Power-Series Decomposition

• Power-series decomposition is a method that rewrites complex series into manageable components such as homogeneous parts, conjugates, and factorized forms.
• It clarifies analytic properties by delineating convergence domains, singularities, and geometric structures in both one-variable and multivariate settings.
• The approach enables efficient computational recovery and analysis through finite-dimensional linear algebra and explicit reconstruction from moments, coefficients, or derivatives.

Power-series decomposition denotes a family of constructions in which a formal or analytic power series, or an object encoded by power-series coefficients, is represented through simpler components such as homogeneous parts, compositional conjugates, inverse-series expansions, infinite products, partition-indexed sums, elementary multivariate series, or polynomial-exponential terms. The literature uses the phrase in several non-equivalent but structurally related senses. In one-variable formal dynamics it includes conjugacy and inverse-series formulas; in $q$-series it includes Euler-type product factorizations; in several complex variables it includes decomposition into series with half-space or wedge domains of convergence; and in moment problems it includes extraction of finitely many polynomial-exponential components from truncated series data (Schreiber, 2023, Schneider et al., 30 Jan 2025, Balakumar, 2021, Mourrain, 2016).

1. Formal setting and elementary operations

A basic setting is the ring of formal power series, or the algebra of convergent complex power series inside a disk of convergence. For involutory one-variable series, the defining functional equation is

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$

with coefficients in a field of characteristic zero (Schreiber, 2023). For multivariate systems,

$\varphi=\sum_{k=0}^{\infty}\varphi_k,$

where each $\varphi_k$ is homogeneous of degree $k$, invertibility under composition is characterized by

$\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$

the invertible series form the formal transformation group $G_\infty(n,K)$ (Zhang, 2022).

In the analytic theory of complex power series, absolute convergence is the central organizing principle. It justifies rearrangement, grouping, Cauchy products, composition, division, differentiation, and change of center, and it yields structural facts such as uniqueness of coefficients, Taylor expansion, the Principle of Identity, and the Principle of Isolated Zeros (1207.1472). A representative change-of-center formula is

$$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$

valid for $z_0$ inside the disk of convergence (1207.1472).

These foundational results delimit what counts as a legitimate decomposition. In formal settings the emphasis falls on algebraic invertibility and composition. In analytic settings the emphasis falls on convergence, rearrangement, and localization of singularities.

2. Conjugacy, inversion, and finite-order series

A central compositional decomposition concerns involutory functions. Every nontrivial involutory function $f\neq \mathrm{id}$ is a compositional conjugate of negative identity: $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$0 Equivalently, all nontrivial involutions lie in the conjugacy class of $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$1 in the group of compositional power series (Schreiber, 2023). This representation is constructive. If $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$2 is involutory, then the even coefficients of $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$3 satisfy the recursion

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$4

while the odd coefficients $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$5 are arbitrary, subject to $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$6 (Schreiber, 2023).

The same paper gives an explicit coefficient decomposition through multivariable Lah polynomials: $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$7 where $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$8 are partial Bell polynomials and $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$9 are orthoinverse polynomials (Schreiber, 2023). Thus the coefficients of an involution are not merely recursively constrained; they admit a closed-form description in a Bell-polynomial/orthoinverse basis.

Inverse-series decomposition is treated from a different angle in the logarithmic form of Lagrange inversion. For

$\varphi=\sum_{k=0}^{\infty}\varphi_k,$0

the compositional inverse is represented through a logarithmic generating expression built from derivatives of $\varphi=\sum_{k=0}^{\infty}\varphi_k,$1, and the coefficient formula

$\varphi=\sum_{k=0}^{\infty}\varphi_k,$2

is recovered in that framework (Dzhumadil'daev, 2016). The accompanying calculus of differential operators uses Bell polynomials and two operator multiplications, called black and white multiplications, to organize higher-order chain-rule structure (Dzhumadil'daev, 2016).

Finite-order multivariate series exhibit a parallel linearization phenomenon. If $\varphi=\sum_{k=0}^{\infty}\varphi_k,$3 is periodic and $\varphi=\sum_{k=0}^{\infty}\varphi_k,$4 is diagonalizable, then $\varphi=\sum_{k=0}^{\infty}\varphi_k,$5 is conjugate to $\varphi=\sum_{k=0}^{\infty}\varphi_k,$6; over $\varphi=\sum_{k=0}^{\infty}\varphi_k,$7, this classifies all periodic series in $\varphi=\sum_{k=0}^{\infty}\varphi_k,$8 (Zhang, 2022). When $\varphi=\sum_{k=0}^{\infty}\varphi_k,$9 with $\varphi_k$0 a primitive $\varphi_k$1-th root of unity, periodicity imposes the recursion

$\varphi_k$2

so the higher homogeneous components are explicitly constrained by the lower ones (Zhang, 2022). This suggests a sharp distinction between arbitrary invertible series and special finite-order classes for which complete conjugacy descriptions are available.

