Polynomial Doubling Techniques

• Polynomial-doubling techniques are recursive methods that use quadratic polynomial mappings to systematically expand algebraic and analytic structures.
• They enable rapid computations in areas such as fast polynomial division, normed algebra constructions, and efficient covering in high-dimensional metric spaces.
• These methods underpin applications ranging from constructing Cayley–Dickson algebras to exact spectral analyses in quantum models and combinatorial optimization.

Polynomial-doubling techniques encompass a diverse range of constructions and algorithms in algebra, analysis, combinatorics, and mathematical physics, all unified by the core idea of recursive doubling—typically via polynomials or polynomial mappings—of algebraic or analytic structures. This entry surveys foundational paradigms including Cayley–Dickson algebraic doubling, Newton–doubling for fast polynomial division, doubling methods for orthogonal polynomials in spectral problems, and the combinatorial regime of doubling constants in additive combinatorics and metric geometry.

1. Algebraic Doubling: The Cayley–Dickson Construction

The Cayley–Dickson construction generates algebras of dimension $2^{N+1}$ from those of dimension $2^N$ using pairs of quadratic polynomials $(f,g)$; these define the product in the new algebra $\mathbb{A}_{N+1}$ via

$$(a,b)*(c,d) = (f(a,b,c,d),\, g(a,b,c,d)), \quad a,b,c,d\in\mathbb{A}_N.$$

To maintain properties such as norm-multiplicativity and involutive anti-automorphisms, $f$ and $g$ must satisfy stringent quadratic identities. There exist precisely eight such "polynomial-doubling products," four of which recover the complex numbers ($\mathbb{C}$), quaternions ($\mathbb{H}$), and octonions ($\mathbb{O}$); these persist as division algebras up to $\dim=8$ (Bales, 2017).

For example, the classical product is

$$f(a,b,c,d) = a c - \bar{b} d, \qquad g(a,b,c,d) = d \bar{a} + b c,$$

which, under the recursive application, successively yields $\mathbb{R}\to\mathbb{C}\to\mathbb{H}\to\mathbb{O}$. For $N\geq4$, these algebras lose normed division properties and introduce zero divisors, but all such algebras remain "twisted group algebras" over the group $(\mathbb{Z}_2^N, \oplus)$.

2. Newton Doubling and Fast Polynomial Arithmetic

In computational algebra, the Newton-doubling method accelerates operations such as polynomial inversion and division. Given $f(x)\in D[x]$ with $f(0)=1$, the Newton iteration doubles the accuracy of $g_k(x)\approx f(x)^{-1} \bmod x^m$ at each step: $$g_{k+1}(x) = 2g_k(x) - g_k(x)^2 f(x) \pmod{x^{2m}}.$$ Starting from $g_0(x)=1$ (mod $x$), iterating $r=\lceil\log_2\ell\rceil$ times yields $g_r(x)\equiv f(x)^{-1} \bmod x^\ell$ for arbitrary $\ell$. This allows division

$a(x) = q(x) b(x) + r(x),\quad \deg\,r<m,$

to be reduced via reversal to a single truncated inversion, and it is performed in $O(M(n))$ time, with $M(n)$ the cost of degree-$n$ polynomial multiplication (Cao et al., 2011). Arbitrary (non-power-of-2) modulus exponents $\ell$ incur no asymptotic overhead.

3. Doubling Metrics and Approximation in Analysis

The "doubling" property of weights or metrics is central to approximation theory and optimization in high-dimensional metric spaces:

• Weights: A nonnegative function $w$ on $[-1,1]$ is doubling if there exists $C_d\geq1$ such that for any interval $I$,

$w(2I) \leq C_d\,w(I).$

For weighted polynomial approximation, this condition enables sharp equivalences between best $L_p$-approximation rates and moduli of smoothness, even when the weights have finitely many singularities. The realization and equivalence results for doubling weights extend classical Jackson- and Bernstein-type theorems, controlling all weighted polynomial approximability through doubling (Kopotun, 2014, Kopotun, 2014).

