## Laurent Polynomials and Superintegrable Maps (2007)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Hone07laurentpolynomials,

author = {Andrew N. W. Hone},

title = {Laurent Polynomials and Superintegrable Maps },

year = {2007}

}

### OpenURL

### Abstract

### Citations

818 |
The arithmetic of elliptic curves
- Silverman
- 1986
(Show Context)
Citation Context ...= 1, x2, x3, x4 ∈ Z with x2|x4, satisfy the divisibility property xm|xn whenever m|n, and correspond to values of the division polynomials of the curve (for a description of these see Exercise 3.7 in =-=[48]-=-). In this sense, an EDS generalizes properties of certain linear recurrence sequences. For example, the Fibonacci numbers are generated by the recurrence Fn+1 = Fn +Fn−1 with initial values F0 = 1, F... |

376 |
The On-Line Encyclopedia of Integer Sequences,” http://www.research.att.com/∼njas/sequences/index.html
- Sloane
(Show Context)
Citation Context ... elliptic theta functions. Somos observed numerically that by taking the coefficients α = β = 1 and initial data x0 = x1 = x2 = x3 = 1, the fourth-order recurrence (2.4) yields a sequence of integers =-=[50]-=-, that is 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, . . . . (2.5) Similarly he noticed that for the Somos-k recurrences xn+k xn = [k/2] ∑ j=1 αj xn+k−j xn+j (2.6) with all coefficients αj =... |

155 |
Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture
- Bressoud
- 1999
(Show Context)
Citation Context ...ely. A little earlier, when Mills, Robbins and Rumsey made their study of the Dodgson condensation method for computing determinants [36] (which produced the famous alternating sign matrix conjecture =-=[2]-=-), they considered the recurrence Dℓ,m,n+1Dℓ,m,n−1 = α Dℓ+1,m,nDℓ−1,m,n + β Dℓ,m+1,nDℓ,m−1,n, (2.7) for α = 1 and observed that it produced Laurent polynomials in the initial data. The equation (2.7) ... |

56 |
Dressing Chains and Spectral Theory of the Schrödinger Operator. Funktsional’nyi Analiz i ego prilozheniya
- Veselov, Shabat
- 1993
(Show Context)
Citation Context ...onsidered were reduced/stationary flows of the KdV hierarchy, whose BTs could be obtained by reduction from the Darboux–Bäcklund transformation for KdV (this is in the same vein as the dressing chain =-=[60]-=- – see also [63]), while the BTs in [25] were derived more directly. In Vadim’s work with Pol Vanhaecke [34], all of the previously knownLaurent Polynomials and Superintegrable Maps 3 examples were u... |

53 |
Integrable systems in the realm of algebraic geometry, volume 1638
- Vanhaecke
(Show Context)
Citation Context ...discrete shifts on the (generalized) Jacobian of the associated spectral curve, thus identifying them as the discrete-time counterparts of algebraically completely integrable systems, as described in =-=[56]-=-, for instance. While there has been subsequent work by Vadim and others on BTs in classical mechanics [8, 13, 39], a lot of the original motivation for studying them came from quantum integrable syst... |

53 |
Memoir on elliptic divisibility sequences
- Ward
(Show Context)
Citation Context ...ot of a quartic [41]. In fact, Somos-4 sequences have an ancestor from the 1940s, in Morgan Ward’s work on elliptic divisibility sequences (EDS), which just correspond to multiples of a point n P ∈ E =-=[61, 62]-=- i.e. this is the special case P0 = ∞, so that z0 = 0, with the further requirement that A = B = 1 in (2.20). The iterates of an EDS, which are generated by (2.4) with coefficients α = (x2) 2 , β = −x... |

48 |
The Laurent phenomenon
- Fomin, Zelevinsky
(Show Context)
Citation Context ...ion. Another interesting and unexpected property of the Poisson maps considered below is that their iterates are Laurent polynomials in the initial data; this is an instance of the Laurent phenomenon =-=[14]-=-. Vadim was an expert on special functions, and orthogonal polynomials in particular (for one of his many contributions in this area, see [35], for instance). However, most of the sequences of (Lauren... |