3. Factorization, powers, and partition-indexed decompositions

A second major meaning of power-series decomposition is multiplicative factorization. Any power series with unit constant term can be written as

$\varphi_k$3

The coefficients $\varphi_k$4 and exponents $\varphi_k$5 are linked by explicit partition sums (Schneider et al., 30 Jan 2025). In one direction,

$\varphi_k$6

where $\varphi_k$7 is the rising factorial. In the other,

$\varphi_k$8

with

$\varphi_k$9

The factorization is therefore explicit in both directions, and a $k$0-analogue replaces the rising-factorial factors by Gaussian $k$1-binomial coefficients (Schneider et al., 30 Jan 2025).

Powers of power series also encode compositions of integers. If

$k$2

then

$k$3

counts generalized compositions of $k$4 into $k$5 parts over an arbitrary commutative ring (Janjic, 2010). The recurrence

$k$6

makes the decomposition recursive, while specific choices of $k$7 recover classical restricted-composition formulas (Janjic, 2010).

In this combinatorial sense, decomposition is neither conjugacy nor inversion. It is a re-expression of global coefficients through partitions, divisor sums, or ordered part structures. A common misconception is that power-series decomposition is necessarily additive or compositional; the $k$8-factorization and generalized-composition formulas show that multiplicative and partition-theoretic decompositions are equally central.

4. Multivariate analytic geometry and singularity-theoretic decompositions

In several complex variables, decomposition is tied to the geometry of domains of convergence. For

$k$9

the domain of convergence $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$0 is a logarithmically convex complete Reinhardt domain, and its logarithmic image

$\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$1

is a convex subset of $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$2 (Balakumar, 2021). An elementary power series is one whose logarithmic domain is a half-space; a simple power series has logarithmic domain equal to an intersection of finitely many half-spaces, that is, a wedge (Balakumar, 2021).

The main structure theorem states that every power series admits a decomposition into elementary power series: $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$3 where each $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$4 has a half-space logarithmic domain and the largest open subset on which all $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$5 and $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$6 converge absolutely is exactly $\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$7 (Balakumar, 2021). A second decomposition uses wedges formed by intersections of pairs of supporting half-spaces. The support function

$\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$8

gives the half-space representation

$\varphi_0=0,\qquad \det(\varphi_1)\ne 0;$9

so the decomposition is controlled by the convex geometry of $G_\infty(n,K)$0 (Balakumar, 2021). The same framework yields the statement that in each fiber of the restriction of the absolute map to the boundary of the domain of convergence there exists a singular point of the power series (Balakumar, 2021).

A singularity-theoretic analogue appears for two-variable formal series in positive characteristic. For $G_\infty(n,K)$1, the Eggers-Wall tree $G_\infty(n,K)$2 organizes the branches of $G_\infty(n,K)$3, and $G_\infty(n,K)$4 admits Eggers decomposition if and only if $G_\infty(n,K)$5 satisfies the Eggers condition: $G_\infty(n,K)$6 for every marked point $G_\infty(n,K)$7 root of $G_\infty(n,K)$8 (Felipe et al., 28 Sep 2025). In characteristic zero this decomposition is always possible; in positive characteristic it may fail, and the tree gives a necessary and sufficient combinatorial criterion (Felipe et al., 28 Sep 2025).

These results place decomposition at the interface of coefficient asymptotics, convex geometry, and singularity theory. The decomposed pieces encode not only local algebra but also the global shape of the convergence domain or the branch structure of a plane curve germ.

5. Reconstruction from moments, coefficients, and derivatives

Power-series decomposition is also a reconstruction problem: given truncated coefficients or moments, recover the minimal structured components that generated them. In the multivariate polynomial-exponential setting, one seeks

$G_\infty(n,K)$9

with polynomial weights $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$0 and frequencies $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$1, from truncated moment data (Mourrain, 2016). The theory is built on the duality between $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$2 and $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$3, on the Hankel operator $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$4, and on the Artinian quotient algebra $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$5 (Mourrain, 2016).

The multivariate Kronecker theorem states that $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$6 has finite rank if and only if $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$7 is polynomial-exponential: $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$8 where $$f(z)=\sum_{p=0}^\infty b_p (z-z_0)^p,\qquad b_p=\sum_{n=p}^\infty a_n \binom{n}{p} z_0^{\,n-p},$$9 is the dimension of the space spanned by $z_0$0 and its derivatives (Mourrain, 2016). The flat extension criterion provides a rank-stability condition ensuring that a truncated Hankel matrix extends uniquely to a full series of the same rank. Algorithmically, one constructs bases of the Artinian Gorenstein algebra by a Gram-Schmidt orthogonalization process, builds multiplication matrices, and extracts frequencies and weights by eigenanalysis (Mourrain, 2016).

