Discrete-time polynomial systems with inputs and outputs: structure, reachability, observability, and minimal realizations

• Page numbers refer to the place where the notation is introduced.Source: Notations · p. 3

1 Introduction

• This revised exposition updates the author’s 1976 thesis and lecture-notes material, correcting errors and adding a sample of later connections.The earlier work was never published as a journal paper, and its difficult typesetting and errors may have contributed to its limited citation and overlooked connection to linearization by duality.Source: §1 · p. 5
• The approach builds realizations from polynomial observables, with dynamics acting on them through a transpose or co-transition map.When the observation space is finite dimensional, this dual representation leads to state-affine realizations.Source: §1 · pp. 5–6
• Polynomial input/output maps capture multiplicative nonlinear effects and provide a structured setting for studying internal realizations of discrete-time behaviors.The introduction contrasts this class with the set-theoretic generality of automata theory and general system theory.Source: §1 · p. 6
• The paper uses algebraic geometry and commutative algebra to study polynomial systems, while keeping its exposition elementary and largely avoiding explicit sheaf-theoretic concepts.It presents system-theoretic applications of algebraic concepts and uses simplified terminology such as “polynomial map.”Source: §1 · pp. 6–7
• The paper treats discrete-time systems only and leaves the interplay between continuous and discrete time for future understanding.Source: §1 · p. 7
• The paper proceeds from intuitive results through algebraic preliminaries and realization theory to finiteness, state-affine systems, realization lattices, and open problems.Source: §1 · p. 7

The Systems Setup

• A polynomial input/output map is a causal, shift-invariant map whose output depends polynomially on past inputs, represented by a Volterra series finite-degree in each variable.Evaluation is well-defined because input sequences have support bounded on the left.Source: §2 · p. 8
• Although polynomial systems may lack observable polynomial realizations, k-systems realize every polynomial input/output map and always induce polynomial input/output maps.A k-system uses a k-space as its state space, with polynomial transition and output maps.Source: §2 · p. 10
• Every polynomial input/output map has a canonical realization, quasi-reachable and algebraically observable, unique up to isomorphism.Quasi-reachability means the reachable states are dense in the state space; algebraic observability distinguishes states through algebraic processing of input/output data.Source: §2 · pp. 10–11
• A map has a finite-dimensional realization exactly when its canonical realization is almost polynomial and it satisfies an algebraic input/output difference equation.The equation has a unique irreducible minimal-order form up to constant multiples, and a qualifying map also satisfies an equation linear in the current output.Source: §2 · p. 11
• Over the real numbers or an algebraically closed field, a realization has minimal dimension exactly when it is weakly canonical; over any field, canonical realizations are minimal.Weakly canonical systems are quasi-reachable and almost all states are indistinguishable from only finitely many other states.Source: §2 · p. 12
• The canonical realization is a polynomial system exactly when its observation algebra is finitely generated, and its minimal embedding dimension equals the smallest number of algebra generators.That dimension need not equal the state-space dimension unless the observation algebra is a polynomial ring.Source: §2 · p. 12
• Over fields containing the rationals, a map has a finite realization exactly when the ranks of its Jacobian matrices J_1(f), J_2(f), ... are uniformly bounded.For linear maps, J_n(f) is the nth principal minor of the behavior (Hankel) matrix.Source: §2 · p. 12
• A bounded map has a finite-dimensional realization exactly when its observation space is finite-dimensional, and any finite k-system realization can be replaced by an observable state-affine system.Bounded means each input has a uniformly bounded degree, while state-affine systems have state variables occurring linearly.Source: §2 · pp. 12–13

State-affine realizations

• For bounded finitely realizable maps, span-canonical state-affine realizations exist, are unique up to a linear coordinate change, and have minimum possible dimension.Span-canonical means observable with reachable states spanning the full state space.Source: §2 · p. 13

Formal power series

• The exponent series’ behavior (Hankel) matrix supports minimal realization procedures, and its row space is isomorphic to the map’s observation space.Under the correspondence with representations, span-reachability and observability match automata-theoretic reachability and observability.Source: §2 · pp. 13–14

