Polynomial Stability Lemma

• Polynomial Stability Lemma is a collection of sharp analytic, algebraic, and combinatorial bounds that quantify root localization, perturbation sensitivity, and structural stability of polynomials across diverse settings.
• It provides explicit control over growth, critical point dynamics, and fixed point behavior in univariate, multivariate, and finite-field contexts, underpinning rigorous analyses in dynamics, combinatorics, and numerical methods.
• These lemmas support practical applications in control theory and approximation, synthesizing classical Szász bounds, Gauss–Lucas insights, and perturbation theories into unified frameworks.

A Polynomial Stability Lemma refers to any sharp analytic, algebraic, or combinatorial statement that certifies, bounds, or quantitatively governs the roots, spectral localization, dynamical fixed point behavior, perturbation sensitivity, or structural properties of polynomials under various constraints. Multiple distinct but deeply connected forms of such lemmas appear in the literature, underlying results in combinatorics, complex analysis, dynamics, algebraic structure theory, random matrix theory, and applied mathematics. The following survey synthesizes the main types, emphasizing structural results, quantification of stability, and paradigm applications as documented in recent primary sources.

1. Classical One-Variable Lemmas and Szász-type Bounds

The prototype of a polynomial stability lemma is the Szász lemma (Knese, 2017), which gives an explicit growth bound for stable univariate polynomials—those with no zeros in the upper half-plane. If $p(z) = 1 + \sum_{k \geq 1} c_k z^k$ is normalized and stable, then for all $z \in \mathbb{C}$,

$$|p(z)| < \exp\left(|z||c_1| + 3|z|^2|c_2|^2 + |c_2|^2\right).$$

Modern refinements, such as Knese's improved Szász (Theorem 1.3), replace the estimate's quadratic term with the exact expression $|p_1|^2 - 2\,\mathrm{Re}(p_2)$, reflecting deeper control via the polynomial's low-degree Taylor data. Bivariate and multivariate generalizations leverage determinantal representations, yielding bounds dependent solely on first and second partials at the origin. Such results are critical for explicit normal family arguments, elucidation of stability-preserving operators, and hyperbolic programming (Knese, 2017).

2. Gauss–Lucas Type Stability and Critical Point Localization

A geometric instance of a polynomial stability lemma is the stability version of the Gauss–Lucas Theorem (Steinerberger, 2018). For $p(z) = \prod_{k=1}^{n+m}(z - a_k)$, if $n$ roots are in the closed unit disk and $m$ roots are outside, separated by $d > 1 + 2m/n$, then all $n-1$ critical points of $p$ lie inside the unit disk and all $m$ outlying critical points are at distance at least $(dn-m)/(n+m) > 1$ from the origin. This bound is sharp. Further, a “pairing" theorem asserts that for large $n$, each exterior root is $O(n^{-1})$-close to a critical point, quantifying the “electrostatic" pull of dominant root clusters and establishing the precise interplay between root geometry and critical-point localization. Such results admit extensions to arbitrary convex sets and elucidate the stability of critical points under root perturbations (Steinerberger, 2018).

3. Combinatorial and Additive Structure: Polynomial Stability in Finite Fields

For sets in finite abelian groups, a Polynomial Stability Lemma (PSL) is a powerful dichotomy tool for additive combinatorics (Moghadam, 4 Dec 2025, Gomez-Perez et al., 2012). In Marton's conjecture (finite fields, odd characteristic), the PSL asserts: For any $A \subset \mathbb{F}_p^n$ with $|A+A| \leq K|A|$, one has either a near-coset structure (a large portion of $A$ fills a large coset) or, via spectral dispersion and a Chang-type span bound, there is a nontrivial quotient reducing the doubling constant by $K^{-C}$ with only polynomial loss in density. Iteration produces polynomial-structured Freiman–Ruzsa-type covers, with the PSL providing non-probabilistic, explicit error control (Moghadam, 4 Dec 2025).

For iterates of polynomials in finite fields, stability takes the form of preservation of irreducibility: The polynomial $f$ is stable if $f^{(n)}(X)$ is irreducible for every $n$. The relevant lemma (Gomez-Perez et al., 2012) gives exact characterizations via discriminants and resultants. For $f \in \mathbb{F}_q[X]$ of degree $d \geq 2$, the parity (square/non-square) of both $\mathrm{Disc}(f)$ and $\mathrm{Res}(f^{(n)}, f')$ alternates with $n$ and $d$, tightly constraining which polynomials can be stable and how often this phenomenon arises.

