Discrete Algebraic Dynamical Systems

• Discrete algebraic dynamical systems are defined by using polynomial, Laurent, and operator methods to encode dynamics in discrete time.
• They capture shift invariance through behavioral submodules and autoregressive modules, offering rigorous algebraic characterizations.
• Extensions include finite deterministic semiring approaches and operator-algebraic discretizations that link continuous differential equations with exact lattice analogues.

Discrete algebraic dynamical systems are discrete-time dynamical systems whose evolution is encoded and analyzed through algebraic structures such as polynomial or Laurent polynomial operators, modules of trajectories, finite-field polynomial maps, semirings of finite deterministic systems, ideals, or birational transformations. The term is used across several related frameworks rather than a single formalism. In one major line, a system is a closed, shift-invariant submodule of a sequence space over $\mathbb{N}^r$ or $\mathbb{Z}^r$, described behaviorally by polynomial or Laurent polynomial laws (Andriamifidisoa et al., 2012, Aandriamifidisoa et al., 15 Jul 2025). In another, finite deterministic systems are organized into a commutative semiring under disjoint union and direct product, so structural hypotheses become polynomial equations over dynamical systems (Gaze-Maillot et al., 2020, Doré et al., 2024). Other strands study polynomial dynamical systems over finite fields, rational and birational maps, exact-solvable discrete systems, and operator-algebraic discretizations of differential equations (Laubenbacher et al., 2011, Viallet, 2015, Tempesta, 2014).

1. Principal formulations

Several algebraic formulations recur in the literature. In the linear behavioral setting, a discrete system over $\mathbb{N}^r$ or $\mathbb{Z}^r$ is specified by a trajectory module $\mathcal{B}$ rather than by an input/output decomposition; in the finite deterministic setting, a system is a pair $(X,f)$ or $(A,f)$ with $f$ a self-map of a finite state set; in the finite-field setting, a polynomial dynamical system is a map $f=(f_1,\dots,f_n):K^n\to K^n$ with polynomial coordinate functions; and in rational or integrable settings, the dynamics is given by polynomial, rational, or piecewise-linear maps whose iteration is studied through factorization, singularities, or symplectic structure (Aandriamifidisoa et al., 15 Jul 2025, Doré et al., 2024, Laubenbacher et al., 2011, Machacek et al., 2021).

Framework	Basic object	Representative sources
Behavioral/module-theoretic	$\mathbf{D}$- or $\mathbb{Z}^r$0-submodule of trajectories	(Andriamifidisoa et al., 2012, Aandriamifidisoa et al., 15 Jul 2025)
Finite deterministic semiring	Isomorphism class of $\mathbb{Z}^r$1	(Gaze-Maillot et al., 2020, Doré et al., 2024, Dennunzio et al., 2022)
Finite-field polynomial dynamics	Polynomial self-map of $\mathbb{Z}^r$2	(Laubenbacher et al., 2011)
Algebraic inference and relations	Ideals, minimal primes, simplicial complexes	(Harrington et al., 2022, Kornyak, 2010)
Rational/integrable dynamics	Birational, solvable, or mutation maps	(Viallet, 2015, Bihun et al., 2016, Machacek et al., 2021)
Operator-algebraic discretization	Rota algebra realization of ODE-type dynamics	(Reyes et al., 2024, Tempesta, 2014)

A plausible implication is that the subject is unified less by a single state-space model than by a common methodological principle: algebraic structure is used to express invariance, composition, solvability, or reconstruction in discrete time.

2. Linear behavioral theory over $\mathbb{Z}^r$3

The 2012 framework of discrete linear algebraic dynamical systems starts from the duality between the vector space $\mathbb{Z}^r$4 of all multi-indexed sequences and the vector space $\mathbb{Z}^r$5 of finitely supported sequences. Via the identifications

$\mathbb{Z}^r$6

the pairing becomes

$\mathbb{Z}^r$7

This pairing defines orthogonals in both spaces, and closed subspaces of $\mathbb{Z}^r$8 for the topology of pointwise convergence are exactly the orthogonals of subsets of $\mathbb{Z}^r$9 (Andriamifidisoa et al., 2012).