Realizability criteria

• A bounded map is finitely realizable if and only if it satisfies an input/output difference equation linear in the output.The paper identifies this as a new result, including for internally-bilinear systems and multilinear input/output maps.Source: §2 · p. 14

Formal power series

• A bounded map has a state-affine realization exactly when its exponent formal power series is rational, enabling constructions between rational expressions and realizations.The exponent series is obtained from the Volterra series and is a power series in noncommutative variables.Source: §2 · pp. 13–14

Finite maps

• Finite maps are exactly those whose span-canonical realizations decompose as cascades of linear systems and memory-free nonlinearities.Finite maps restrict the total degree of the Volterra series to be finite.Source: §2 · p. 14

Quasi-reachable realizations

• The quasi-reachable realizations of a fixed map form a complete lattice under simulation, with the canonical realization minimal and the input space as the largest state space.The join of two realizations is a subsystem of their parallel connection.Source: §2 · p. 14

Normal realizations

• Normal realizations also form a complete lattice, and any two reachable, observable normal polynomial realizations that are abstractly canonical are isomorphic.Every quasi-reachable realization has a normalization.Source: §2 · p. 14

3 Algebraic Preliminaries

• The section defines a k-space directly as the set of k-points of an affine k-scheme, with k an arbitrary infinite field.Source: §3 · p. 15, §3.1 · p. 16

3.1 𝑘-Reduced Algebras

• A k-reduced algebra embeds into the algebra of k-valued functions on its homomorphisms to k, with elements acting by evaluation.Equivalently, its k-radical—the intersection of the kernels of all homomorphisms to k—is zero.Source: §3.1 · pp. 16–17
• If A is an integral domain integral over B, with B finitely generated over k and Q(A) finitely generated over Q(B), then A is a finite B-module and finitely generated over k.Source: §3.1 · p. 19

3.2 The Zariski Topology

• The sets V(S) of k-points where every element of S vanishes form the closed sets of the Zariski topology on X(A).V(S) is unchanged when S is replaced by the ideal it generates.Source: §3.2 · p. 20
• Closed subsets of X(A) correspond inclusion-reversingly to closed ideals of A, while a closed subset is irreducible exactly when its vanishing ideal is prime.An ideal I is closed precisely when I = I(V(I)).Source: §3.2 · pp. 21–22

3.3 𝑘-Spaces

• A k-space is a set equipped with a k-algebra of polynomial functions whose evaluations identify its points with X(A(X)); varieties are those with finitely generated function algebras.Every k-space is isomorphic to one of the form (X(A), ι(A)) for a k-reduced algebra A.Source: §3.3 · pp. 22–23
• Polynomial maps between k-spaces correspond bijectively, with arrows reversed, to homomorphisms between their k-algebras of polynomial functions.Consequently, the category of k-spaces is dual to that of k-reduced k-algebras.Source: §3.3 · p. 24
• A polynomial map is dominating exactly when its transpose is injective, and it is a closed immersion exactly when its transpose is surjective.Source: §3.3 · p. 25
• For maps of varieties, images of constructible sets are constructible over algebraically closed fields, while dominating maps to irreducible targets contain an open subset under the stated field conditions.For real-closed fields, real-constructible sets map to real-constructible sets; a dominating map to an irreducible target contains a strong-topology-open set over R.Source: §3.3 · p. 26
• Principal open subsets are represented by localization maps, and an almost-variety is a k-space reached by a dominating polynomial map from a variety.The localization map identifies the corresponding k-space homeomorphically with the principal open set.Source: §3.3 · pp. 26–27
• Every irreducible almost-variety has a principal open subset that is a variety.Specifically, its localized coordinate algebra a^-1A(X) is finitely generated.Source: §3.3 · p. 27
• Canonical state spaces are generally almost-varieties, and the section points to an input/output map whose canonical state space provides an example.Source: §3.3 · p. 27
• For an almost-variety X(A), the example’s principal open variety has complement consisting of one point.Thus X(A) is the disjoint union of that variety and the extra point.Source: §3.3 · p. 27

