynamics relative to the generic behaviour expected. As an example, when we iterate a rational map in dimensions, we typically expect the maximum of the degrees of numerator and denominator to grow exponentially with the number of iterations. If we work projectively in dimensions, the map transforms to a polynomial map of the projective space, with components that are homogeneous of degree . If we let be the degree of , we can define the algebraic entropy to be [3]
[TABLE]
Typically, . However, in some special cases, systematic cancellations may occur between numerator and denominator that may lead to slow (i.e. subexponential) growth whence , the case of vanishing entropy. This property in itself may be taken as a definition of integrability and it has received much recent attention, also in connection to other related integrability properties/detectors ([3, 5, 8, 9, 11, 16, 17, 23, 25, 32, 35]).
In this paper, we investigate algebraic entropy of partial difference equations in the plane, or lattice equations. Consider a square lattice whose sites have coordinates . Field variables are defined on each site of the lattice (see Figure 1 of Section 2) and we assume that on each lattice square there is an equation which is multi-affine in the variables, i.e. affine in each variable (e.g. see equation (10) of Section 2). This equation allows to solve for the field variable on any vertex of the square as a rational function of the field variables on the other three vertices. Coefficients in this equation might depend on the lattice site , in which case the equation is termed non-autonomous and autonomous otherwise. Historically, it has often been the case that the former has been obained from the latter for a class of equations via a deautonomizing process that maintains the integrability signature of singular confinement [5, 8, 11]. This includes non-autonomous versions of the Adler-Bobenko-Suris (ABS) equations, modified Korteweg-De Vries (mKdV) and sine-Gordon (sG) equations [5, 6, 26, 28].
The extension of algebraic entropy to lattice equations was begun in [33]. In the main, lattice equations have been studied by specifying initial conditions along a staircase (cf. Figure 5). These initial condtions are taken to be affine in an indeterminate variable and the lattice rule allows an evolution of parallel staircases to the initial one (e.g. upwards in Figure 5). Denoting by the degree of the polynomial in the indeterminate on this evolving sequence of staircases, Viallet proposed to use the same definition (1) for the algebraic entropy of the lattice equation. Numerous numerical investigations support the idea that, when not exponential, grows quadratically (or linearly in special cases). Again, the lattice equations where this subexponential growth occurs typically have other properties associated with integrability (e.g. singularity confinement [6]).
Our approach here is to consider directly the degree growth of the partial difference equation without taking a one-dimensional reduction of the degrees. Instead of boundary values on a staircase, we take corner boundary values along the boundary of the first quadrant with the boundary values at most affine in an indeterminate. Using the lattice equation, we can calculate the field variables inside the quadrant starting from the vertex. We record the degree in the indeterminate of the numerator (or denominator) of the field variable at each site, denoted . In practice – see Section 2 below – we write each field variable in projective coordinates, that is we introduce and follow the evolution of and via a projective version of the lattice rule. Then , where is the degree of and and is the degree of their greatest common divisor (which is cancelled in numerator and denominator of the associated field variable to given the reduced degree ). Generically, one expects no common factor to and and the ‘magic’ is that for certain lattice equations, the common factors grow fast enough that grows subexponentially.
We summarise our main results:
- One generically has for any multi-affine rule (6) that
[TABLE]
which corresponds to exponential degree growth – see Section 3. However, some lattice rules distinguish themselves by a systematic factorization of and and the appearance of common factors. This can lead to sub-exponential growth of if there is sufficient exponential growth of .
- For a large class of lattice equations – see Table 1 above the double-line division and Table 2 – we find that for the projective version of these rules iterated on any lattice block, the iterates and at the top right corner share a common factor. We denote this factor by and find it depends only on the variables in the lattice block at the sites and . The factor is homogeneous of degree in these variables with
[TABLE]
This local factorization provides the ‘engine’ that leads to continuous factor generation and sub-exponential growth.
Suppose the values are generated from the following recurrence involving the aforementioned :
[TABLE]
and the boundary values in the first quadrant are chosen to agree with those of our lattice equation’s when or . Then we conjecture for a class of lattice equations specified in Table 2 that this remains true on iteration of both (4) and the corresponding lattice equation. Consequently,
[TABLE]
- If the previous conjecture is true for the considered lattice equations, the reduced degree sequence satisfies the the following linear partial difference equation with constant coefficients:
[TABLE]
If the corner boundary degrees and are bounded polynomially of degree in their respective coordinates and , then the sequence of (53) is bounded polynomially of degree along each lattice diagonal. Therefore for fixed , i.e. zero algebraic entropy.
-
The use of periodic staircases is a standard way to derive integrable maps from integrable lattice equations. A consequence of (5) is that it automatically implies quadratic degree growth (at most) for these staircase reductions, as has been found experimentally. Thus, we regard (5) as a master equation to describe the vanishing lattice algebraic entropy.
-
We also provide examples of lattice maps that also develop a factor over a lattice block, but of lesser degree than (3). We show explicitly that, subject to a postulated recurrence for similar to (4), their degree growth is exponential on the lattice and in reduction.