41 |
Integrable maps
- Veselov
- 1991
(Show Context)
Citation Context ...elow satisfy nonlinear equations instead of linear ones. The theory of discrete integrable maps has seen a great deal of activity in the past twenty years. The situation was much clarified by Veselov =-=[57, 58, 59]-=- who introduced integrable Lagrange correspondences – a natural discrete-time analogue of Liouville integrable continuous flows – which induce (generically multi-valued) shifts on the associated Liouv... |

37 | Cluster algebras and Poisson geometry
- Gekhtman, Shapiro, et al.
- 2003
(Show Context)
Citation Context ...m, xn }2 = (n − m) xmxn, (2.9) which has Casimirs un = xn−1xn+1 (xn) 2 , n = 1, 2. (2.10) This bracket is of the ‘log-canonical’ type that has previously been found in the context of cluster algebras =-=[18]-=-; it is natural to consider it as a Poisson bracket on the field of rational functions C(x0, x1, x2, x3). (The reason for the subscript 2 on the bracket will become apparent in the next section.) The ... |

36 |
On Bäcklund transformations for many-body systems
- Kuznetsov, Sklyanin
- 1998
(Show Context)
Citation Context ...s obtained for the Toda lattice by Pasquier and Gaudin [38], Kuznetsov and Sklyanin identified a special class of time-discretizations for integrable Hamiltonian systems which they referred to as BTs =-=[32]-=-, by analogy with Bäcklund transformations for evolutionary PDEs. In the setting of finite-dimensional systems with a Lax pair, BTs were identified as explicit Poisson maps which preserve the same set... |

36 |
The periodic Toda chain and a matrix generalization of the Bessel function recursion relations
- Pasquier, M
- 1992
(Show Context)
Citation Context ...eserve the same integrals as the original continuous system (see [52] for the state of the art in integrable discretizations). Building on results obtained for the Toda lattice by Pasquier and Gaudin =-=[38]-=-, Kuznetsov and Sklyanin identified a special class of time-discretizations for integrable Hamiltonian systems which they referred to as BTs [32], by analogy with Bäcklund transformations for evolutio... |

36 | The Problem of Integrable Discretization: Hamiltonian Approach
- Suris
- 2003
(Show Context)
Citation Context ... possible (e.g. Poisson structure, Lax pair, etc.). However, in general such a time-discretization will be implicit, and it will not preserve the same integrals as the original continuous system (see =-=[52]-=- for the state of the art in integrable discretizations). Building on results obtained for the Toda lattice by Pasquier and Gaudin [38], Kuznetsov and Sklyanin identified a special class of time-discr... |

33 | Primes in elliptic divisibility sequences
- Einsiedler, Everest, et al.
(Show Context)
Citation Context ...nerates to an expression in terms of the hyperbolic sine. The arithmetical properties of EDS and Somos sequences – in particular the distribution of primes therein – are a subject of current interest =-=[10, 11, 49]-=-. Some of these properties are discussed in the book [12] (see section 1.1.20, for instance), where it is suggested that such bilinear recurrences should be suitable generalizations of linear ones, wi... |

31 |
The intersection of recurrence sequences
- SCHLICKEWEI, SCHMIDT
- 1995
(Show Context)
Citation Context ...operties of integer sequences generated by linear recurrences have been the subject of a great deal of study in number theory, and nowadays they find applications in computer science and cryptography =-=[12]-=-. However, the theory of nonlinear recurrence sequences is still in its infancy. Clearly, a kth-order nonlinear recurrence relation of the form xn+k = F (xn, xn+1, . . . , xn+k−1) (2.1) is just a part... |

28 |
On the analogue of the division polynomials for hyperelliptic curves
- Cantor
- 1994
(Show Context)
Citation Context ...zations of linear ones, with many analogous features. Based on the appearance of higher-order Somos recurrences in the work of Cantor on the analogues of division polynomials for hyperelliptic curves =-=[6]-=- (see also [40] for analytic formulae), it was conjectured in [27] that every Somos-k sequence should correspond to a discrete linear flow on the Jacobian of such a curve (with an associated discrete ... |

