Polynomial Input-Gain Functions

• Polynomial input-gain functions are algebraic mappings that transform system inputs through polynomial operations, playing a key role in control, integration, and combinatorial problems.
• They enable explicit system design by providing robust, polynomial bounds that ensure stability and efficiency, as seen in variance reduction for numerical integration.
• Their structural properties, including degree multiplicativity and decomposability, support analytical techniques such as small-gain theorems and model reduction in nonlinear systems.

Polynomial input-gain functions are algebraic mappings, typically defined over commutative groups or fields, that describe how system inputs are transformed through polynomial operations to affect system outputs or state transitions. These functions arise in diverse mathematical and engineering contexts, including deterministic and randomized numerical integration, robust/nonlinear control, network spectral analysis, functional equations, and combinatorial enumeration. They are central in both explicit system design—where input-to-output relationships are crafted for stability, robustness, or efficiency—and in abstract structural theorems where polynomiality implies deep regularity properties for gain, attenuation, or transformation processes.

1. Definition and Foundational Properties

Polynomial input-gain functions generalize the concept of classical gain (linear amplification or attenuation) by allowing the gain to depend polynomially on the state or input variables. Formally, in control systems, a polynomial input-gain function γ: ℝ₊ → ℝ₊ may take the form γ(s) = a₁ s + a₂ s² + ... + a_k s^k, where the coefficients a_j ≥ 0. In operator-theoretic and functional equations on groups, a polynomial input-gain is often described as a function f: S → H that vanishes under a high-order difference operator—characterizing f as a generalized polynomial of bounded degree (Almira, 2022).

The polynomial structure offers desirable algebraic closure properties. For generalized monomials, which are diagonal evaluations of symmetric multi-additive functions, the composition f(P(x)) (where P is a classical polynomial) produces higher-degree polynomials, with degree multiplicativity: deg(f(P(x))) = deg(f)·deg(P) (Gselmann et al., 2022). This aligns with homogeneity and decomposition criteria inherent in polynomial functional equation theory.

2. Polynomial Gain Functions in Quasi-Monte Carlo and Lattice Rule Integration

In randomized quasi-Monte Carlo (RQMC) integration, polynomial input-gain functions arise as variance reduction factors (gain coefficients) in lattice rules for high-dimensional numerical integration. Here, the integration error's variance is decomposed as a sum weighted by gain coefficients, which quantify the improvement over Monte Carlo methods (Baldeaux et al., 2010). Specifically, for functions of bounded variation of order α, the estimator variance satisfies:

$$\mathrm{Var}[\hat{I}(f)] = \sum_{k \neq 0} T_{2\alpha+1,\gamma}(k) \cdot \omega_k(f)$$

where $T_{2\alpha+1,\gamma}(k)$ are the polynomial gain coefficients dependent on the integration lattice, and $\omega_k(f)$ depends on the Walsh coefficients of the integrand.

Selecting lattice parameters to minimize the total polynomial input-gain (quality criterion $B(q, \alpha, \gamma)$) yields nearly optimal integration rules with variance decay $$\mathrm{Var}[\hat{I}(f)] = O(N^{-(2\alpha+1)+\delta})$$ for any $\delta > 0$, and implementation is accelerated via FFT methods and circulant matrix analysis. Small gain coefficients directly measure efficiency improvements and are tightly connected to the system's polynomial input-gain structure.

3. Polynomial Input-Gain Functions in Stability and Robustness Analysis

In nonlinear control theory—especially for input-to-state stability (ISS), integral ISS (iISS), and nonlinear $\mathcal{L}_2$-gain frameworks—polynomial input-gain functions are prominent in both the specification and the analysis of robust system behavior under disturbances (Kellett et al., 2014, Shiromoto et al., 2015). Here, ISS properties defined via comparison functions α, β, σ can typically employ polynomials. For example:

$$\int_0^t |x(\tau)|^2 d\tau \le \max \{ \beta(|x(0)|), \gamma(\int_0^t |w(\tau)|^2 d\tau) \}$$

with $\gamma(s) = a_1 s + a_2 s^2$ is representative of polynomial input-gain bounds. Small-gain conditions for stability of interconnections are concrete in the polynomial setting: e.g., for quadratic gains, the requirement reduces to $\bar{\gamma}_1 \bar{\gamma}_2 < 1$.