This paper is organized as follows. In section 2, we give a setting to measure the complexity of a certain lattice equation. Using the equation, one can easily write down each vertex in projective coordinates and obtain the corresponding rules in projective coordinates. In this section, we also give a list of integrable lattice equations that we consider in this paper. In the next section, we explore growth of degrees over . Initial values are given as polynomials in on the horizontal and vertical axes of the square in the first quadrant. We calculate an upper bound for degrees of multi-linear lattice equations. In section 4, we present a conjecture that seems to hold for equations in the Adler-Bobenko-Suris (ABS) list and their non autonomous versions whose parameters are given in strips and , non-autonomous and other equations in the Hietarinta-Viallet list. Based on the factorization at the point , we give a recursive formula to build a common divisor of the the numerator and denominator of each vertex. This gives an upper bound for the actual degree of each vertex. By considering two types of initial values, we are able to show the polynomial growth of these equations. In particular, using initial values as ratios of degree-one-polynomials, we can explain the quadratic growth of the equations which was given numerically by Viallet [7, 33]. We also study growth of degrees of the discrete Korteweg–de Vries (KdV) equation [21], Lotka-Volterra (LV) equation [18], modified-modified Volterra (mLV) equation [19], Tzitzeica (Tz) equation [1] and the equation given by Mikhailov and Xentinidis (MX equation) [20]. We will show that even these equations do not have any common factor at the point , one can bring them to the previous cases by shifting the starting point. In section 5, we present a similar conjecture for some non-integrable equations in the paper [12]. This conjecture then helps us to prove that these equations have exponential growth. In section 6, we apply results given in section 4 and section 5 to staircase boundary conditions. Especially, when the staircase boundary values are periodic, we obtain results for growth of degrees of mappings.
2. The setting and some preliminary results
In this section, we set up a methodology to measure the complexity of certain lattice equations upon iteration, following aspects of the foundational papers [12, 31, 33]. The methodology is valid for both autonomous and non-autonomous equations. We are given a square lattice (see Figure 1) whose sites have coordinates . Field variables are defined on each lattice site and are related via an equation which is multi-affine in these variables, i.e. affine in each variable:
[TABLE]
Coefficients in this equation may depend most generally on the lattice site , called the non-autonomous case, accounting for the subscript on . In the autonomous case, the rule is the same on each lattice square. Generically, one can solve (6) for and obtain
[TABLE]
where and are multi-affine in their arguments with no common factor. We assume that each of the three arguments appears in at least one of and (so, in particular, both and are non-zero and not both constant). Now we write each field variable in projective coordinates, that is we introduce etc. We obtain the projective version of (7)
[TABLE]
where and are homogeneous polynomials of degree . For example, for the non-autonomous version of the lattice rule given by
[TABLE]
we obtain its projective version as
[TABLE]
This example encapsulates the general property:
Observation 1**.**
The projective coordinates and at the top right vertex of Figure 1 are homogeneous polynomials (8) and (9) of degree , where each term of these polynomials includes exactly one projective coordinate from each of the remaining three vertices of the square.
Table 1 is the list of non-autonomous and autonomous lattice equations that we will consider in this paper. For ease of notation in listing them, we denote and . The equations satisfy the assumption of the form (7) above.
In this table, we use the name and , for equations given by Adler-Bobenko-Suris (ABS) [2] with parameters given on strips [28, 29], i.e. lattice parameters depends on and depend on only. Also given are other well-known discrete integrable systems such as KdV, MX, LV, mLV and Tz equations [1, 2, 13, 18, 19, 20]. The non-autonomous mKdV and sG were introduced in [26]. The non-autonomous KdV equation was studied in [15]. In particular, the autonomous KdV equation can be obtained from two copies of autonomous [21]. The LV equation can be obtained from the autonomous KdV equation via a Miura transformation [22]. The rest of the list in Table 1 denoted by E comprises lattice equations introduced by Hietarinta and Viallet [12]. Their degree growth was studied using a staircase initial condition. Among these equations, equations E21 and E22 can be reduced to autonomous versions of and in the ABS list. Equations above the double line division in Table 1 are known (or suggested) to be integrable via one or more standard properties (consistency, Lax pair, singularity confinement, numerically-observed low degree growth using a staircase initial condition). Equations below the double line were shown in [12] to have exponential degree growth using a staircase initial condition. We will study (or restudy) all equations in Table 1 with corner boundary values.
In addition to Table 1, we also consider a non-autonomous version of the lattice equation [34] that recently appeared in [7]:
[TABLE]
It has been checked numerically that this equation has polynomial growth and it follows the same degree pattern as seen originally in [7, 34]. Moreover, can be obtained from by setting all coefficients that depend on and to be [math]. Recall that the significance of is that all the equations in the ABS list can be obtained from it by choosing the parameters appropriately.
In this paper, we derive the projective versions of the non-autonomous and autonomous rules of Table 1 in the way described above (some can be found in [24]). We prescribe corner initial conditions or corner boundary conditions, as previously considered in e.g. [12, 31] and also in [24]. That is, we prescribe the values and and and with , on the borders of the first quadrant of the lattice. We work out from the origin using these initial conditions and the projective lattice rule to generate the top right entries in each square (the rule can depend on the lattice site for non-autonomous equations). It is clear that and with , are both multinomial expressions in the variables given by the initial conditions and , , and and , . In the non-autonomous case, these expressions will also contain the parameters from the lattice rules involved. Furthermore, one may want to put the corner boundary conditions at a point other than the origin to sample all possible arrangements of lattice rules. For the non-autonomous , for example, one can start with one of the following points to sample all possible parities at the corner. However, we can always bring the corner point to the origin by shifting so for the rest of our paper, we use the origin as a starting point and initial values are given on the border of the first quadrant.