3.4 Dimension

• Almost-varieties are finite-dimensional, and chains of irreducible closed subsets are bounded in length by the dimension.Source: §3.4 · p. 28
• A dominating polynomial map between irreducible almost-varieties of dimensions n and m admits principal-open varieties and a dominating map to Y1 × k^(n−m) with fibers of size at most s.When k is algebraically closed, the map is onto and every fiber has exactly s elements; over R, the varieties are differentiable manifolds and the differentials have full rank.Source: §3.4 · p. 29

4.1 Volterra Series

• The two monoid operations on proper input sequences underpin the framework: addition defines polynomial structure, while concatenation supports finiteness conditions.Source: §4.1 · p. 31
• Volterra series have finite degree in each individual variable and form an integral-domain k-algebra.Their degree in each variable is bounded, though the overall degree may be infinite.Source: §4.1 · pp. 31–32
• A Volterra series has finite overall degree if and only if its support is finitely generated.Equivalently, its exponent series uses only finitely many variables.Source: §4.1 · p. 33
• For a k-subalgebra B of Volterra series, its transcendence degree equals the supremum of the transcendence degrees of its finite truncations B_t.Source: §4.1 · p. 34

4.2 Construction of Ω and Γ

• The input space Ω is identified with U[z], the set of polynomials in z with coefficients in U = k^m.Ω is constructed from the union of finite input sequences, identified with sequences having finitely many nonzero entries.Source: §4.2 · p. 35
• Polynomial maps from U[z] to k^p correspond bijectively to ordered p-tuples of Volterra series.The construction embeds U[z] as a dense subset of Ω, where polynomial maps are restrictions of polynomial maps on Ω.Source: §4.2 · p. 37

4.3 Abstract Response Maps and Systems

• Causal, constant-structure input/output maps correspond bijectively to response maps.Source: §4.3 · p. 38
• Every response map has an abstractly canonical realization, unique up to a unique isomorphism among abstractly canonical realizations.Abstractly canonical means reachable and observable.Source: §4.3 · pp. 39–40

4.4 Polynomial Response Maps and 𝑘-Systems

• A response map is polynomial exactly when it is realized by a k-system, whose state space is a k-space and whose transition and output maps are polynomial.The free realization on Ω supplies a k-system realizing each polynomial response map.Source: §4.4 · pp. 41–42

4.5 Quasi-Reachability

• A k-system is quasi-reachable when its quasi-reachable state set X_R equals its state space X.Lemma 4.7 introduces equivalent statements characterizing this condition.Source: §4.5 · p. 42
• For dim Σ=n<∞, quasi-reachability is attained by time n, ordinary reachability need not be bounded-time, and every system contains a closed quasi-reachable subsystem.The example over R has X_t={x∈R | x≥−t}≠X_R=X for every t≥0.Source: §4.5 · p. 43

4.6 Algebraic Observability

• A system is algebraically observable when its observation algebra equals A(X), equivalently its observability map is a closed immersion; this implies abstract observability.This definition requires each state coordinate to be polynomial in the basic observables; algebraic observability is stronger than abstract observability.Source: §4.6 · p. 44
• Every k-system dominates an algebraically observable system.The system is constructed from the original system’s observation algebra.Source: §4.6 · p. 44

4.7 Existence and Uniqueness of Canonical Realizations

• Every polynomial response map has a canonical k-system realization, and any two canonical realizations of that map are uniquely isomorphic.Canonical means quasi-reachable and algebraically observable.Source: §4.7 · pp. 45–46

5 Finiteness Conditions

• The observation space, algebra, and field characterize finiteness properties of the canonical system and connect to input/output equations and a Jacobian criterion.The section also gives a response map with terms of increasing lag-dependent degree that has no finite-dimensional realization.Source: §5 · pp. 46–47

5.1 The Observables of 𝑓

• The observables L_f, A_f, and Q_f are the space, algebra, and field generated by basic observables and serve as system invariants, approximated by observability and reachability chains.Source: §5.1 · pp. 47–48