28 |
Viallet C., Singularity confinement and chaos in discrete systems
- Hietarinta
- 1998
(Show Context)
Citation Context ...act that in that case they have nonzero algebraic entropy. Recall that for a rational map, the algebraic entropy is defined as lim n→∞ (log dn)/n, where dn is the degree of the nth iterate of the map =-=[21]-=-. Usually finding this limit requires extensive calculations of the corresponding sequence of rational functions of the initial data, or of the iterates of the projectivized form of the map. However, ... |

27 |
Alternating-sign matrices and descending plane partitions
- Mills, Robbins, et al.
- 1983
(Show Context)
Citation Context ...h the integrality of the particular sequence (2.5) follows immediately. A little earlier, when Mills, Robbins and Rumsey made their study of the Dodgson condensation method for computing determinants =-=[36]-=- (which produced the famous alternating sign matrix conjecture [2]), they considered the recurrence Dℓ,m,n+1Dℓ,m,n−1 = α Dℓ+1,m,nDℓ−1,m,n + β Dℓ,m+1,nDℓ,m−1,n, (2.7) for α = 1 and observed that it pro... |

27 | Perfect matchings and the octahedron recurrence
- Speyer
- 2007
(Show Context)
Citation Context ...s an ordinary difference reduction of the partial difference equation (2.7): it has been noted by Propp that if xn satisfies (2.4) then Dℓ,m,n = x2n+m satisfies the discrete Hirota equation (see also =-=[51]-=- for another reduction). Many more examples of this Laurent property have begun to emerge quite recently as an offshoot of the theory of cluster algebras due to Fomin and Zelevinsky (see [15] and refe... |

26 |
The on–line encyclopedia of integer sequences, e-mail: sequences@research.att.com
- Sloane
(Show Context)
Citation Context ... elliptic theta functions. Somos observed numerically that by taking the coefficients α = β = 1 and initial data x0 = x1 = x2 = x3 = 1, the fourth-order recurrence (2.4) yields a sequence of integers =-=[50]-=-, that is 1,1,1,1,2,3,7,23, 59,314, 1529,8209,83313,... . (2.5) Similarly he noticed that for the Somos-k recurrences xn+k xn = [k/2] ∑ j=1 αj xn+k−j xn+j (2.6) with all coefficients αj = 1, if all k ... |

24 |
Action-angle variables and their generalizations
- Nekhoroshev
- 1972
(Show Context)
Citation Context ...rk 3. The situation whereby an integrable system has more independent conserved quantities than the number of degrees of freedom is known as non-commutative integrability (in the sense of Nekhoroshev =-=[37]-=-), because not all these quantities can be in involution with one another. In this example, J1 Poisson commutes with both J2 and J3, but {J2, J3}1 ̸= 0. The terminology ‘superintegrable’ is applied in... |

23 | The Many Faces of Alternating–Sign Matrices
- Propp
(Show Context)
Citation Context ...erved that it produced Laurent polynomials in the initial data. The equation (2.7) thus became known within the algebraic combinatorics community, where it is referred to as the octahedron recurrence =-=[43]-=-, while in the theory of integrable systems it is known as a particular form of the discrete Hirota equation [68] (the bilinear equation for the taufunction of discrete KP). The Somos-4 recurrence (2.... |

19 | Primes generated by elliptic curves
- Everest, Miller, et al.
- 2003
(Show Context)
Citation Context ...nerates to an expression in terms of the hyperbolic sine. The arithmetical properties of EDS and Somos sequences – in particular the distribution of primes therein – are a subject of current interest =-=[10, 11, 49]-=-. Some of these properties are discussed in the book [12] (see section 1.1.20, for instance), where it is suggested that such bilinear recurrences should be suitable generalizations of linear ones, wi... |

19 |
The strange and surprising saga of the Somos sequences
- Gale
- 1991
(Show Context)
Citation Context ..., 6, 7, but denominators appear for k = 8. Various direct proofs that the terms of the sequence (2.5) are all integers were found at the beginning of the 1990s, when various other examples were found =-=[17, 47]-=-, but a deeper understanding came from the realization that the recurrence (2.4) has the Laurent property: its iterates are all Laurent polynomials in the initial data (and in α, β) with integer coeff... |

19 | Elliptic Curves and Related Sequences
- Swart
- 2003
(Show Context)
Citation Context ... unpublished work of several number theorists2 – see the discussion of Zagier [67], and the results of Elkies quoted in [4]. The algebraic part of the construction is described in the thesis of Swart =-=[53]-=- (who also mentions unpublished results of Nelson Stephens), and van der Poorten has recently presented another construction based on the continued fraction expansion of the square root of a quartic [... |