The decisive operator-theoretic statement is that multiplication by a polynomial on $\mathbb{N}^r$0 has as functorial adjoint the polynomial operator in the shift on $\mathbb{N}^r$1. For

$\mathbb{N}^r$2

the shift action is

$\mathbb{N}^r$3

Thus $\mathbb{N}^r$4 and $\mathbb{N}^r$5 become $\mathbb{N}^r$6-modules. In the vector-valued setting, a polynomial matrix $\mathbb{N}^r$7 acts on $\mathbb{N}^r$8, and its kernel

$\mathbb{N}^r$9

is a $\mathbb{Z}^r$0-submodule. Such kernels are called autoregressive modules (Andriamifidisoa et al., 2012).

The main characterization theorem states that a subset $\mathbb{Z}^r$1 is a discrete algebraic dynamical system iff it is a kernel $\mathbb{Z}^r$2, equivalently iff it is of the form $\mathbb{Z}^r$3 for a finite nonempty $\mathbb{Z}^r$4, equivalently iff it is a closed $\mathbb{Z}^r$5-submodule of $\mathbb{Z}^r$6 (Andriamifidisoa et al., 2012). In this formulation, time invariance is encoded by shift invariance, and system laws are constant-coefficient multidimensional difference equations.

3. Laurent-series extension and bidirectional systems over $\mathbb{Z}^r$7

The 2025 Laurent-series framework extends the one-sided $\mathbb{Z}^r$8 theory to the full lattice $\mathbb{Z}^r$9 by replacing polynomial operators with Laurent polynomial operators and one-sided power series with Laurent series: $\mathcal{B}$0 This makes forward and backward shifts legal within the same operator ring and is used to model bidirectional discrete systems, including noncausal filters and multidimensional spatial masks (Aandriamifidisoa et al., 15 Jul 2025).

The scalar product remains coefficientwise: $\mathcal{B}$1 with matching by equal multi-indices. The shift action becomes

$\mathcal{B}$2

and multiplication on $\mathcal{B}$3 is again adjoint to shift on $\mathcal{B}$4. For a monomial $\mathcal{B}$5, this is the pure bidirectional shift

$\mathcal{B}$6

The framework preserves the duality and adjointness principles of the 2012 theory (Aandriamifidisoa et al., 15 Jul 2025).

Within Willems’ behavioral viewpoint, a system is a triple $\mathcal{B}$7 with $\mathcal{B}$8, $\mathcal{B}$9, and $(X,f)$0 a $(X,f)$1-submodule. Autoregressive systems are those of the form

$(X,f)$2

for a Laurent polynomial matrix $(X,f)$3. A representative example is

$(X,f)$4

which yields $(X,f)$5 and therefore parity-two-periodic solutions. The data-processing case study uses

$(X,f)$6

so that $(X,f)$7 averages neighboring samples and realizes a noncausal two-sided filter (Aandriamifidisoa et al., 15 Jul 2025).

The Laurent-series construction is explicitly purely formal: arbitrary $(X,f)$8-supports are allowed, and no topology, convergence structure, grading restriction, or support cone condition is imposed. The paper does not develop a full orthogonal-closure or annihilator theory in the stronger Malgrange/Oberst style, leaving topological properties for future work (Aandriamifidisoa et al., 15 Jul 2025).

4. Finite deterministic systems, semirings, and equation solving

For finite deterministic discrete-time systems, the basic object is a pair $(X,f)$9 with $(A,f)$0 finite and $(A,f)$1. Up to graph isomorphism, these systems form a commutative semiring under disjoint union and direct product. In this setting, structural hypotheses are expressed as polynomial equations such as

$(A,f)$2

where the $(A,f)$3 and $(A,f)$4 are known systems and the $(A,f)$5 are unknown systems (Dennunzio et al., 2022, Doré et al., 2024).

A coarser invariant is the profile

$(A,f)$6

where $(A,f)$7 counts states at height $(A,f)$8, namely at distance $(A,f)$9 from a limit cycle. Profiles themselves form a commutative semiring with

$f$0

This abstraction preserves enough arithmetic that general polynomial solvability over profiles is undecidable, while a single linear equation with constant right-hand side is $f$1-complete (Gaze-Maillot et al., 2020).