5.2 Finite Realizability and Minimality

• Finite realizability is equivalent to the canonical system being almost polynomial, while weak canonicity is equivalent to minimality and irreducibility when k = R or k is algebraically closed.A realization is minimal exactly when its dimension equals the dimension of the canonical realization.Source: §5.2 · p. 51, §5.3 · p. 52

5.3 Polynomial Canonical Systems

• The canonical system is polynomial exactly when A_f is a finitely generated k-algebra.Source: §5.3 · p. 52
• Given finite generators of A_f, the section describes a procedure for obtaining a polynomial canonical realization.The procedure uses polynomial relations among the generators to construct the transition and output maps.Source: §5.3 · pp. 52–53
• The example’s canonicalization identifies x and −x as unobservable states, yielding an algebraically observable system whose complete reachability becomes quasi-reachability.The resulting system is polynomial; the canonical system is polynomial for a wide class of response maps, including those defined by recursive polynomial equations.Source: §5.3 · p. 53

5.4 Bounded Maps

• Boundedness means the degree in past inputs is uniformly finite, so no input is raised above a fixed power.Linear, multilinear, and internally-bilinear response maps are given as bounded examples.Source: §5.4 · p. 54
• Theorem 5.4 equates bounded finite realizability, bounded realizability by a polynomial system, and a polynomial canonical realization.The theorem also identifies absence of intrinsic nonlinear feedback as an essential characteristic of bounded maps.Source: §5.4 · pp. 54–55

5.5 Input/Output Equations

• Theorem 5.5 gives equivalence criteria involving rational and affine difference equations; an integral difference equation implies a polynomial canonical system and a recursive equation.It also states that an integral equation together with a possibly different rational equation implies a recursive equation.Source: §5.5 · p. 56
• For a finitely realizable map, a minimal-order irreducible algebraic difference equation is unique up to scalar, and a rational equation determines and reconstructs the map.The passages also state that two polynomial input/output maps satisfying the same rational equation coincide.Source: §5.5 · p. 59
• A polynomial canonical system does not imply that its response map satisfies an integral difference equation.The passage identifies this implication as false by counterexample.Source: §5.5 · p. 59

5.6 Jacobian Condition

• In characteristic zero, finite realizability is equivalent to a uniform bound on the ranks of the Jacobians J_t(f), or equivalently vanishing of all (s + 1)-minors for some s and every t.The stated setup assumes a zero-characteristic field; the passage notes that the criterion may be generalized in positive characteristic using spaces of differentials.Source: §5.6 · pp. 59–60

5.7 Some Examples and Counterexamples

• For f0 from Example 5.7, Σ_f0 is not a polynomial system, and no polynomial realization of f0 makes every pair of reachable states distinguishable.Source: §5.7 · pp. 61–62
• Under Lemma 5.4's onto reachability and maximal-separation conditions, any quasi-reachable realization with distinguishable reachable states is isomorphic to the canonical realization.Over R or an algebraically closed field, the passage says the onto condition can be replaced by dim Σ_f = n.Source: §5.7 · p. 62
• The canonical bilinear response system Σ1 has dimension 2n + 1 but satisfies an affine difference equation of order n + 1.Thus the order of a difference equation can be strictly lower than the dimension of the canonical realization.Source: §5.7 · p. 63
• The number of equations needed to represent a polynomial canonical realization is the smallest number of algebra generators, and its state variety need not be affine space.For f3, at least three equations are needed for the canonical realization, although a noncanonical realization uses two; the canonical state variety is a singular surface.Source: §5.7 · pp. 63–64
• Homogeneous response maps can be realized by converting them to multilinear maps, but this construction can require dimension s even when the canonical realization has a single state.The section also gives finitely realizable polynomial responses that satisfy no strictly recursive equation of the stated finite-lag form.Source: §5.7 · pp. 64–65

6 State-Affine Systems

• State-affine systems are introduced as natural realizations for bounded input/output maps.The section develops them through representations of Volterra, or exponent, series.Source: §6 · p. 66