18 | Bäcklund transformations for finite-dimensional integrable systems: a geometric approach
- Kuznetsov, Vanhaecke
(Show Context)
Citation Context ...the Darboux–Bäcklund transformation for KdV (this is in the same vein as the dressing chain [60] – see also [63]), while the BTs in [25] were derived more directly. In Vadim’s work with Pol Vanhaecke =-=[34]-=-, all of the previously knownLaurent Polynomials and Superintegrable Maps 3 examples were unified via an algebro-geometric approach, which explained the deeper meaning of BTs as discrete shifts on th... |

18 | Hirota’s difference equations
- Zabrodin
- 1997
(Show Context)
Citation Context ...lgebraic combinatorics community, where it is referred to as the octahedron recurrence [43], while in the theory of integrable systems it is known as a particular form of the discrete Hirota equation =-=[68]-=- (the bilinear equation for the taufunction of discrete KP). The Somos-4 recurrence (2.4) is an ordinary difference reduction of the partial difference equation (2.7): it has been noted by Propp that ... |

17 |
Growth and integrability in the dynamics of mappings
- Veselov
- 1992
(Show Context)
Citation Context ...elow satisfy nonlinear equations instead of linear ones. The theory of discrete integrable maps has seen a great deal of activity in the past twenty years. The situation was much clarified by Veselov =-=[57, 58, 59]-=- who introduced integrable Lagrange correspondences – a natural discrete-time analogue of Liouville integrable continuous flows – which induce (generically multi-valued) shifts on the associated Liouv... |

15 | Discrete Painlevé equations - Grammaticos, Nijhoff, et al. - 1999 |

14 | Cluster algebras IV: coefficients
- Fomin, Zelevinsky
(Show Context)
Citation Context ...see also [51] for another reduction). Many more examples of this Laurent property have begun to emerge quite recently as an offshoot of the theory of cluster algebras due to Fomin and Zelevinsky (see =-=[15]-=- and references). The exchange relations in a cluster algebra of rank k are typified by a recurrence of the form xn+k xn = c1 M1(xn+1, . . . , xn+k−1) + c2 M2(xn+1, . . . , xn+k−1) (2.8) for suitable ... |

14 |
The law of repetition of primes in an elliptic divisibility sequence
- Ward
- 1948
(Show Context)
Citation Context ...ot of a quartic [41]. In fact, Somos-4 sequences have an ancestor from the 1940s, in Morgan Ward’s work on elliptic divisibility sequences (EDS), which just correspond to multiples of a point n P ∈ E =-=[61, 62]-=- i.e. this is the special case P0 = ∞, so that z0 = 0, with the further requirement that A = B = 1 in (2.20). The iterates of an EDS, which are generated by (2.4) with coefficients α = (x2) 2 , β = −x... |

14 |
Periodic fixed points of Bäcklund transformations and the KdV equation
- Weiss
- 1986
(Show Context)
Citation Context ...educed/stationary flows of the KdV hierarchy, whose BTs could be obtained by reduction from the Darboux–Bäcklund transformation for KdV (this is in the same vein as the dressing chain [60] – see also =-=[63]-=-), while the BTs in [25] were derived more directly. In Vadim’s work with Pol Vanhaecke [34], all of the previously knownLaurent Polynomials and Superintegrable Maps 3 examples were unified via an al... |

13 |
Elliptic curves and quadratic recurrence sequences
- Hone
(Show Context)
Citation Context ...ion of the relations ℘(z) = λ, ℘(z0) = λ − u0. The coefficients α, β and also J are given as elliptic functions of z by α = ℘ ′ (z) 2 , β = ℘ ′ (z) 2 (℘(2z)− ℘(z)), J = ℘ ′′ (z). From this it follows =-=[27]-=- that the solution to the initial value problem for the Somos-4 recurrence (2.4) can be written in terms of the Weierstrass sigma function as xn = AB n σ(z0 + nz) σ(z) n2 (2.20) for suitable A, B. The... |

