Residues of skew rational functions

LFANT); Xavier Caruso (IMB

REVIEW 2 major objections 4 minor 1 cited by

Residues of skew rational functions

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Pith's one-line read Skew rational functions over $K[X;\theta]$ admit Taylor-like expansions and a theory of residues with a residue formula and a change-of-variables rule.

desk verdict A real first systematic theory of residues for skew rational functions, with detailed proofs, but the main residue formula is false as stated over imperfect fields; fixable with a perfectness or separable-poles hypothesis. read the letter →

arxiv 1908.08430 v2 pith:HUDVIMB2 submitted 2019-08-22 math.RA cs.SC

Xavier Caruso (IMB , LFANT) This is my paper · ORCID

classification math.RAcs.SC MSC 16S36

keywords skewpolynomialsOreringsrationalfunctionsTaylorexpansionsresidueformulachangeofvariablessectionoperatorslinearisedReed-Solomoncodes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to carry the analytic side of rational functions—Taylor expansion, residues, the residue theorem, substitution rules—into the noncommutative setting of skew polynomials. In the ring $K[X;\theta]$, multiplication obeys $X a = \theta(a) X$, so evaluating at a point is not immediate; the paper shows that every element of the fraction field nevertheless expands in powers of $X^r - z$ around each nonzero separable closed point, and in Laurent series around $0$ and $\infty$. The coefficient of the degree $-1$ term defines the skew residue, and the paper proves a residue formula and a change-of-variables formula for endomorphisms of the fraction field. If the construction works, skew rational functions inherit a residue calculus that reduces to the classical one in the commutative case and should be usable in coding theory.

What carries the argument

The engine is the Taylor-like isomorphism $\tau_N$ for the completion $\hat{A}_N$ of the Ore algebra at a separable central polynomial $N$. It is built by a skew Hensel construction: one finds $C\in\hat{C}_N$, $C\equiv 1\pmod N$, such that $N(CX)=0$, so $\tau_N$ sends $X$ to $CX$ and $N$ to $T$; the inverse is then an isomorphism $\hat{A}_N\simeq(A/NA)[[T]]$. The section operators $\sigma_j(f)=\sum_i a_{j+ir}Y^i$ split $A$ over $C=K[Y^{\pm 1}]$ and satisfy $\sigma_j\circ\gamma_C=N_j(C)(\gamma_C\circ\sigma_j)$, which is what connects residues under change of variables. In the canonical case ($\operatorname{char}K\nmid r$), the preferred $\tau_{z,\mathrm{can}}$ corresponds to the derivation $\partial_{\mathrm{can}}=r^{-1}Y^{-1}$ acting by $X\mapsto r^{-1}X^{1-r}$, and its divided powers give the Taylor coefficients.

What would settle it

Take $K=\mathbf{C}$ with $\theta$ complex conjugation, $r=2$, and $f=(X^2-z)^{-2}$ for a nonzero $z$; compute the coefficient of $T^{-1}$ in $\sigma_1(TS_z(f))$ for two different $z$-admissible isomorphisms $\tau_z$, for instance the canonical one and one conjugated by a nontrivial $V\in(C/NC)[[T]]$ with $V\equiv 1\pmod T$. If the coefficient changes, the partial residue at a double pole is choice-dependent, so the residue formula cannot extend beyond simple poles without extra structure.

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Extended reading notes

Core claim

Over a field $K$ with an automorphism $\theta$ of finite order $r$, set $Y=X^r$, let $F=K^{\theta}$, and let $A=K[X^{\pm 1};\theta]$. The paper's central claim is that every skew rational function $f\in \operatorname{Frac}(A)$ has a Taylor expansion $TS_z(f)\in (A/NA)((T))$ at any nonzero $z$ in a separable closure $F^s$, where $N$ is the minimal polynomial of $z$; the expansion is constructed from a noncommutative Hensel-type isomorphism $\tau_N:\hat{A}_N\to (A/NA)[[T]]$. The skew residue $sres_z(f)$ is the coefficient of $T^{-1}$, and its $j$-th partial residue is $sres_{z,j}(f)=\iota_z\circ\sigma_j(sres_z(f))$. The main theorems are: the sum of all $0$-th partial residues is $0$ (Theorem 3.2.1); if $f$ has at most a simple pole at every nonzero point, then $\sum_z sres_{z,j}(f)=0$ for every $j$ (Theorem 3.2.2); and under an endomorphism $\gamma_C(X)=CX$ the residue transforms as $\gamma(sres_{\gamma_\star z}(f))=sres_z(\gamma(f)\,d\gamma(Y)/dY)$, up to an explicit conjugation by a norm-one element of $C'$ when $C$ is not central (Theorems 3.3.2 and 3.3.7). When $\operatorname{char}K\nmid r$, a canonical choice $\tau_{z,\mathrm{can}}$ exists and makes the canonical residues computable through divided powers of the canonical derivation.