6.1 Recognizable Series

• The behavior matrix characterizes recognizable series: finite rank is equivalent to recognizability, and canonical representations are minimal with dimension equal to that rank.Canonical representations are unique up to a unique representation isomorphism.Source: §6.1 · p. 67

6.2 State-Affine Systems

• State-affine realizations of bounded polynomial response maps correspond bijectively to finite-dimensional representations, preserving span-reachability and observability.Span-reachability corresponds to accessibility, and observability to reducedness.Source: §6.2 · pp. 69–70
• Bounded, finitely realizable response maps have a span-canonical state-affine realization, unique up to isomorphism, whose dimension equals the behavior-matrix rank.Such a realization is span-canonical exactly when it is minimal.Source: §6.2 · p. 70

6.3 Finite Response Maps and Cascades of Linear Systems

• A polynomial response map is bounded with a cascade span-canonical realization exactly when it is realizable by a cascade or is a finitely realizable finite map.These three conditions are equivalent.Source: §6.3 · p. 72

6.4 Rationality

• An exponent series is rational if and only if it is recognizable.Rationality also yields finite realizability of every degree-truncated response map for bounded, finitely realizable maps.Source: §6.4 · pp. 72–73

7.1 The Lattice QR( f)

• QR(f) is a complete lattice, with the free realization as its largest element and the canonical realization as its smallest.The ordering is described through corresponding subalgebras ordered by inclusion.Source: §7.1 · p. 74

7.2 Examples Using the Lattice Construction

• Joins can be constructed as fibre products; examples show that joins may be more complex than the original systems while the canonical realization is their meet.One example has a cusp-shaped join, and another identifies the canonical realization as the meet.Source: §7.2 · pp. 74–75
• A chain of polynomial realizations has a nonpolynomial meet, and its cofinality property rules out a minimal polynomial realization for the example response.Any proposed polynomial realization is either incomparable with one system in the chain or lies above one of its members, so it is not minimal.Source: §7.2 · pp. 76–77

7.3 Some Relevant Sublattices

• For finitely realizable f, the minimal-dimensional realizations form a complete sublattice, and their observation fields are algebraic over the canonical observation field.Minimal realizations correspond to subalgebras algebraic over A_f that satisfy the stated realization conditions.Source: §7.3 · p. 77
• Over an algebraically closed field of characteristic zero, a quasi-reachable almost-polynomial system is quasi-canonical exactly when some open dense state-space subset has no indistinguishable pair.The text notes that the “if” direction does not hold in general over the reals, though it does for restricted classes such as state-affine systems.Source: §7.3 · p. 78
• The quasi-canonical realizations form a complete sublattice with a largest element, constructed by intersecting the canonical observation field with the Volterra-series algebra.This largest realization is obtained by restricting the maps defining the free realization.Source: §7.3 · p. 78
• The reachable-distinguishable realizations form a complete sublattice, while the abstractly observable realizations form a complete join-semilattice of QR(f).The classes consist, respectively, of systems with pairwise distinguishable reachable states and abstractly observable systems.Source: §7.3 · p. 78
• Closure of AO(f) under joins yields a largest abstractly observable realization, which is initial among all realizations, while an example shows AO(f) need not be closed under meets.The meet of the observable systems Σ_t is not observable.Source: §7.3 · p. 79

7.4 Normal realizations

• The integral closure of Σ1 equals the intersection of integrally closed subalgebras containing its state algebra and satisfying (7.2), when Σ1 ≤ Σ2 and Σ2 is normal.This construction defines the canonical normal realization as the integral closure of Σ_f.Source: §7.4 · p. 81
• NOR(f) is a complete lattice: its meets are those in QR(f), and its joins are integral closures of joins in QR(f).The integral closure of a realization is well-defined, independent of the containing realization used in its construction, and preserves the realization order.Source: §7.4 · pp. 81–82
• A one-to-one-as-schemes morphism from a finite-dimensional quasi-reachable realization to a normal realization is an open immersion.In particular, the natural morphism from a polynomial realization in AO(f) ∩ NOR(f) to the canonical normal realization is an open immersion; if the source is reachable, Σ2 ≤ Σ1.Source: §7.4 · p. 83
• Over an algebraically closed field of characteristic zero, any two abstractly canonical normal polynomial realizations are isomorphic as k-systems.The result specializes to a polynomial coordinate change for reachable, abstractly observable systems with state space k^n.Source: §7.4 · pp. 82–83