• Metric Spaces: A metric space is doubling if each ball of radius $r$ can be covered by $\lambda$ balls of radius $r/2$, with the smallest such $\lambda$ giving the doubling dimension $d=\log_2\lambda$. This property underpins algorithmic approaches in optimization, such as polynomial-time approximation schemes (PTAS/EPTAS) for the Maximum Scatter TSP in $d$-dimensional doubling metrics, where algorithmic performance crucially depends on the exponential scaling in $d$ induced by recursive packing and covering arguments (Kozma et al., 2015).

4. Polynomial-Doubling in Orthogonal Polynomial Systems and Physics

A general "doubling method" constructs new families of orthogonal polynomials by coupling a given family with a quadratic spectral transformation. For a sequence $\{P_n(x)\}$ of real orthogonal polynomials (with recurrence coefficients $A_n, C_n$), the doubled system is constructed via

$$Q_{2n}(x) = t^+_n\,R_n(\pi(x)) + t^-_{n-1}\,R_{n-1}(\pi(x)),\qquad Q_{2n+1}(x) = x\,R_n(\pi(x))$$

where $R_n$ is the "symmetrized" version of $P_n$, $\pi(x)=\tau_2 x^2+\tau_0$ is a quadratic spectral map, and the $t_n^\pm$ encode "hopping" parameters. The constraints required for the doubled system correspond to the recurrence and orthogonality structure.

This framework yields exact diagonalization of inhomogeneous Su-Schrieffer-Heeger (SSH) models. For instance, the standard SSH chain coincides with doubling Chebyshev polynomials; inhomogeneous models correspond to Krawtchouk or $q$-Racah polynomial doublings, allowing explicit construction of spectra and eigenstates in these quantum systems (Crampé et al., 19 Nov 2025). The method has further generalizations to multivariate and $q$-deformed setups.

5. Polynomial Doubling in Additive Combinatorics

In additive combinatorics, polynomial-doubling refers to the relationship between the doubling constant of a set and its structural properties. For $A\subset\mathbb{F}_2^n$, the doubling constant $K=|A+A|/|A|$ quantifies additive growth under sumsets. The polynomial Freiman–Ruzsa theorem asserts that if $K$ is bounded, $A$ is covered by at most $K^C$ cosets of a subspace $V$ of size at most $|A|$, where $C$ is universal.

Recent algorithmic advances yield efficient (classical and quantum) algorithms that construct such subspaces and coverings, with complexities polynomial in $K$, and provably optimal dependence on $K$ in the number of cosets and runtime. The analysis combines sampling, random projection (yielding Freiman isomorphisms), combinatorial covering lemmas, quadratic form learning via Fourier analysis, and energy increment arguments, all carefully tracking the polynomial dependencies (Arunachalam et al., 2 Sep 2025).

6. Extensions, Applications, and Theoretical Significance

Polynomial-doubling techniques permeate domains where recursive structure, dimension doubling, and scale control are central:

• In algebra, the Cayley–Dickson paradigm underpins the structure of real normed division algebras and their extensions as twisted group algebras.
• In computational mathematics, Newton–doubling and related recursive doubling schemes are foundational for quasi-linear time algorithms in polynomial arithmetic, matrix powering, and effective symbolic computation (Bostan et al., 2023).
• In approximation theory, doubling weights yield a precise moduli-approximation dictionary in $L_p$ and quasi-normed settings, generalizing classical unweighted results to broad families of weights, including those with singularities.
• In combinatorics and discrete optimization, the doubling property enables efficient covering, packing, and dimension reduction arguments, with implications for both algorithm design and combinatorial structure theorems.
• In mathematical physics, polynomial-doubling methods yield exactly solvable models with analytic characterizations of spectra and states, enabling systematic studies of multi-band, inhomogeneous, or $q$-deformed quantum systems.

Polynomial-doubling thus represents not an isolated methodology, but a suite of principles crossing deep algebraic, analytic, and algorithmic domains, unified by recursive leveraging of polynomial (typically quadratic) structure for scale amplification, structure preservation, and algorithmic efficiency.