Algorithmically, the connected case is substantially more tractable. A connected FDDS is a weakly connected functional digraph with one cycle and rooted in-trees attached to it. The 2024 root-and-division paper develops polynomial algorithms for division and $f$2-th roots when the sought solution is connected, leading to an efficient solution of

$f$3

for connected $f$4. The key representation is the unroll, which converts a connected system into a finite set of infinite rooted trees encoding the periodic branch and the transient attachments; sufficiently deep finite cuts determine whether a connected quotient or root exists (Doré et al., 2024).

The 2022 equation-solving pipeline separates the original DDS equation into a $f$5-abstraction and an $f$6-abstraction. The $f$7-abstraction keeps only cardinalities and becomes an equation over $f$8, solved via reduced multi-valued decision diagrams. The $f$9-abstraction keeps only the periodic subsystem $f=(f_1,\dots,f_n):K^n\to K^n$0, decomposed into unions of cycles $f=(f_1,\dots,f_n):K^n\to K^n$1, and uses explicit cycle-product formulas such as

$f=(f_1,\dots,f_n):K^n\to K^n$2

The transient part, called the $f=(f_1,\dots,f_n):K^n\to K^n$3-abstraction, is explicitly deferred to future work (Dennunzio et al., 2022).

5. Dependency graphs, ideals, and symmetry-based structure theory

In finite-field polynomial dynamics, a polynomial dynamical system is a map

$f=(f_1,\dots,f_n):K^n\to K^n$4

over a finite field $f=(f_1,\dots,f_n):K^n\to K^n$5. Its dependency graph has an edge $f=(f_1,\dots,f_n):K^n\to K^n$6 when $f=(f_1,\dots,f_n):K^n\to K^n$7 depends on $f=(f_1,\dots,f_n):K^n\to K^n$8. The central structural theorem is

$f=(f_1,\dots,f_n):K^n\to K^n$9

with Boolean/max-min matrix multiplication on adjacency matrices. As a consequence, if the dependency graph of $\mathbf{D}$0 is acyclic, then some iterate $\mathbf{D}$1 is constant, so $\mathbf{D}$2 has a unique fixed point and no periodic orbit of length greater than $\mathbf{D}$3 (Laubenbacher et al., 2011). This is a direct structure-to-dynamics result.

A different structural program reconstructs signed dependency graphs from continuous-valued observations of monotone discrete-time systems

$\mathbf{D}$4

For one coordinate $\mathbf{D}$5, pairwise order constraints are encoded in ideals

$\mathbf{D}$6

and minimal local wiring diagrams correspond exactly to minimal primes of $\mathbf{D}$7. Under monotonicity and sufficiently dense sampling, with probability $\mathbf{D}$8 the true signed local wiring diagram is eventually uniquely recovered; bounded input and output noise are handled by modified ideals involving $\mathbf{D}$9 when sign information is ambiguous (Harrington et al., 2022).

A broader combinatorial-algebraic framework represents local rules as systems of discrete relations on abstract simplicial complexes. Relations $\mathbb{Z}^r$00 admit extensions, proper consequences, and canonical decompositions into lower-dimensional factors and a principal factor. The domains of irreducible components define a simplicial complex attached to the system. In deterministic settings with a finite orbit decomposition under a symmetry group, the framework argues that moving soliton-like structures arise inevitably when a trajectory returns to the same orbit class (Kornyak, 2010).

6. Rational, solvable, and integrable algebraic dynamics

For birational discrete systems, one influential viewpoint is that singularities organize the factorization pattern of iterates. After finitely many steps, homogeneous coordinates of iterates stabilize into factored forms whose blocks satisfy generalized Hirota-type recurrences. These recurrences determine exact degree growth and algebraic entropy. Integrable examples such as the McMillan map have quadratic growth and zero entropy, while nonintegrable but still structured examples such as Hietarinta–Viallet-type maps exhibit higher-degree recurrences and positive entropy (Viallet, 2015).