Region-dependent gains—where local and global input-gain functions (possibly of differing polynomial forms) govern system stability for different operating domains—are synthesized using piecewise polynomial functions and small-gain theorems, with explicit construction and verification via polynomial inverses and composition conditions (Shiromoto et al., 2015).

4. Polynomial Input-Gain Functions in Functional Equations on Groups

Functional equation theory provides structural characterization of polynomial input-gain functions in abstract algebraic models. Aichinger's equation offers a canonical test: a function $f: S \to H$ on a commutative group is polynomial of degree ≤ m if and only if

$$f(x_1 + \dots + x_{m+1}) = \sum_{i=1}^{m+1} g_i(x_1,\dots,x_{m+1})$$

with each $g_i$ not depending on x_i (Almira, 2022). This decomposition implies that input-gain mappings governed by such recurrence relations must be polynomial.

Applications span probability theory (characterization of independence via the Ghurye-Olkin and Skitovich-Darmois equations), operator theory, and modeling—where input-gain functions satisfying structured equations (e.g., Wilson's, Kakutani-Nagumo-Walsh equations) must be polynomial and hence admit explicit decomposition, degree control, and stability analysis.

5. Graph-Theoretic Input-Gain Functions and Tutte Polynomial Invariants

In combinatorial theory, input-gain functions extend beyond numerical values to polynomial invariants encoding coloring, flow, and enumeration properties of graphs—particularly in weighted gain graph settings (Forge et al., 2013, Hameed et al., 2020). The total dichromatic polynomial

$$Q(\Phi, h)(\mathbf{u}, v, z) = \sum_{S \subseteq E} v^{|S|-n+b(S)} z^{c(S)-b(S)} \prod_{W \in \mathcal{W}_{Tb}(S)} u_{h/S(W)}$$

and its specializations count proper colorations and lattice points in orthotopes, yielding piecewise polynomial functions of input bounds.

Characteristic polynomials of skew gain graphs further encode network input-gain behavior in their spectra; coefficients are formed as sums of products of edge gains and symmetrizing anti-involution terms, with weighted graphs as a special case (Hameed et al., 2020):

$$a_i(\Phi_f) = \sum_{L \in \mathcal{L}_i} (-1)^{K(L)} \Bigg[ \prod_{K_2 \in L} \prod_{\vec{e} \in K_2} g(\varphi(\vec{e})) \Bigg] \Bigg[ \prod_{C \in L} ( \varphi(C) + f(\varphi(C)) ) \Bigg ]$$

6. Polynomial Input-Gain in Nonlinear Model Reduction and Control Design

Polynomial input-gain functions are central to nonlinear model reduction via balanced truncation for control-affine polynomial systems. Analytic energy functions are expressed as sums of homogeneous polynomial terms (via Kronecker products), and explicit transformation equations (solved degree-by-degree using tensor algebra) ensure input-normal/output-diagonal balancing (Corbin et al., 29 Oct 2024):

$$z = \Phi(x) = T_1 x + T_2 (x \otimes x) + \dots + T_{d-1} (x^{\otimes d-1})$$

Algebraic conditions (involving unique diagonal monomial identification and sparse tensor representations) yield scalable algorithms with computational complexity $O(d n^{d+1})$, with direct application to high-dimensional (mass-spring-damper, Duffing oscillator) systems through open-source software.

In feedback linearization with singular input-gain matrices, polynomial structures (or their regularized inverses) allow prioritized semilinear forms and lexicographically optimized control, achieving task hierarchy and robust stability via admissible pseudoinverse constructions and general Lyapunov-based guarantees (An et al., 2023).

7. Practical, Structural, and Enumerative Implications

The theory and applications of polynomial input-gain functions yield several strong implications:

• Explicit degree control and decomposability enable robust design, model reduction, and control synthesis across domains.
• Gain structures adhering to polynomial forms allow efficient implementation (including FFT-based algorithms) and sharp variance reduction or stability bounds.
• Universality in Tutte invariants and functional equation solutions place polynomial input-gain functions at the core of structural results in combinatorics, probability, algebra, and systems theory.
• Region-dependent, piecewise polynomial gains facilitate nuanced local/global stability, modular analysis, and adaptation in nonlinear systems.

The algebraic richness and analytic versatility of polynomial input-gain functions make them foundational tools for modern research in high-dimensional integration, nonlinear control, algebraic system theory, and network enumeration.