13 |
Superintegrability of the Calogero-Moser systems,” Phys
- Wojciechowski
- 1983
(Show Context)
Citation Context ... and J3, but {J2, J3}1 ̸= 0. The terminology ‘superintegrable’ is applied in the even more special situation that the number of independent integrals is one less than the dimension of the phase space =-=[64]-=-, as is the case here. Upon applying the Diophantine Laurentness Lemma to the case of initial data (1, 1, 1, 1), and choosing integer β (with β ̸= 0 to avoid the degenerate case of a fixed point) we g... |

12 | p-adic properties of division polynomials and elliptic divisibility sequences
- Silverman
(Show Context)
Citation Context ...nerates to an expression in terms of the hyperbolic sine. The arithmetical properties of EDS and Somos sequences – in particular the distribution of primes therein – are a subject of current interest =-=[10, 11, 49]-=-. Some of these properties are discussed in the book [12] (see section 1.1.20, for instance), where it is suggested that such bilinear recurrences should be suitable generalizations of linear ones, wi... |

11 | Separation of variables and Bäcklund transformations for the symmetric Lagrange top
- Kuznetsov, Petrera, et al.
(Show Context)
Citation Context ...e discrete-time counterparts of algebraically completely integrable systems, as described in [56], for instance. While there has been subsequent work by Vadim and others on BTs in classical mechanics =-=[8, 13, 39]-=-, a lot of the original motivation for studying them came from quantum integrable systems (Baxter’s Q-operator). This idea has proved extremely effective (see e.g. [33, 35]), and will no doubt continu... |

10 |
Papageorgiou V., Do integrable mappings have the Painlevé property
- Grammaticos, Ramani
- 1991
(Show Context)
Citation Context ...the Laurent property (while inserting another coefficient α in front of the xn+3xn+1 term does not, unless c = 2). These recurrences also satisfy the singularity confinement test that was proposed in =-=[19]-=- as an analogue of the Painlevé test for discrete equations: if an apparent singularity is reached (in this case, corresponding to the situation that one of the iterates vanishes), then it is always p... |

8 |
Periodicity of Somos sequences
- Robinson
- 1992
(Show Context)
Citation Context ..., 6, 7, but denominators appear for k = 8. Various direct proofs that the terms of the sequence (2.5) are all integers were found at the beginning of the 1990s, when various other examples were found =-=[17, 47]-=-, but a deeper understanding came from the realization that the recurrence (2.4) has the Laurent property: its iterates are all Laurent polynomials in the initial data (and in α, β) with integer coeff... |

7 | An in nite set of Heron triangles with two rational medians
- Buchholz, Ratbun
- 1997
(Show Context)
Citation Context ...elliptic curve E from a Somos-4 or Somos-5 sequence was previously understood in unpublished work of several number theorists2 – see the discussion of Zagier [67], and the results of Elkies quoted in =-=[4]-=-. The algebraic part of the construction is described in the thesis of Swart [53] (who also mentions unpublished results of Nelson Stephens), and van der Poorten has recently presented another constru... |

6 | Sigma function solution of the initial value problem for Somos 5 sequences
- Hone
(Show Context)
Citation Context ...4) can be written in terms of the Weierstrass sigma function as xn = AB n σ(z0 + nz) σ(z) n2 (2.20) for suitable A, B. There is an analogous formula for the general solution of the Somos-5 recurrence =-=[28]-=-, which has an additional dependence on the parity of n. 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Figure 1. A family of orbits for the nonlinear recurrence (2.13) associated with Somos-... |

6 |
Mangazeev V.V., Sklyanin E.K., Q-operator and factorised separation chain for Jack polynomials
- Kuznetsov
(Show Context)
Citation Context ...itial data; this is an instance of the Laurent phenomenon [14]. Vadim was an expert on special functions, and orthogonal polynomials in particular (for one of his many contributions in this area, see =-=[35]-=-, for instance). However, most of the sequences of (Laurent) polynomials treated below satisfy nonlinear equations instead of linear ones. The theory of discrete integrable maps has seen a great deal ... |