4. Fixed Point Stability in Real Polynomial Dynamics

A canonical nonlinear-dynamical Polynomial Stability Lemma, developed in (Franco, 2017, Franco-Medrano et al., 2015), gives explicit algebraic conditions for the (local) stability of fixed points in real polynomial maps. Any degree-$m$ polynomial map with all real fixed points is topologically conjugate to a canonical form

$$g_m(x) = x + (-1)^{m-1} s x \prod_{i=1}^{m-1} (x - x_i),$$

where $s$ is the sign of an affine scaling. The linear stability of a fixed point $x_k$ is controlled by the Product Position Function (PPF): $P_k = (-1)^{m-1} s \prod_{i \neq k}(x_k - x_i).$ The fixed point $x_k$ is asymptotically stable if and only if $-2 < P_k < 0$. Transitions at $P_k = -2$ yield period-doubling, generating “stability bands” which fully characterize the dynamical regime as parameters vary. Exact band boundaries in degrees $2$ and $3$ correspond to known iterated-map bifurcation sequences (e.g., logistic map, cubic generalized maps), while high-degree cases are subject to numerical boundary determination (Franco, 2017, Franco-Medrano et al., 2015).

5. Stability Under Coefficient Perturbations and GCDs

Quantitative polynomial stability with respect to coefficient perturbations is formalized in (Remm, 2022). If the coefficients $a_k$ of a degree-$n$ polynomial $p$ are altered by at most $\varepsilon$, exact bounds are given on the motion of each root $\alpha_i$, controlled by the distance to its neighbors and the magnitude of the derivative $p'(\alpha_i)$: $$|\widetilde{\alpha}_i - \alpha_i| \leq \frac{\sum_{k=0}^{n} |\Delta a_k|\, |\alpha_i|^k}{|p'(\alpha_i)| - \sum_{k=1}^n k |\Delta a_k| |\alpha_i|^{k-1}}.$$ If the roots are well-separated, this yields $O(\varepsilon)$ root stability. The same formalism governs the stability of $\gcd(f,g)$ under small perturbations of $f$ and $g$, yielding explicit $O(\varepsilon^{1/\mu})$ root movement for $\mu$-fold common roots, and applies directly to eigenvalue perturbation theory in linear algebra (Remm, 2022).

6. Functional-Analytic and Algorithmic Stability

Stability issues arise in numerical analysis for least-squares polynomial approximation. The relevant lemma (Xu et al., 14 Jul 2024) quantifies: For degree $m$ polynomials sampled on $n$ i.i.d. points (uniform or Jacobi weights), stability of the least-squares solution is ensured (i.e., condition number $O(1)$) with high probability if and only if $n \gtrsim m^2 \log n$ (uniform case), as determined by the coherence $K(m+1)$ of the orthonormal basis. This bound is both necessary and sufficient: if $n \ll m^2$, the estimator becomes exponentially ill-conditioned (Xu et al., 14 Jul 2024).

7. Applications in Control and High-Order S-Lemmas

In control and switched systems, the “high-order” S-Lemma and its polynomial generalizations (HS-Lemma/NHS-Lemma) (Zhang et al., 2014) function as polynomial stability lemmas certifying positivity or copositivity in Lyapunov-theoretic contexts. For homogeneous polynomials $f,g$ of equal degree $k$, under structural nondegeneracy conditions,

$$f(x) \text{ copositive with } g(x) \implies \exists\, \xi \geq 0,\, f(x) - \xi g(x) \geq 0\,\, \forall x.$$

Such certificates are essential for polynomial Lyapunov function construction and stability analysis of nonlinear or switched dynamical systems of degree greater than two (Zhang et al., 2014).

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Table: Summarized Types of Polynomial Stability Lemmas

Lemma Type	Structural Setting	Main Application
Szász/Stability Inequalities	One/multi-variate stable polynomials	Growth control, normal families, analysis
Gauss–Lucas Stability	Root/critical point geometry	Localization, perturbation, random polynomials
Additive/Fourier–analytic	Subsets in finite fields, iterates	Structure of sets, irreducibility, Freiman–Ruzsa
Dynamical/Fixed Point	Real polynomial maps with real fixed pts	Bifurcation analysis, chaos, canonical conjugacy
Perturbation Theoretic	Coefficient/Root perturbations	Numerical stability, GCD, eigenvalue bounds
Norm/Algorithmic	Least-squares, random sampling	Numerical analysis, conditioning, approximation
S-Lemma (High-Order)	Polynomial inequalities, Lyapunov theory	Nonlinear control, switched systems

8. Concluding Remarks and Open Directions

No single “Polynomial Stability Lemma” is universal; instead, the term denotes a set of quantitative, sharp bounds and dichotomies governing the qualitative and quantitative stability properties of polynomials in various analytic, algebraic, and dynamical contexts. Key trends include sharpness of constants, transfer of structural dichotomies (e.g., via Fourier or root localization) to global performance bounds, and a unification of combinatorial and analytic perspectives. Unresolved questions remain regarding higher-dimensional analogs, optimality of iterative dichotomies, and the algebraic complexity of stability loci for general (nonlinear) semialgebraic parameter spaces (Moghadam, 4 Dec 2025, Franco-Medrano et al., 2015, Zhang et al., 2014).