Load-bearing premise

The load-bearing premise is that $f$ has at most a simple pole at every nonzero closed point; if any nonzero pole is higher, the partial residues entering the formula are not canonically defined.

Editorial extensions

If this is right

The $0$-th partial residue at any point is independent of choices and equals the classical residue of the differential form $\sigma_0(f)\,dY$; the skew theory therefore contains the commutative residue theory as its $0$-th section.
Under the simple-pole condition on all nonzero points, every partial residue $sres_{z,j}(f)$ is canonical and Theorem 3.2.2 gives $\sum_{z\in F^s\sqcup\{\infty\}} sres_{z,j}(f)=0$ for all $j$.
For endomorphisms with $C\in\operatorname{Frac}(Z)$, the change-of-variables formula holds without a conjugating factor; for general $C$, Theorem 3.3.7 makes the conjugation by $U$ explicit from the norm-one equation $(\mathrm{id}\otimes\theta)(U)/U=C/\sqrt[r]{N_{C/Z}(C)}$.
When $\operatorname{char}K\nmid r$, canonical residues satisfy $sres_{z,j,\mathrm{can}}(f)=r\zeta^{-j}\rho_\zeta(y^{j+r-1}\sigma_j(f)\,dy)$ with $\zeta^r=z$, so they are computable by ordinary residue calculations.
The paper announces that this residue calculus describes the duals of linearised Reed-Solomon codes, including Gabidulin codes, in a forthcoming companion article.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the simple-pole hypothesis is dropped, the ambiguity in $\tau_z$ can be viewed as gauge freedom; one could develop a residue calculus in which the chosen Taylor isomorphism is part of the data, and the paper's Eq. (21) suggests the naive sum carries no information for $j\neq0$, so a truly noncommutative residue theorem needs an extra global constraint.
A testable extension is whether the change-of-variables law is functorial under composition of endomorphisms: the conjugating elements $U$ should satisfy a cocycle condition, and checking that would turn the residue machinery into a natural transformation.
In characteristic dividing $r$, Proposition 1.4.4 rules out a canonical derivation with vanishing $p$-th power; an extension could invoke divided-power or Frobenius-descent data, and small finite-field computations could test whether a modified residue formula still holds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuine first systematic theory of residues for skew rational functions, with detailed proofs and an honest account of what is canonical and what is not. But the main residue formula is false as stated. The proof of Theorem 3.2.1 sums classical residues only over F^s, and over an imperfect field that sum can miss a purely inseparable pole with nonzero residue. The theorem needs a perfectness or all-poles-separable hypothesis, or a definition of residues at inseparable points.

What is genuinely new: the Taylor-like expansions around separable points (Theorems 2.2.1 and 2.2.5), the residue definitions 3.1.1–3.1.3, and the change-of-variables theorems 3.3.2 and 3.3.7. The author is careful about choice-dependence: he states explicitly that residues at higher-order poles are not canonical, and he isolates the canonical case p∤r cleanly. The algebraic machinery (endomorphism classification, section operators, derivations) is well organized, and the proofs are thorough. I did not find gaps elsewhere.

The soft spot is load-bearing, and the stress-test note lands. In Theorem 3.2.1, Proposition 3.1.5 identifies sres_{z,0}(f) with the classical residue of σ0(f)dY, and then the proof applies the classical residue formula over F^s. But the classical formula is over the algebraic closure, not the separable closure. Over F=F_p(t), take r=1, θ=id, f=Y^{p-1}/(Y^p-t). No z∈F^s is a pole, so all finite sres_{z,0}(f) are 0. At infinity the expansion is \tilde X + t\tilde X^{p+1}+\cdots, giving sres_{\infty,0}(f)=-1. Yet f dY has residue 1 at the purely inseparable point Y^p=t. The missing term is exactly what would make the total sum zero. So Theorems 3.2.1 and 3.2.2 are false for arbitrary fields. This is not a minor caveat—it is the paper's advertised residue formula. The likely fix is short: assume F perfect or char 0, or sum over all closed points and define residues there. In the intended coding-theory applications, finite fields are perfect, so the theorem is safe there.

Who should read this: algebraists working on Ore and skew polynomials, and coding theorists using linearized Reed–Solomon codes. The paper deserves a serious referee; the flaw is real but localized and fixable. I would ask the author to add the missing hypothesis and to discuss inseparable residues. If repaired, this is a solid contribution, though not a paradigm shift.