8.1 The Canonical State-Space

• A direct realization using rational transition and output maps may have poles at some state-input configurations, although it determines the response for generic input sequences.In characteristic zero, the state can be represented using as few as n + 1 coordinates, where n is the canonical realization’s dimension.Source: §8.1 · p. 84
• Theorem 8.1 guarantees a decomposition of the canonical state-space into quasi-affine varieties, with the first component a principal open subset; under its stated conditions, the decomposition covers the image of a polynomial realization morphism.On these pieces, transitions and outputs can be specified by polynomial predicates and rational functions defined wherever the predicates hold.Source: §8.1 · p. 85

8.2 Unconstrained Realizations

• The bilinear single-output unconstrained-realization result does not extend to bilinear two-output or trilinear single-output responses, whose observation algebras are not polynomial rings.A separate example shows that f5 has a four-dimensional canonical realization but no unconstrained realization of dimension four: the assumed realization would force an algebra of transcendence degree at least five.Source: §8.2 · pp. 86, 87, 89, 90
• For homogeneous response f4, the number of generators r can be arbitrarily larger than the canonical dimension n = 3, showing that unconstrained realizations need not attain the canonical dimension.The example’s system is algebraically observable, and its quasi-reachable set is specified by equations without linear terms.Source: §8.2 · p. 88
• An unconstrained-realization passage describes A(XΣ) as a polynomial ring in four variables and says this contradicts a preceding fact.The supplied passage does not identify that preceding fact.Source: §8.2 · p. 90

8.3 Generalizations

• Generalized responses allow non-shift-invariant maps and arbitrary k-spaces for inputs and outputs; for U = k^m, bounded degree in each fixed number of recent inputs characterizes them.The response extends polynomial maps on finite input histories to a completion of U*, and shift-invariant responses recover the earlier definition.Source: §8.3 · pp. 90–91
• Every generalized polynomial response has a canonical generalized k-system realization, unique up to isomorphism.Here, canonical means that the reachability map dominates and the realization is algebraically observable.Source: §8.3 · p. 92
• A generalized canonical realization can have a nonirreducible state space, while nonaffine state spaces add no new polynomial responses when inputs and outputs remain affine.In the example, the canonical state space is the union of two lines; the nonaffine-state-space observation follows from affinization.Source: §8.3 · p. 92

8.4 Suggestions for Further Research

• Open questions include bounded-time reachability, field-dependent criteria for minimality, and whether local canonical structure can help construct minimal realizations.The text also identifies effective algebraic computation and explicit criteria for other system-theoretic problems as research needs.Source: §8.4 · p. 93
• Polynomial equalities and inequalities can encode whether a finite response prefix has a realization of specified dimension and degree, yielding a decision procedure for polynomial realizability.The passage notes that the Tarski–Seidenberg-based approach has impractically high computational complexity.Source: §8.4 · pp. 93–94
• The bounded-map realization algorithm is explicit but numerically unstable as presented; a stable modification is proposed as a direction for further work.The proposed modification would use slightly more algebraic operations.Source: §8.4 · p. 94
• Further directions include weighing state-affine simplicity against dimension and controllability, studying topological approximation and genericity, and clarifying links to continuous-time theory.The section also raises possible extensions to finite fields, rings, automata, analytic realizations, and category-theoretic models.Source: §8.4 · pp. 94–95

9 Addendum: Connection to Some More Recent Work

• The addendum points to representative connections between the preceding results and later research, with an intentionally incomplete selection reflecting the author's reading.The work summarized in the preceding sections was completed in 1976, and the bibliography reflects that date.Source: §9 · p. 95
• The addendum's purpose is to indicate how selected later papers relate to the preceding sections, rather than provide a complete survey.Source: §9 · p. 95
• Throughout the preceding sections, f denotes a polynomial response map; L_f, A_f, and Q_f denote its observation space, algebra, and field, and Σ_f its canonical realization.Source: §9 · p. 95