A related finite-degree problem is power-sum decomposition of polynomials: $z_0$1 For quadratic $z_0$2 and $z_0$3, an algorithm based on low-order partial derivatives, random restriction to an $z_0$4-dimensional subspace, singular-value analysis, and standard tensor decomposition succeeds for $z_0$5 generic components, and more generally the method handles $z_0$6 generic degree-$z_0$7 polynomials for any $z_0$8 (Bafna et al., 2022). This is a polynomial rather than a power-series problem, but it is closely related through its use of derivative subspaces and structured low-rank recovery.

The unifying theme is that decomposition can be reduced to finite-dimensional linear algebra once the correct algebra of moments, derivatives, or Hankel structure has been identified.

6. Computational and dynamical realizations

Algorithmic work has isolated classes of power series for which composition and decomposition are computationally efficient. A composition sequence is a fixed sequence of basic operations—addition, scalar multiplication, powering, root, inversion, exponential, logarithm—applied iteratively starting from $z_0$9. For power series built by such sequences, both composition $f\neq \mathrm{id}$0 and the inverse decomposition problem can be carried out in $f\neq \mathrm{id}$1 or $f\neq \mathrm{id}$2, depending on whether exponentials or logarithms occur (0804.2337). This yields fast change-of-basis algorithms for Euler, Bernoulli, Fibonacci, Laguerre, Hermite, Jacobi, Krawtchouk, Meixner, and Meixner-Pollaczek polynomial bases (0804.2337).

In nonlinear ODEs, spectral power series decompose trajectories into exponentially weighted modes determined by the eigenvalues of the linearized system. For

$f\neq \mathrm{id}$3

the ansatz

$f\neq \mathrm{id}$4

leads to the linear system

$f\neq \mathrm{id}$5

with $f\neq \mathrm{id}$6 assembled from lower-order coefficients (Basor et al., 2023). The coefficients obey explicit decay estimates, and numerical experiments show that truncations such as $f\neq \mathrm{id}$7 or $f\neq \mathrm{id}$8 closely match numerical integration for $f\neq \mathrm{id}$9 (Basor et al., 2023).

In power-system simulation, decomposition appears in Taylor-series model reduction. Around an equilibrium $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$00,

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$01

and the higher-order tensors are compressed by CP tensor decomposition (Osipov et al., 2019). On the 140-bus, 48-machine NPCC system, the tensor-based reduction yields RMS rotor angle error $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$02 degrees, compared with $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$03 degrees for linear reduction, while both reduced models run in about $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$04 s versus $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$05 s for the full model (Osipov et al., 2019). Here power-series decomposition is not purely symbolic; it is a surrogate-model architecture that trades tensor rank and Taylor order against runtime and fidelity.

7. Functional-analytic and operator-valued extensions

The notion of decomposition extends beyond scalar series to spaces, valuations, and matrices over formal power series rings. For valuations on lattice polygons with values in $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$06, every $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$07-equivariant valuation decomposes into dilative components: $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$08 The simple part is classified by $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$09-invariant formal power series $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$10 satisfying

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$11

while the low-dimensional part is determined by the value on a point and by the value on a segment (Boroczky et al., 6 Oct 2025). This is an explicit graded decomposition in a formal-power-series target.

In functional analysis, power series spaces of infinite type also admit decomposition theorems. For

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$12

a Pełczyński-Vogt decomposition result states that if a locally convex Hausdorff space $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$13 is mutually complemented with $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$14, then

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$15

under the stated nuclearity and stability assumptions (Debrouwere et al., 28 Nov 2025). Combined with Gabor-frame methods, this yields the sequence-space representation

$$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$16

for multiplier spaces of Gelfand-Shilov spaces of Beurling type (Debrouwere et al., 28 Nov 2025).

An operator-theoretic analogue occurs for matrices over formal power series rings. A normal matrix over $$f\circ f=\mathrm{id}, \qquad f(x)=\sum_{n\ge 1} f_n \frac{x^n}{n!}, \quad f_1\neq 0,$$17 is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the associated spectral projections have entries in the ring (Dai et al., 9 Feb 2026). The resulting algorithm reduces diagonalizability to a splitting test for the minimal polynomial and a membership test for the spectral projections (Dai et al., 9 Feb 2026).

Taken together, these extensions show that power-series decomposition is not confined to scalar expansions. It also governs graded valuations, sequence-space structure, and spectral decompositions over formal power series rings.