To probe the phenomenon of cancellation at the heart of algebraic entropy, following [12, 31, 33], we take the initial values and and and , where , as polynomials in an indeterminate . 111The specialisation to univariate polynomials also enables concepts like greatest common divisor, which are not always defined in the multinomial case. Using the lattice rule, one can calculate and with . We factor and (over ) and define to be their greatest common divisor so we can write
[TABLE]
As usual, is taken to be a monic polynomial and and are relatively prime if and only if . Suppressing the -dependence for ease of presentation, define the non-negative integers
[TABLE]
The key issue (for algebraic entropy) relates to the growth of the degree of . Because and are considered projectively, when , they can be replaced by their barred versions. This corresponds to the cancellation of from the numerator and denominator of the rational function . Correspondingly, is replaced by the lesser as the (true) degree at the vertex. This was observed previously in [12, 31]. One manifestation of this factorization process (leading to a diagnostic for detecting it) is that if the corner initial conditions are taken to be integers, the resulting integers and after cancellation of common integer factors would be smaller in magnitude than generically expected. This is equivalent to saying in the non-projective setting of (7), and with corner initial conditions taken in , that some lattice rules give a lower height for the rational number than generically expected. This is closely connected to the concept of Diophantine integrability for lattice maps, explored in the last part of [8].
In typical entropy calculations, the is discarded and only its degree is noted. For what follows in the next section, we find it useful to also observe some of its internal structure as we proceed iteratively away from the origin. We have the following properties:
Proposition 2**.**
For the projective lattice rules of Observation 1, we have the following:
** 2. 2.
{\gcd}_{l,m}(w)\;{\gcd}_{l+1,m}(w)\;{\gcd}_{l,m+1}(w)\;\big{|}\;{\gcd}_{l+1,m+1}(w)* which implies when and are not both [math] that*
2a.
2b. {\gcd}_{l,m}(w)^{3}\;\big{|}\;{\gcd}_{l+1,m+1}(w)\implies g_{l+1,m+1}\;\geq\;3\,g_{l,m}
Proof.
It is straightforward from Observation 1 that 1. and 2. hold. The statements of (2a) and (2b) follow easily from the first statement in 2. ∎
Remark 3**.**
If and share the same degree, (16)-(18) become:
[TABLE]
If this degree equality of the two components holds at each of the 3 vertices used in (14)-(15), then generically this is retained with
[TABLE]
Part 2. of Proposition 2 illustrates the nesting and propagation of the greatest common divisors that occurs for growing and growing when there starts to be a non-trivial at some point. At any lattice point in the first quadrant, the of and includes all the greatest common divisors of the polynomials at the lattice points to its left and/or below it. This leads to great multiplicities of factors in the . We remark that [12] used cancellation (i.e. non-trivial ) after steps on the diagonal as a criterion to detect integrable rules and Part 2b. of Proposition 2 shows that once this occurs, it propagates at an exponential rate.
With respect to statement 2. of Proposition 2, we remark that once we factor out the product from and , we are left with of (8) and of (9), but now with the barred arguments as defined by (14) and (15). These polynomials in the barred variables may have a further non-trivial . We define
[TABLE]
so that
[TABLE]
Remark 4**.**
Computationally, it follows that at each vertex, it is most efficient to record the triple , from which one can reconstruct and if necessary. One takes barred variables as the arguments in the right-hand side of (8) and (9) and checks the greatest common divisor, , of the resulting and . This gives of (23) and of (24) whereas is updated via
[TABLE]
3. Growth of ambient degrees before cancellation
Our approach in this section is to identify a linear partial difference equation satisfied by the upper bound on the degree at each vertex, which also represents the generic degree at each vertex before common factors and cancellations are considered. This enables the exponential growth of these degrees to be verified. Once again, the results hold for both autonomous and non-autonomous equations.
3.1. Solution of recurrence for the upper bound on degrees
Part 1. of Proposition 2 shows that an upper bound for the degree is provided by the non-negative integer sequence satisfying
[TABLE]
The value satisfies this same recurrence at any vertex if either or contains the term comprising a maximal degree polynomial in from each of the other 3 vertices. The argument of Remark 3 above suggests that, generically, when the boundary values are chosen with both components at each vertex having the same degree in .
For initial numerical experiments, we take two cases of such boundary values, namely:
[TABLE]
For Case I boundary values, we have and for . For Case II boundary values, we have for .
Our numerical experiments on the equations of Table 1 use Maple to calculate the polynomials and for (i.e. a square of 144 lattice sites based at the origin) and Case I or Case II boundary values with random integer coefficients chosen in the interval . For ABS equations, parameters are also taken randomly with integer values such that the corresponding equations are not degenerate. They confirm the agreement where is calculated from (16) with (8) and (9) and from (26), with the corresponding same boundary values in each case. Consequently, we call the ambient degree at the vertex (i.e. before any analysis of a possible common factor and cancellation of this factor). If, furthermore, there is actually no non-trivial for the two polynomials and (so no cancellations possible), then gives the true degree throughout the lattice.
The solution of (26) with Case I boundary values can be represented as (part of) a semi-infinite array , with the indices and measured vertically, respectively to the right, with respect to the origin at the bottom left corner so . The solution for Case II boundary values corresponds to removing the first column and the last row of this array.
[TABLE]
A formula for the double-indexed sequence is given by the following theorem.