9.1.1 The Koopman operator

• The Koopman operator acts linearly on observables, while this work reconstructs its canonical state space algebraically from polynomial observables and studies their finiteness properties.Koopman work often uses Banach or Hilbert spaces and emphasizes spectral approximation; here the key objects are the observable space, algebra, and field.Source: §9.1.1 · pp. 95–96

9.1.2 Systems with inputs

• With inputs, exact finite-dimensional Koopman models are generally not linear in the input, and bilinear lifted models arise naturally.The paper relates this issue to state-affine systems, which are linear in the state and polynomial in the input; bilinear systems are a special case.Source: §9.1.2 · p. 96

9.1.3 Exact finite-dimensional linear embeddings

• For discrete-time polynomial systems, finite-dimensional observability is characterized through boundedness, behavior-matrix rank, and affine input/output difference equations.When the observation space is finite dimensional, the paper’s state-affine realization has a state space naturally dual to that space.Source: §9.1.2 · p. 96, §9.1.3 · p. 97

Continuous-time and algebraic realization

• Bartosiewicz developed continuous-time realization theory for algebraic varieties and rational systems, while Grossman and Larson based input/output realization on bialgebras.Wang and Sontag related rational realizations to algebraic input/output differential equations and studied lower bounds on observable realization dimension under analyticity assumptions.Source: §9.2 · p. 97

Rational and equation-based realization

• Němcová and van Schuppen gave existence conditions and minimality results for rational realizations, while other work treated input/output equations using skew polynomial rings and rational dynamical systems.The latter included realization of differential-algebraic input/output equations and existence of observable and real realizations.Source: §9.2 · p. 97

Observability and reachability

• Commutative-algebra methods have also produced conditions for polynomial-system observability, reachability, and controllability, complementing this work’s algebraic observability and quasi-reachability.The cited studies address global and local observability at an initial state as well as reachability and controllability.Source: §9.2 · p. 97

Field scope

• Some results here are stated only over algebraically closed fields or over the reals using ad hoc arguments; real algebraic geometry offers tools to revisit them over the reals.The paper also assumes an infinite field, placing finite-field polynomial-system applications outside its scope.Source: §9.2 · p. 97

Rational formal power series

• The treatment of state-affine systems relies on rational formal power series in noncommuting variables and the rank of the associated behavior matrix.Berstel and Reutenauer (2011) provide a modern account of that theory.Source: §9.3 · p. 97

Switched and parameter-varying systems

• Formal-series methods characterize input/output representations and minimal realizations for classes including hybrid, discrete-time switched, and affine linear parameter-varying systems.The cited switched-system theory uses a finite-rank condition on a generalized Hankel matrix and gives uniqueness of minimal realizations up to isomorphism.Source: §9.3 · p. 98

Weighted automata

• Hankel-matrix factorizations underpin spectral learning of weighted automata, including hidden Markov models as special cases.Related work also introduces a canonical form for weighted automata and applies it to approximate minimization.Source: §9.3 · p. 98

Identifiability

• The observation field captures which state functions are determined by input/output data, linking realization observables to parameter identifiability.Identifiability work has used observation algebras and fields, input/output equations, and fields of identifiable functions.Source: §9.4 · p. 98

Field scope

• Polynomial dynamical systems over finite fields have been used in gene-regulatory-network modeling and reverse engineering, but are outside this work’s infinite-field scope.The paper notes that the algebraic viewpoint is similar.Source: §9.4 · p. 98

• The work revisits questions framed elsewhere as lifting nonlinear dynamics to linear actions on observables, finite-dimensional observable spaces, state-space realization from input/output equations, and Hankel-matrix rank conditions.Source: §9.5 · p. 98
• Making these connections explicit is intended to help readers coming from the related areas.Source: §9.5 · p. 98
• A fuller account of these relations and their continuous-time counterparts is deferred to future work.Source: §9.5 · p. 98