Multivariate Polynomial Decomposition

Updated 22 August 2025

Multivariate polynomial decomposition is the process of expressing a polynomial in several variables as a structured composition of simpler functions, leveraging concepts from algebraic geometry and tensor methods.
Algorithmic approaches incorporate techniques such as factorization, differentiation, and tensor decompositions to efficiently recover underlying components, particularly in the tame case.
Advances in invariant decomposition, positivity, and optimization link these methods to applications in system identification, quantum physics, and statistical analysis.

Multivariate polynomial decomposition is the paper of expressing a polynomial in several variables as a composition or structured combination of simpler functions—often with a univariate or low-dimensional internal structure—and developing both the theoretical underpinnings and practical algorithms to obtain such representations. This topic lies at the intersection of algebraic geometry, computational algebra, tensor methods, and applications in areas such as symbolic computation, optimization, system identification, and quantum physics.

1. Foundational Principles and Decomposition Types

A central question is, given a multivariate polynomial $f$ , can it be written as a composition $f = g \circ h$ where $g$ and $h$ are polynomials (or rational functions) and $h$ has lower complexity or a certain structured form? In the univariate case, functional decomposition is characterized by classical results (such as Ritt's theorems) with well-understood uniqueness and algorithmic properties in the "tame" case. In the multivariate setting, decompositions can be:

Functional (Composition) Decomposition: $f(x) = g(h(x))$ , with $h$ a multivariate mapping, and $g$ typically univariate or simpler.
Direct Sum and Diagonalization: Decomposition as a sum of polynomials, each depending on a disjoint subset of variables, possibly after a linear change of variables (Fang et al., 3 Mar 2025).
Decoupled/Tensor-based Decompositions: Expressing a polynomial vector $f(u)$ as $W g(V^T u)$ with $g$ a vector of univariate functions, $W, V$ matrices, thus "decoupling" cross-variable interactions (Dreesen et al., 2014, Hollander et al., 2016, Usevich et al., 2017).
Product of Linear Forms: Writing $f(x)$ as a product $\prod l_i(x)^{\alpha_i}$ , associated with Waring decomposition or symmetric tensor decomposition (Koiran et al., 2018), and sometimes solving via Lie algebraic methods.

The nature of the decomposition and its uniqueness, as well as the complexity of finding it, depend critically on the field's characteristic and the structure of the polynomial.

2. Field Characteristic, Tame vs. Wild Behavior, and Additive Polynomials

The arithmetic of the base field, notably its characteristic $p$ , exerts a strong influence:

Tame Case: If $p$ does not divide the degrees of the composing polynomials, decompositions are generally unique up to well-understood ambiguities (linear or trivial transformations). Classical algorithms (e.g., using the separation of variables via $h(x)-h(y)\mid f(x)-f(y)$ ) yield polynomial-time solutions (Giesbrecht, 2010).
Wild Case: When $p$ divides the degree of a factor, especially for additive polynomials (those $f$ with $f(x+y) = f(x)+f(y)$ , i.e., $f(x) = \sum a_i x^{p^i}$ ), decomposition is highly non-unique. The number of inequivalent decompositions can be super-polynomial or even exponential in degree. The structure of decompositions is tightly linked to the vector space of roots over $\mathbb{F}_p$ , and counting decompositions relates to Gaussian binomial coefficients (Giesbrecht, 2010).

An explicit criterion for composability is: $f(x)=g(h(x)) \iff h(x)-h(y)\mid f(x)-f(y).$

For additive polynomials of degree $n=p^v$ , the number of bidecompositions can be as large as

$\prod_{\alpha=1}^{v'} \frac{p^v-p^\alpha}{p^{v'}-p^\alpha},$

where $v'$ depends on the dimension of a chosen subspace related to the decomposition.

3. Algorithmic Approaches and Complexity

Significant algorithmic progress covers both the exact and noisy cases:

Separated Factors/Factoring: For functional decomposition, the core algorithm often involves factoring $f(x) - f(y)$ or its analogs and reconstructing $h$ from divisors, yielding efficient algorithms in the tame regime. For additive polynomials, the algorithm recursively peels off indecomposable factors based on vector space structure (Giesbrecht, 2010).
Reduction and Lifting: For multivariate $f$ , reducing to a univariate case (e.g., via substitutions that make $f$ monic in one variable), decomposing, and lifting back to the multivariate context using power series reversion and Newton iteration (Giesbrecht, 2010).
Recombination via Darboux Polynomials: Decomposition of rational functions via the logarithmic derivative approach, leveraging the property that Darboux polynomials decompose multiplicatively under derivation, reduces combinatorial selection of factors to solving a linear algebra problem (the kernel of a coefficient matrix) (Chèze, 2010). This is especially effective for sparse polynomials via Newton polytope techniques.
Differentiation and Homogenization: Algorithms such as those by Ye, Dai, Lam and Faugère, Perret use differentiation to recover the "right factor space" and homogenization to facilitate handling nonhomogeneous cases, with provable success for quartic homogeneous polynomials over $\mathbb{C}$ and high probability over large finite fields (Zhao et al., 2010).
Tensor/CP Decomposition Approaches: By stacking Jacobian matrices (first-order information) or polynomial coefficients into a three-way tensor, CP decomposition can recover the transformation matrices $V, W$ and the univariate nonlinearities $g_i$ . Uniqueness is assured under Kruskal-type rank conditions, and weighted CPD generalizations accommodate approximate/noisy data (Dreesen et al., 2014, Hollander et al., 2016, Usevich et al., 2017).