Theorem 5**.**
*Let denote the integer sequence solution satisfying (26)
for all with boundary values and for . For , let be the coefficient of in the Taylor expansion of the following function around [math], i.e,*
[TABLE]
We have for all ,
[TABLE]
Proof.
It is easy to see from (26) that is what results from applying the recurrence with prescribed initial conditions given on the axes as in the statement of the theorem. On the other hand, (30) also shows as the Taylor coefficients of for all and for all . It is adequate to prove that satisfies the same recurrence (26) as . We have
[TABLE]
On the other hand, we also have
[TABLE]
Equating coefficients of , we obtain
[TABLE]
This implies
[TABLE]
which is the same recurrence given by (26).
We now calculate using the following formula for ,
[TABLE]
Expanding , we get
[TABLE]
This gives us the formula given by (31). ∎
We mention that from the particular solution of (31), we can generate the general solution for the recurrence (26) with corner boundary values [25, Proposition 6].
3.2. Exponential growth of the ambient degree
We remark that the sequence is well known as the Delannoy numbers. Apart from the ad hoc approach to our solutions above, an alternative way to study the recurrence (26) is as a linear partial difference equation with constant coefficients, for which the method of bivariate generating functions applies [4, Chapter 6.4]. Generating functions for the full double sequence and diagonal (central) sequence with Case I boundary conditions are known:
[TABLE]
These lead to the asymptotics
[TABLE]
whence the diagonal entropy
[TABLE]
4. Integrable lattice rules-vanishing entropy
In this section, we identify linear recurrences for the degree of the greatest common divisor at each vertex for the lattice equations given in section 2. Together with the recurrence (26), this helps to furnish us with a linear partial difference equation for , whose growth is ultimately our interest. Sometimes, we can solve this recurrence and conclude polynomial growth of degree in the vertex coordinates and , or exponential growth.
4.1. Exponentially growing gcd: the inherited gcd and the spontaneous gcd
As mentioned in the previous section, we use Maple to calculate the polynomials and for and Case I or Case II boundary values (27)-(28) with random integer coefficients chosen in the interval . Now, however, we are interested in the common factor of these polynomials so we follow the procedure outlined in Remark 4 of Section 2. For all autonomous versions of equations given above KdV in Table 1, the degree array of the (see (17) for Case I boundary values (27) is given by
[TABLE]
and for Case II boundary values (28) is
[TABLE]
The zero entries in the first row and column of and reflect the generic case of no common factors in the boundary values, i.e.
[TABLE]
We note that these arrays give a lower bound for the non-autonomous version of these equations, i.e. the degree of the gcd is sometimes higher than for their autonomous versions. One explanation for this is the freedom of coefficients of the original equations, whence there might be extra common factors appearing at some vertices when we iterate those rules. Therefore, it leads to even bigger cancellations which does not affect the polynomial growth of these equations.
The entries of and illustrate the properties Part 2.(a,b) of Proposition 2. The numbers on the diagonal in both cases appear to be growing exponentially with an exponent . This is close to the rate of exponential growth of the maximal degree on the diagonal (32) and exceeds the minimum exponential growth of expected by Part 2.(b). This is because of spontaneous new additions to that augment the product inherited from the 3 other vertices of the lattice square. We call the latter product the inherited and we use the term spontaneous for of (2) and (25) for . Defining
[TABLE]
we have from (25):
[TABLE]
Using (37), we can calculate the array of spontaneous degrees corresponding to (33) and (34) – see the left hand side of, respectively, Figures 2 and 3. Using (29) and the values (33) and (34), we can also calculate the reduced degree values (18) in each case - see the right hand side of Figures 2 and 3. These arrays are found to be the same for all rules given above the KdV equation of Table 1.
The spontaneous degrees reveal more apparent structure than their unbarred counterparts. In (38), , once a non-zero entry appears on a diagonal . In (39), for . We also observe from our numerical experiments:
Observation 6**.**
For the autonomous lattice equations given above the KdV equation of Table 2 and the boundary conditions Case I and Case II, we have found that
[TABLE]
for . This leads to
[TABLE]
For autonomous KdV and MX, it holds for .
In the two figures, the two numbers enclosed by boxes in the left array and by circles in the right array illustrate the relationship (40). In the right array, the three numbers in bold and the three numbers in underlining/italic in illustrate the recurrence (41). It will turn out that (41) is a consequence of Theorem 9 that covers both autonomous and non-autonomous equations of the next section.
4.2. Polynomial growth of integrable lattice rules
We note that sustained occurrences of a spontaneous as we iterate the rule in the first quadrant is a necessary condition for subexponential growth of . As seen in (34), the first gcd appears at the point for Case II initial values. It is actually related to an important observation of [12, 34] which provides a mechanism for this and one that can be verified quickly for a rule. From [12, 34], it is known that for autonomous equations in the ABS list and for any lattice square ], we obtain a common factor of and of (8)-(9) for arbitrary initial values at the 5 corner sites , see Figure 4. This factorization property over any lattice square was used to search for new integrable equations in [12]. For many equations, as indicated in the figure, the common factor can actually be written in terms of coordinates and at just the 2 sites and and might depend also on lattice parameters or the parity of and . We omit all factors that do not depend on variables and .