6 |
A geometric approach to singularity confinement and algebraic entropy
- Takenawa
(Show Context)
Citation Context ... ζ = (3 + √ 5)/2 is the square of the golden mean, while log ζ turns out to be the value of the algebraic entropy for (4.4). Note that while the calculation of the algebraic entropy is quite involved =-=[21, 55]-=-, it is quite straightforward to calculate the growth of heights from (4.6). Similarly to the previous example, the only confined singularities that appear in (4.6) are isolated zeros. These examples ... |

5 | The cube recurrence. Electronic - Carroll, Speyer - 2004 |

5 |
Diophantine non-integrability of a third-order recurrence with the Laurent property
- Hone
(Show Context)
Citation Context ...ly, we have noted the close connection between the Laurent property and the notion of singularity confinement as introduced in [19]. (For other examples of confined maps with the Laurent property see =-=[29, 30]-=-.) This connection seems to persist for rational maps that do not themselves have the Laurent property. For example, consider the second-order equation 3 un+1 = (un) 2 + 1 , (4.1) un−1un which is supe... |

5 | Singularity confinement for maps with the Laurent property
- Hone
(Show Context)
Citation Context ...ly, we have noted the close connection between the Laurent property and the notion of singularity confinement as introduced in [19]. (For other examples of confined maps with the Laurent property see =-=[29, 30]-=-.) This connection seems to persist for rational maps that do not themselves have the Laurent property. For example, consider the second-order equation 3 un+1 = (un) 2 + 1 , (4.1) un−1un which is supe... |

5 |
Sklyanin E.K., Quantum Bäcklund transformation for DST dimer model
- Kuznetsov, Salerno
(Show Context)
Citation Context ...s in classical mechanics [8, 13, 39], a lot of the original motivation for studying them came from quantum integrable systems (Baxter’s Q-operator). This idea has proved extremely effective (see e.g. =-=[33, 35]-=-), and will no doubt continue to bear fruit for a long time to come. The last time I saw Vadim was in Leeds in April 2005, when he invited me to give one in the series of Quantum Computational seminar... |

5 |
Problems posed at the St Andrews Colloquium
- Zagier
- 1996
(Show Context)
Citation Context ...uction of a sequence of points P0 + nP on elliptic curve E from a Somos-4 or Somos-5 sequence was previously understood in unpublished work of several number theorists2 – see the discussion of Zagier =-=[67]-=-, and the results of Elkies quoted in [4]. The algebraic part of the construction is described in the thesis of Swart [53] (who also mentions unpublished results of Nelson Stephens), and van der Poort... |

4 | Recursion relation of hyperelliptic PSI-functions of genus two
- Matsutani
(Show Context)
Citation Context ...ear ones, with many analogous features. Based on the appearance of higher-order Somos recurrences in the work of Cantor on the analogues of division polynomials for hyperelliptic curves [6] (see also =-=[40]-=- for analytic formulae), it was conjectured in [27] that every Somos-k sequence should correspond to a discrete linear flow on the Jacobian of such a curve (with an associated discrete integrable syst... |

3 |
Non-autonomous Hénon–Heiles systems, Phys
- Hone
- 1998
(Show Context)
Citation Context ...ady constructed an analogous BT for the non-autonomous case of this system, as well as deriving the explicit formula for the generating function of the canonical (contact) transformation in that case =-=[22]-=-. During my viva voce examination a few months earlier, Allan Fordy had ⋆ This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The full collectio... |

3 |
Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the sl(2) Gaudin magnet
- Hone
(Show Context)
Citation Context ...hat was suggested by Vadim, concerning Bäcklund transformations (BTs) for finite-dimensional integrable Hamiltonian systems. This turned out to be very fruitful, resulting in three joint publications =-=[23, 24, 25]-=-. Vadim’s presence in Rome was immensely stimulating for me, because he succeeded in posing just the right question, at the precise moment when I had the necessary tools available to answer it. The sp... |

2 |
Discrete chaos, Chapman and Hall/CRC
- Elaydi
- 2000
(Show Context)
Citation Context ...s is the quadratic map defined by the recurrence xn+1 = x 2 n + c (2.2) with a parameter c, which is a prototypical model of chaos. However, note that the special cases c = 0, −2 are exactly solvable =-=[9]-=-, and in these cases one can also argue that (2.2) is integrable in the sense of admitting a commuting map (see [57] and references). The theory of linear recurrence sequences relies heavily on the fa... |