In many cases, complexity is polynomial time for well-structured ("nice") families of polynomials and quasi-polynomial or exponential otherwise, especially in the presence of field characteristic obstacles or for general rational functions.

4. Extensions: Rings, Modules, and Special Structures

$S_n$ -module Decomposition: The decomposition of polynomial rings as $S_n$ -modules can be described combinatorially by enumerating multiset tableaux of a prescribed shape and content, with explicit connections to representation theory, invariant rings, and generating sets (Orellana et al., 2019).
ANOVA and Dimensional Decomposition: In numerical and statistical contexts, the ANOVA decomposition splits $f$ into components corresponding to subsets of variables, facilitating sparse recovery of functions with low effective dimension and interpretability of variable dependencies (Potts et al., 2019).
Generalized Polynomial Dimensional Decomposition (GPDD): For stochastic problems with dependent variables, GPDD offers a hierarchical expansion of $y(X)$ in terms of measure-consistent orthogonal polynomials, organizing terms by interaction degree and yielding efficient truncated approximations (Rahman, 2018).
Infinite-Dimensional Decomposition in Uncertainty Quantification: MDFEM uses infinite-variate decomposition for elliptic PDEs with stochastic inputs, allowing the reduction of computational complexity through anchored decompositions and quasi-Monte Carlo integration (Nguyen et al., 2018).

5. Factorization and Special Decompositions

Product of Linear Forms: Efficient "black-box" algorithms exist for detecting and constructing factorizations into products of linear forms, using Lie algebraic structures, simultaneous diagonalization, bivariate projections, and geometric zero-set methods. This underpins approaches to Waring decomposition/symmetric tensor decomposition (Koiran et al., 2018).
q-Integer Linear Decomposition: For $q$ -hypergeometric contexts, multivariate polynomials can be uniquely decomposed into products of univariate polynomials in $q$ -integer monomial arguments by geometric methods (Newton polytopes) or iterative bivariate reductions, with efficient algorithms applicable over unique factorization domains (Giesbrecht et al., 2020).
Handelman Decomposition: For nonnegativity certification and polynomial optimization on polytopes, Handelman's theorem enables decompositions into nonnegative sums of affine monomials in the constraints, facilitating PTAS algorithms in fixed dimensions (Loera et al., 2016).

6. Advanced Topics: Polynomial-Exponential Decomposition and Hankel Methods

Polynomial-Exponential Decomposition from Moments: Any function representable as a finite sum of polynomial-exponential terms $\sigma(y) = \sum \omega_i(y) e^{\langle \xi_i, y \rangle}$ can be reconstructed from its truncated multi-index moment/Hankel matrix using eigen-decomposition of companion pencils and the duality between Artinian Gorenstein algebras and Hankel operators. Kronecker's theorem is generalized: finite-rank multivariate Hankel operators correspond precisely to such decompositions (Mourrain, 2016, Harmouch et al., 2017). Efficient algorithms exploit orthogonalization and eigen-analysis to recover frequencies and weights.
Structured Hankel Decomposition and Error Correction: SVD-based extraction of algebra bases, eigen-analysis of multiplication operators, and explicit weight recovery formulae provide robust decomposition, even with noisy moments. Rescaling and Newton-type iterations address numerical stability and error correction (Harmouch et al., 2017).

7. Decomposition with Invariance, Positivity, and Quantum-inspired Structures

Invariant and Positive Polynomial Decomposition: Recent frameworks incorporate group invariance and positivity directly, generalizing tensor network decompositions from quantum physics to multivariate polynomial settings. Invariant decompositions use group actions and Jordan algebra structures to characterize possible decompositions, with corresponding notions of separable and sum-of-squares (sos) decompositions. Rank inequalities, border rank approximations, and undecidability phenomena in cyclically invariant scenarios reveal deep connections to tensor rank theory and computational complexity (Cuevas et al., 2021).
Jordan Algebras and Simultaneous Direct Sum Decomposition: For sets of polynomials, the simultaneous direct sum decomposition is governed by the structure of the shared center algebra, a special Jordan algebra derived from their Hessians. Existence of a complete set of orthogonal idempotents in this algebra is equivalent to the possibility of such a decomposition (Fang et al., 3 Mar 2025).

The landscape of multivariate polynomial decomposition encompasses a diverse array of algebraic, algorithmic, and analytic tools. The field characteristic, algebraic invariants (centers, idempotents, Lie algebras), combinatorial models, and connections to tensor decompositions all play fundamental roles. The interplay between uniqueness, computational complexity, and application requirements drives ongoing research, particularly in high-dimensional, noisy, or structured data scenarios. Notably, methods that couple algebraic geometry (e.g., Gröbner bases, Hankel operators) with numerical linear algebra (SVD, eigenvalue problems) are central to modern algorithmic advances, enabling practical decomposition and factorization of polynomials in both symbolic and data-driven settings.