We find that this property also holds for many non-autonomous versions lattice equations. Table 2 gives the details of for integrable equations in Table 1 above the KdV equation. In the appendix, we give the factor for the non-autonomous of (13). It should be noted that for all equations in Table 2, and in the appendix, the common factor is ‘quartic’ in terms of the off-diagonal variables of the first square.
In summary, we have the following proposition.
Proposition 7**.**
Consider the lattice rules of Table 1 which are mentioned in Table 2 and the non-autonomous of (13). For the projective version of these rules iterated on any lattice block (Figure 4), the iterates and at the top right corner share a common factor that is given in Table 2 or the appendix. This factor depends only on the variables in the lattice block at the sites and and is homogeneous of degree in these variables. Extending the lattice block to the right and upwards to a lattice block with top right vertex at , we find the divisibility property
[TABLE]
In the case that and are taken as polynomials of equal degree in an indeterminate , we have:
[TABLE]
Proof.
The existence and form of for the indicated equations follows from direct computations using maple. ∎
This Proposition is important as it helps us to see how the new gcd appears at each vertex. Assuming corner boundary values as polynomials in an indeterminate , such as (27) and (28), we can assume that at the point :
- •
{\gcd}_{l,m}(w)\;{\gcd}_{l+1,m}(w)\;{\gcd}_{l,m+1}(w)\;\big{|}\;{\gcd}_{l+1,m+1}(w),
- •
where all quantities belong to the ring of integer polynomials . The first statement is part 2 of Proposition 2. From standard divisibility results, suppressing the dependence for brevity, we have :
[TABLE]
Motivated by the form of the divisor on the rhs of (43), we define the sequence by the recurrence:
[TABLE]
Suppose we take Case II boundary values of (28) for such that if or , and we define if or , so that shares the same boundary conditions. We then compare the results of iterating the lattice equations of Table 2 plus the non-autonomous , to find (which involves factoring and at every site) to calculated directly by iterating (44) (which avoids needing to factor at each site). We find that, in fact,
[TABLE]
where is a lattice-dependent constant that depends on the equation for all the autonomous equations and non-autonomous . For other non-autonomous equations, sometimes instead of having (up to a constant factor), is actually a non-trivial divisor of .
We now take a closer look at the rhs of (43). Recalling the definitions of barred variables from (14) and using the form of in Proposition 7, we can write
[TABLE]
where is similar in form to but using the bar variables. As a result of the inheritance property from Proposition 2, we can write
[TABLE]
Here is a divisor of Therefore, using both (46) and (47) in (43):
[TABLE]
which via (25) implies:
[TABLE]
Alternatively, we can also rewrite (43) using just (47) in the denominator and replacing in the numerator using (the downshift) of (25):
[TABLE]
and this too is a divisor of .
Motivated by the first term on the rhs of (50), we consider the following recurrence on each lattice block:
[TABLE]
Extensive numerical investigations for and a range of boundary conditions leads us to the following
Conjecture 8** **(Enabling Conjecture).
Suppose the values for the recurrence (51) in the first quadrant are chosen to agree with when and . Then we conjecture for the lattice equations of Table 2 plus (non-autonomous) that this remains true on iteration of both (51) and the lattice equation:
[TABLE]
*where is a lattice-dependent constant. From (50), our conjecture is equivalent to saying that , (up to a lattice constant). *
The conjecture means that calculating directly for the lattice rules can be replaced by using (51) and the only dependence on the particular lattice rule is through the factor from Table 2. Evaluating still necessitates iterating the lattice rule. If, however, we are only interested in the degree growth through the lattice, the uniformity of across the different rules, expressed by (42), allows a statement for all lattice equations with the same form of .
Recall the reduced degrees of (20) with (37). The conjecture leads to:
Theorem 9**.**
For all the lattice equations given in Table 2 and (non-autonomous) , the reduced degree sequence of (20) satisfies the the following linear partial difference equation with constant coefficients:
[TABLE]
if conjecture (8) is true. If the corner boundary degree differences and are bounded polynomially of degree in their respective coordinates and , then the sequence of (53) is bounded polynomially of degree along each lattice diagonal. Therefore for fixed . In particular, the corner boundary values of Case I (Case II) of (27)-(28) produce linear (quadratic) growth for .
Proof.
Taking degrees in (51), satisfies the following recurrence for
[TABLE]
where the first term comes from (42). Adding this to (21) and its downshifted version gives (53). Alternatively, conjecture (8) gives (40) and writing and using (21) and (37) to replace and , and then downshifting the result to replace will give again (53).
Let so that if, say, ,
[TABLE]
with an analogous expression for the growth of reduced degree along diagonals below the main diagonal.
Equation (53) gives
[TABLE]
which is equivalent to . Iterating this downward in gives
[TABLE]
which can be downshifted in :
[TABLE]
Adding both sides from (57) onwards gives
[TABLE]
which expresses in terms of boundary values. Substitution in (55) yields:
[TABLE]
Suppose and where are polynomials with constant coefficients of respective degrees . Then by standard results, the sum is a polynomial in of degree . Consequently, is bounded by a polynomial in of degree , and so is .
For Case I boundary values, boundary values are constant on both axes. Explicitly, we have and for all and . It implies that and for all . Hence and are constant so and we have linear growth along the diagonal. In fact, directly from (58), for all so the coefficient of linear growth is . This is what is observed in the right hand side of figure 2.