Another exact-solvability mechanism starts from a solvable discrete-time system for polynomial coefficients $\mathbb{Z}^r$01, forms the monic polynomial

$\mathbb{Z}^r$02

and then treats its zeros $\mathbb{Z}^r$03 as new dynamical variables. The key identity

$\mathbb{Z}^r$04

generates new solvable descendants, and repeated coefficient-zero reinterpretation yields higher “generations” of solvable discrete-time systems (Bihun et al., 2016).

Cluster-inspired mutation dynamics provide a further algebraic class. For $\mathbb{Z}^r$05,

$\mathbb{Z}^r$06

define a real-valued extension of rank-two cluster mutation on $\mathbb{Z}^r$07, preserving the log-canonical symplectic form $\mathbb{Z}^r$08. Its tropical companion

$\mathbb{Z}^r$09

preserves $\mathbb{Z}^r$10 and the piecewise quadratic first integral

$\mathbb{Z}^r$11

When $\mathbb{Z}^r$12, the tropical map is periodic; when $\mathbb{Z}^r$13, both tropical and non-tropical dynamics are unbounded in the regimes stated in the paper (Machacek et al., 2021).

A compact biological example is the Wolbachia-infection model, where the two-sex recurrence collapses after one iterate to the invariant diagonal and reduces to

$\mathbb{Z}^r$14

The fixed-point structure changes at $\mathbb{Z}^r$15, and for $\mathbb{Z}^r$16 the map is bistable with an unstable threshold equilibrium separating extinction from persistence (Özdinç et al., 2023). For periodic-orbit certification in algebraic maps, the Poincaré–Miranda methodology combines algebraic elimination, Sturm sequences, and box-sign conditions; it proves, for example, exactly two 5-cycles and three 6-cycles for the Lotka–Volterra-type map $\mathbb{Z}^r$17 (Gasull et al., 2018).

7. Operator-algebraic discretization and exact lattice analogues

A different branch treats discrete algebraic dynamical systems as exact lattice realizations of differential equations. In the Rota-algebra framework, a delta operator $\mathbb{Z}^r$18 with basic polynomials $\mathbb{Z}^r$19 is paired with a deformed product

$\mathbb{Z}^r$20

so that $\mathbb{Z}^r$21 satisfies the Leibniz rule

$\mathbb{Z}^r$22

This makes $\mathbb{Z}^r$23 into a Rota differential algebra and allows continuous and discrete equations to be treated as different realizations of the same abstract system (Tempesta, 2014, Reyes et al., 2024).

At the categorical level, covariant functors send Rota algebras to categories of linear or nonlinear dynamical systems with polynomially varying coefficients. For the forward-difference realization, an analytic solution

$\mathbb{Z}^r$24

is transported to the exact discrete solution

$\mathbb{Z}^r$25

In the linear constant-coefficient case, the paper proves that the Picard–Vessiot Galois group of the discretized equation coincides with that of the original differential equation (Tempesta, 2014).

The 2024 Frobenius-type theory specializes this program to second-order linear equations near ordinary or regular singular points. Its basic discrete equation is

$\mathbb{Z}^r$26

and it yields exact lattice analogues of Airy, Hermite, and Bessel equations through expansions in the basic-polynomial basis $\mathbb{Z}^r$27. For $\mathbb{Z}^r$28, this produces explicit recurrences such as the discrete Airy equation

$\mathbb{Z}^r$29

and a discrete Hermite equation with polynomial-factorial solutions (Reyes et al., 2024).

A plausible implication is that this operator-algebraic line treats discretization itself as a problem in discrete algebraic dynamics: the goal is not numerical approximation, but the construction of exact discrete evolution laws preserving derivation properties, solution spaces, and, in some cases, differential-Galois symmetry. Across the wider subject, open issues remain recurrent: full topological duality in the Laurent-series framework, the transient $\mathbb{Z}^r$30-abstraction for DDS equations, general root and division problems beyond connected systems, nonlinear theories beyond monotonicity, and broader solution-preserving discretizations (Aandriamifidisoa et al., 15 Jul 2025, Dennunzio et al., 2022, Doré et al., 2024, Harrington et al., 2022).