Note on expanding implicit functions into formal power series by means of multivariable Stirling polynomials

Alfred Schreiber

arxiv: 2307.02638 · v3 · submitted 2023-07-05 · 🧮 math.CO

Note on expanding implicit functions into formal power series by means of multivariable Stirling polynomials

Alfred Schreiber This is my paper

Pith reviewed 2026-05-24 07:51 UTC · model grok-4.3

classification 🧮 math.CO

keywords implicit functionsformal power seriesBell polynomialsStirling polynomialspower series expansionmultivariable polynomialsTaylor coefficients

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The pith

A formal power series for the implicit function y(x) solving f(x,y)=0 can be constructed with coefficients depending only on the Taylor coefficients of f.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit formal power series expansion for y(x) where f(x,y(x))=0, with each coefficient of y expressed solely in terms of the coefficients f_{m,n} from the bivariate series for f. It achieves this by applying partial Bell polynomials together with their orthogonal companions. A sympathetic reader would care because the construction supplies a direct, non-recursive route from the given coefficients of f to those of y without solving the implicit relation order by order.

Core claim

Given a bivariate formal power series f(x,y) whose Taylor coefficients f_{m,n} are known, there exists a unique formal power series y(x) satisfying f(x,y(x))=0 whose own coefficients are determined exclusively by the f_{m,n} through the action of partial Bell polynomials and their orthogonal companions.

What carries the argument

Partial Bell polynomials and their orthogonal companions (equivalently, multivariable Stirling polynomials), which convert the coefficients f_{m,n} directly into the coefficients of the series for y.

If this is right

The coefficients of y(x) become explicit algebraic expressions in the f_{m,n} rather than the result of iterative substitution.
No auxiliary data beyond the Taylor coefficients of f are required to obtain the full series for y.
The same polynomials furnish a uniform method for any f whose series is given, independent of the particular orders at which the coefficients vanish.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique may extend to implicit relations in more than two variables by replacing the partial Bell polynomials with their higher-dimensional analogues.
The resulting coefficient formulas could be implemented directly in computer-algebra systems to perform series reversion or implicit solving without numerical iteration.
Because the construction is purely formal, it remains valid over any ring in which the relevant factorials are invertible.

Load-bearing premise

The bivariate function f admits a formal power series expansion at the origin and the implicit-function problem admits a unique formal power-series solution y(x).

What would settle it

A concrete choice of coefficients f_{m,n} for which the series produced by the Bell-polynomial formula fails to satisfy f(x,y(x)) identically zero through all orders when substituted back into f.

read the original abstract

Starting from the representation of a function $f(x,y)$ as a formal power series with Taylor coefficients $f_{m,n}$, we establish a formal series for the implicit function $y=y(x)$ such that $f(x,y)=0$ and the coefficients of the series for $y$ depend exclusively on the $f_{m,n}$. The solution to this problem provided here relies on using partial Bell polynomials and their orthogonal companions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit formula for implicit-function series coefficients via partial Bell polynomials and their orthogonal companions, but the abstract supplies no derivation or sample computation to check it.

read the letter

The main thing to know is that this note claims an explicit, non-recursive expression for every coefficient in the formal power series y(x) solving f(x,y)=0, written solely in terms of the Taylor coefficients f_{m,n} through partial Bell polynomials and their orthogonal companions. That is the stated advance over the usual recursive or Lagrange-inversion approaches. If the derivation is correct, the result is a clean combinatorial identity that stays inside the ring of formal power series and uses only standard properties of those polynomials. The stress-test note is right that the claim follows once the orthogonality relation is granted and the usual non-degeneracy conditions on f hold. The paper does that part cleanly. The soft spot is the complete absence of any worked example, explicit formula for low degrees, or comparison with existing formulas in the abstract; without those, it is impossible to judge whether the expression is shorter or more useful than what is already in the literature on multivariable Lagrange inversion or Faà di Bruno-type formulas. The full manuscript presumably contains the details, but on the supplied text alone the soundness cannot be confirmed. This is a narrow technical note aimed at people who already work with Bell polynomials and formal generating functions. A reader who needs coefficient formulas for implicit series in enumeration problems might find it worth checking. It is not a major reorganization of the field, but the explicitness claim is the sort of thing that deserves a referee to verify the algebra and the literature comparison. I would send it to peer review rather than desk-reject.

Referee Report

0 major / 3 minor

Summary. The paper claims to establish an explicit formal power series expansion for the unique solution y(x) to the implicit equation f(x,y)=0 (with f(0,0)=0 and f_y(0,0)≠0), where each coefficient of y depends only on the given bivariate Taylor coefficients f_{m,n} of f; the construction relies on partial Bell polynomials together with their orthogonal companions (multivariable Stirling polynomials).

Significance. If the claimed coefficient formulas are correct, the note supplies a direct, non-iterative expression for the implicit-function series in the ring of formal power series. This is potentially useful for symbolic computation and for combinatorial enumeration problems that reduce to implicit equations; the reliance on standard Bell-polynomial identities is a positive feature for verifiability.

minor comments (3)

The title refers to 'multivariable Stirling polynomials' while the abstract and introduction speak of 'partial Bell polynomials and their orthogonal companions.' A brief clarifying sentence relating the two families (e.g., via the known change-of-basis between Bell and Stirling polynomials) would remove any potential confusion for readers.
The manuscript should include at least one fully worked low-order example (e.g., the first three coefficients of y(x) for a concrete f such as f(x,y)=y-x-y^2) so that the claimed dependence on the f_{m,n} can be verified by direct substitution.
Notation for the orthogonal companion polynomials is introduced without an explicit reference or self-contained definition; adding a short appendix or a displayed identity that states the orthogonality relation used would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of significance, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring response or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard Bell polynomial identities

full rationale

The paper assumes the standard formal implicit-function setup (f admits a power series with f(0,0)=0 and nonzero linear term in y) and supplies an explicit formula for the coefficients of the unique formal solution y(x) in terms of the given f_{m,n} by composing with partial Bell polynomials and their orthogonal companions. These polynomials are independently defined combinatorial objects whose orthogonality relations are external to the paper; each coefficient of y at order k is thereby a finite, explicit combination of f_{m,n} with m+n ≤ k+1. No step equates the output series to a fitted parameter, renames a known result, or reduces the central claim to a self-citation whose content is unverified. The claim that the coefficients depend exclusively on the f_{m,n} therefore follows directly from the explicit construction rather than by construction from the inputs themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the algebraic closure properties of the formal power series ring under composition and the known inversion formulas for partial Bell polynomials; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The ring of formal power series in two variables admits a unique solution y(x) to f(x,y)=0 when the constant term condition is satisfied.
Required for the existence of the target series y(x).

pith-pipeline@v0.9.0 · 5583 in / 1237 out tokens · 35153 ms · 2026-05-24T07:51:02.765862+00:00 · methodology

Note on expanding implicit functions into formal power series by means of multivariable Stirling polynomials

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)