For Case II boundary values, boundary values are affine on both axes in and one can deduce that . Hence and we have quadratic growth in along the diagonal for . Consequently, the second difference of the latter is necessarily constant and can be calculated. We have for
[TABLE]
This value for the second difference is observed in the right hand side of Figure 3. ∎
The coefficient in (54) and (42) proves crucial in the proof of Theorem 9. It is a consequence of the fact that is quartic in its arguments. It will be shown in section 5 that when instead is quadratic in its arguments, the growth of the reduced degree can be exponential.
We remark that some alternative degree equations equivalent to (53) are given in [30]. These equivalent forms prove useful for identifying and studying linearisable lattice equations.
4.3. Polynomial growth of other integrable lattice equations
Consider the integrable equations given in Table 1 that do not appear in Table 2. Although these equations have different factorization patterns compared to the other integrable equations mentioned in the previous subsection, we will show that Conjecture 8 and Theorem 9 still apply.
We start with the KdV equation. By using direct calculation, with arbitrary corner boundary values, we find that the first non-trivial gcd of and appears at the vertices and where and . At the vertex , the gcd is given as follows
[TABLE]
It is noted that this formula is quite large if we write it in terms of the boundary values. Similarly, for mLV equation, we first see the non trivial gcd of and at and . Using Gröbner bases, one can find that
[TABLE]
For the MX equation, at the point the common factor is , a quadratic rather than a quartic factor. However at the vertex we also obtain (60). Since, the MX equation can be seen as a degeneration of the Tz equation [20], it suggests that Tz might behave similarly. By using Maple, we know that at the point , the common factor is ‘quadratic’ which is not enough for our Conjecture 8. Extending to the square, we find that
[TABLE]
Therefore, for these equations one can try to consider as a factor which plays the similar role as the given in the previous section. In general, for , we have the following table of which is a common factor of and .
We recover the setup of the previous subsection by taking . That is, we replicate the situation of Figure 4, except for the fact that the squares over which the factorization emerges starts one column in and one row up from the previous subsection. Numerical experiments support that Conjecture 8 holds for and the fact that is again quartic here implies Theorem 9, yielding polynomial growth for these equations.
We will end this subsection by consider the LV equation. The first non trivial gcd of and appears at the vertex which is earlier than other integrable equations. However, at the vertex the gcd is given as follows
[TABLE]
which does not seem to be enough for Conjecture 8. Extending to the square, one can see that
[TABLE]
This plays the similar role of in the previous subsection. In fact, numerical results show that Conjecture 8 holds for where
[TABLE]
This conjecture again helps us to prove polynomial growth of LV equation.
5. Non integrable equations with factorization
In this section, we consider the equations of Table 1 below the double-line division. These autonomous equations taken from [12] also have some factorizations at the top right corner of any lattice square, as in Figure 4, but the degree of the factor is insufficient to prevent exponential growth of the reduced degrees as found heuristically in [12].
5.1. Equations with ‘quadratic’ factorization
We consider the equations E20, E26, E27, E28 and E30 of Table 1 and present their factors in Table 4. Replicating the approach of section 4.2, we have again that (43) holds and, as before, we find numerically that (44) reproduces up to a lattice-dependent constant, analogous to (45). Because the of Table 4 are quadratic in their arguments rather than quartic, we have (46) and (47) but without the exponent on each . Consequently, we were led to consider the following recurrence
[TABLE]
for and if or . Note that the rhs of (61) is based upon the divisor of (50) after inserting (47) without the exponent on each in the denominator and excepting the factor .
Numerical experiments for different trials of boundary values on lattice squares lead us to
Conjecture 10** (Enabling Conjecture).**
Suppose the values for the recurrence (61) in the first quadrant are chosen to agree with when and . Then for , we find is a divisor of given by the recurrence (61). For lattice rules E20, E26 and E27, we actually find agrees with up to a lattice-dependent constant.
The conjecture leads to:
Theorem 11**.**
If the enabling conjecture 10 holds, the reduced degrees of the lattice equations of Table 4 are bounded below by the sequence satisfying the following linear partial difference equation with constant coefficients:
[TABLE]
Consequently, the equations of Table 4 have non-vanishing entropy.
Proof.
It follows directly from the formula (61) that the degree of satisfies the recurrence
[TABLE]
where . Denote . Using (62) we obtain for
[TABLE]
which forms a Pascal’s triangle. It implies that and . Thus, we also have and . Therefore, we get . In particular, along the diagonal we obtain . Since for non-constant initial values at and . Note that in the case of constant initial values, we have where . It shows that grows exponentially. Hence, grows exponentially. ∎
We note that in the case where initial values are linear, we have found the following.
- •
We can write (62) as follows
[TABLE]
- •
It leads to .
- •
For , we have
[TABLE]
This can be proved by induction.
5.2. Equation
Finally, we consider given in Table 1. This equation has ‘bigger factorization’ than the non-integrable equations mentioned in the previous subsection. However, we will show that the factorization is not big enough to allow this equation to have vanishing entropy. It can be checked, with reference to Figure 4, that the common factor of and is
[TABLE]
It is important to note that factorization appears first at the point , where . We know that are devisors of . We first used the test for as described in subsection 4.2 and we found that is a divisor of the actual . However, we found that
[TABLE]
It implies that
[TABLE]
Therefore, we predict that
[TABLE]
up to a constant. We have tested this recurrence and obtain the following conjecture.
Conjecture 12**.**
Given Case II boundary values, let for or and let be defined by the following recursive formula
[TABLE]
Then up to a constant factor.
Using this conjecture, we have the following corollary
Corollary 13**.**
Given Case II boundary values, then equation has exponential growth along diagonals, i.e. going from to .
Proof.
Using the recursive formula (66), we have . This shows that grows exponentially along the diagonal. ∎
We note that in the case where initial values are linear, for one can prove
[TABLE]
The proof was done by using induction. We note that the Lotka-Volterra equation which is given as follows
[TABLE]
is a special case of , however the Lotka-Volterra equation has the vanishing entropy.
6. Application to staircase boundary values for lattice equations and to maps from periodic staircase reductions
Thus far, we have considered our lattice equations with corner boundary values which, upon iteration, gives all values of the field in the positive quadrant of the lattice. An alternative and commonly-considered case is when the boundary values are considered on a staircase that zig-zags through the vertices of . From this initial staircase, if well-posed, use of the lattice rule allows values of the field variable to be calculated on a sequence of parallel disjoint staircases which cover the lattice. Entropy of the lattice equation in this case has been addressed in [33] . If we now consider a periodic staircase or a travelling-wave solution to the lattice equation, we can reduce the integrable lattice equation, or partial difference equation, to an ordinary difference equation or map. Entropy of birational maps has been considered in [17, 23, 32, 35], including those obtained from reduction of lattice maps.
The purpose of this section is to show how Theorem 9 and the degree relation (53) lend themselves to these other contexts as particular cases of our corner boundary value problem and reproduce degree growth results obtained previously in these contexts. This lends further weight to our Conjecture 8 on which Theorem 9 relies.
6.1. Staircase boundary values
The degree growth for staircase initial conditions was considered in [5, 14, 33]. We consider firstly the simplest case of a staircase with unit horizontal and vertical step sizes and we choose its orientation to be decreasing to the left and downwards – see Figure 5.
The methodology of the aforementioned references is to ascribe affine polynomials of the same degree in for the and values at each point of the staircase, as we did with Case II corner boundary values, with the generic assumption that the polynomials at each site are coprime. By symmetry, it is seen that on each diagonal of slope parallel to the initial staircase, we can assign a degree , using (21), which gives the degree of and along each diagonal. We take for the initial staircase and see that , , etc. Assuming our Conjecture 8 for the equations of Table 2, a nontrivial only arises via the factor that appears over each lattice block of the iterated lattice rule and the degree of the follows (54). Symmetry again shows that we can define the reduced degrees , for each diagonal with for . The first genuinely-reduced degree arises when since a of degree arises as anticipated by (54), so
[TABLE]
To proceed, we use (53) adapted to a corner created at one edge of the original staircase, as indicated in Figure 5. From (53) with our special case reflecting the equality of degree along each diagonal of slope , we have the fourth order linear ordinary difference equation:
[TABLE]
The associated characteristic equation is
[TABLE]
with general solution:
[TABLE]
The four initial conditions given above mean that we have
[TABLE]
This sequence with quadratic growth has been identified numerically for staircases like that of Figure 5 for various equations including (non-autonomous) , and others ([34, Section 6], [14, Table 1], [33, Section 8, 9] [7]). The method involves fitting in each case a rational univariate generating function to the first few computer-generated values of . Finding a rational generating function is equivalent to the sequence satisfying a linear recurrence with integer coefficients.
Our working above shows that the recurrence (71) and solution (72) are an automatic consequence for the lattice equations covered by Theorem 9 with boundary values given on the staircase of Figure 5.
The staircase of Figure 5 is built from the head-to-foot repetition of the finite generating staircase of one step to the right and one step down. More generally, consider any staircase that is constructed from the head-to-foot repetition of a finite staircase, which is assumed to be well-posed. Without loss of generality, we assume the generator staircase starts with a horizontal step and ends with a vertical step and spans a horizontal width from left to right on the lattice and a vertical height from bottom to top between its starting and ending points (i.e. positive (negative) means a rising (descending) staircase from left to right). We call this a -repeating staircase – Figure 5 has . Viallet [33] gives other simple examples, which repeat just one step of horizontal length and one of vertical height .
Note the orientation of the corner is a clockwise rotation of to that we considered earlier and our south-easterly propagation of parallel staircases entails solving for the bottom right vertex of our quad rather than the top right. However, one checks that all the equations of Table 2 are form-invariant under the transformations which correspond to the clockwise -rotation, namely:
[TABLE]
6.2. Applications to mappings obtained as reductions of periodic staircases
In this section we show that our conjectures still hold for mappings obtained reductions of lattice equations via the staircase method. In particular, we consider the reductions where and . We first present the reduction. Then, we give consequences of conjectures (8), (10) and (12) for the corresponding maps.
Recall that the reduction of lattice equations gives us a -dimensional map. The reduction is described as follows. We introduce the travelling-wave ansatz:
[TABLE]
This ansatz imposes the periodicity condition
[TABLE]
and, conversely, this periodicity condition implies that all fields on the line are the same, provided and are coprime. Hence field variables are now defined on each parallel line of slope . We make a choice of origin on the lattice, at which we place . Since , one can always find and such that for each . This set of points joined from left to right (increasing by ) and down to up (decreasing by ) yields the staircase of initial conditions. The lattice equation (6) specialises with (73) to the ordinary difference equation for :
[TABLE]
where for consistency we must impose periodicity on the parameters:
[TABLE]
By solving from this equation – which is certainly possible if is multilinear in its arguments – we obtain the map
[TABLE]
where the parameters are . For example, the reduction of a lattice equation is illustrated in Figure 7. We observe there the staircase of initial conditions and the periodicity of it and later iterates of . The ordinary difference equation arising from the lattice rule is:
[TABLE]
6.3. Reductions of integrable lattice equations
The partial difference equation (53) for the reduced degrees that holds over a lattice square specialises via (73) and (7) to the difference equation setting. Now
[TABLE]
and (53) reduces to the ordinary difference equation of order :
[TABLE]
The associated characteristic equation is
[TABLE]
Now for
[TABLE]
By assumption, and are coprime so that precisely one of the set is even and there is no divisor greater than one common within the set. Hence
[TABLE]
Here is the th cyclotomic polynomial. Because of the triple eigenvalue and single eigenvalue , the solution of (75) has a part that is generally quadratic in supplemented with a period-2 part:
[TABLE]
If the last part of (76) is present, its roots come in complex conjugate pairs with and coprime. Each pair of roots leads to a term in the solution of (75) of the form
[TABLE]
where and are two arbitary constants determined by initial conditions. This part of the solution is periodic: with period . In summary, the solution to (75) will be linear or quadratic in , supplemented by a periodic part with period that is the of together with all possible divisors of .
The fact that the case in (75) recovers (70) for the repeating staircase we discussed above highlights something. The equation (75) will be obtained whenever the degrees on a staircase are periodic:
[TABLE]
with and coprime. This does not necessarily require the fields to be periodic on the staircase as in the periodic reduction but certainly (74) implies (77). A case in point is when a staircase is repeating as defined above and the degrees of the fields assigned on the staircase respect this repetition. In fact, saying that (77) holds on a staircase forces its geometry to be repeating as a byproduct.
So we have proved:
Theorem 14**.**
Suppose Conjecture (8) is true for all the lattice equations given in Table 2 and (non-autonomous) . Consider a periodic staircase of boundary values on which the assigned degrees have the periodicity property (77), which includes the case of (non-autonomous) maps obtained from the periodic reduction of the lattice equation. The resulting degree as a function of n is the sum of a polynomial part, at most quadratic in , and an additional (necessarily bounded) periodic part in .
Note the presence or absence of the periodic part can be inferred from taking the first or second difference of : if the first (second) difference is constant, then is linear (quadratic) in with no periodic part. Otherwise, the first or second difference will be itself a periodic function of .
We note that numerous numerical studies [12, 14, 32, 33] of individual reductions of lattice equations relevant to Theorem 14 (e.g. ) have exhibited the linear or quadratic degree growth, sometimes with an additional periodic part as evidenced by a non-constant second difference. Our point here is that the truth of conjecture (8) will guarantee this, uniformly in and .
6.4. Reductions of non-integrable lattice equations
We also apply our results in section 5 to reductions of non-integrable lattice equations.
For equations with ‘quadratic’ factorization (given in Table 3), the recurrence (62) becomes as follows
[TABLE]
This give us the corresponding characteristic equation
[TABLE]
For equation E17, the recursive formula (66) reduces to the following equation
[TABLE]
The characteristic equation of this linear recurrence is
[TABLE]
We note that for , equation (81) and hence equation (79) have at least one real root in . This implies that the sequence grows exponentially for both cases.
7. Discussion
We have presented two conjectures that have been used to obtain some results of growth of degrees of some integrable and non-integrable lattice equations including a non-autonomous version of . In particular, subject to a ocnjecture we have shown that a non-autonomous version of which has been introduced recently in [7] has vanishing entropy. Given an autonomous equation or a non-autonomous equation on the square, by looking at the factorization locally at the top right corner of any square, one might be able to predict the integrability in the sense of having vanishing entropy. However, the question here is ‘how big is that factor in order to have vanishing entropy’. We have given some examples in which the algebraic entropy is vanishing. It seems that for these equations, the common factor needs to be ‘quartic’ in terms of the off diagonal variables of the first square. In addition, this factor needs to have the divisible property, i.e. .
Furthermore, we also gave some examples where the common factor is ‘quadratic’ in terms of the off diagonal variables of the first square. These equations turn out to be non-integrable as they have non-vanishing entropy. Therefore, it is worth studying this problem in the future.
In addition, there are some integrable equations where the factorization behaves in a more complicated way such as Tzitzeica and Lotka-Volterra equations. With these equations we need to go beyond the square. We have not been able to provide any recursive formula of the gcd for these equations. Thus, it might be interesting to investigate this problem further. Furthermore, there are non-linear affine equations which are integrable in the sense of possessing a Lax pair [27]. It would be worth to study growth of degrees of these equations as well.
Acknowledgements
This work was supported by the Australian Research Council. We would like to thank Joshua Capel for useful discussions.
Appendix
In this Appendix, we give a factor for the autonomous and non-autonomous equations [7, 34] – see Figure 4. We distinguish cases for the non-autonomous equation of (13) based on the parity of and . We denote and . The factor at the point of is given by cf. [25, 34, 35]
[TABLE]
The factor at the top right corner of any square of the non-autonomous is given as follows.
- •
Case 1 where , we have
[TABLE]
- •
Case 2 where , we obtain
[TABLE]
- •
Case 3 where , we obtain
[TABLE]
- •
Case 4 where , we obtain
[TABLE]
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