Referee Report

2 major / 4 minor

Summary. The paper develops a theory of Taylor expansions and residues for skew rational functions over K[X;θ], where θ is an automorphism of K of finite order r. It constructs Taylor-like expansions around separable closed points and at 0 and ∞ via isomorphisms of completed Ore algebras, defines partial skew residues as coefficients of the degree −1 terms, and proves a residue formula (Theorems 3.2.1 and 3.2.2) and change-of-variables rules (Theorems 3.3.2, 3.3.4, and 3.3.7). A canonical choice of expansions is singled out when the characteristic of K does not divide r.

Significance. If the residue formula were valid in the stated generality, this would be a valuable first systematic treatment of analysis with skew polynomials. The paper is explicit and checkable: the invariants are defined concretely, the section operators σ_j provide a useful toolbox, and the canonical case p∤r is carefully separated from the general noncanonical case with an honest discussion of choices. However, the main residue theorems are false over imperfect fields as stated, so the central advertised result needs a substantive repair before the paper can be accepted.

major comments (2)

[§3.2, Theorem 3.2.1] The statement is false for arbitrary fields because the sum is taken only over z∈F^s. The proof invokes the classical residue formula with z ranging over F^s, but over an imperfect field the classical residue formula is a sum over all closed points, and purely inseparable poles can contribute. Concretely, take r=1, θ=id, K=F=F_p(t), and f=Y^{p-1}/(Y^p−t)∈Frac(A). The denominator has no root in F^s, so every finite sres_{z,0}(f) is zero, while TS_∞(f)=X̃+tX̃^{p+1}+⋯ gives sres_{∞,0}(f)=−1 by Definition 3.1.3; hence the left-hand side is −1, not 0. In the algebraic closure, the unique root α of Y^p−t has classical residue 1, which is exactly the omitted inseparable contribution. The theorem needs an explicit perfectness (or characteristic-zero) hypothesis, or the summation must instead be over all closed points with a residue definition that includes inseparable points.
[§3.2, Theorem 3.2.2] This theorem inherits the same defect as Theorem 3.2.1. The hypothesis that f has at most a simple pole at all points z∈F^s, z≠0, does not exclude purely inseparable poles, and the proof again applies Proposition 3.1.5 and the classical residue formula only at F^s-points. The counterexample above satisfies the simple-pole hypothesis vacuously, since it has no finite F^s-poles, and still violates the conclusion because the sum equals sres_{∞,0}(f)=−1. The advertised skew residue formula is therefore currently established only under an added perfectness hypothesis or after extending the residue definition to inseparable points.

minor comments (4)

[§3.1, Definition 3.1.1] In the displayed formula expressing sres_z(f) in terms of its sections, the upper limit of the sum is written as p−1; since r is the order of θ, it should be r−1.
[§3.2, proof of Theorem 3.2.2] The last sentence of the proof says that sres_{z,j}(f) has to vanish; what is proved is that the sum over z∈F^s⊔{∞} of sres_{z,j}(f) vanishes. This wording should be corrected.
[§1.4.2] The characteristic p is used before it is defined; the paper should state at the first use that p denotes the characteristic of K.
[§3.3.1, Theorem 3.3.2(ii)] The statement says that γ(f) has a single pole at f; it should say that γ(f) has a single pole at z.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and reduces only to the classical residue theorem as an external benchmark.

full rationale

The derivation chain is self-contained. The classification of endomorphisms and derivations (Section 1.3, 1.4), the Hensel-style construction of the Taylor isomorphism tau_N (Theorem 2.2.1), the non-uniqueness statement up to V-conjugation (Proposition 2.2.3), and the canonical choice of tau_z when p does not divide r (Theorem 2.2.5) are all proved in the paper from standard Ore-polynomial algebra. The residue formula in Theorem 3.2.1 is obtained by writing sres_{z,0}(f) as the classical residue rho_z(sigma_0(f)dY) via Proposition 3.1.4 and then applying the classical residue theorem to the forms theta^j(sigma_0(f))dY; the paper itself states that the theorem 'reduces immediately to the classical setting'. Theorem 3.2.2 proceeds in the same way under the simple-pole hypothesis, which is used only to ensure that the partial skew residues are independent of the auxiliary choice of Taylor isomorphism; it is not used to force the vanishing conclusion. The change-of-variables theorems in Section 3.3 are proved from the classical change-of-variables rule for residues together with the Taylor-isomorphism machinery. No parameter is fitted, no uniqueness theorem is imported from the authors' prior work, and the only self-reference is the forthcoming applications paper [5], which plays no role in the proofs. Thus there is no circular step; a possible boundary issue over imperfect fields would be a correctness concern, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard algebra (Ore fraction fields, Hilbert 90, Hensel's lemma, classical residues) plus three explicit domain assumptions: theta has finite order r, N is separable, and p does not divide r for the canonical variants. No free parameters are fitted to data.

assumptions (7)

standard math Ore's fraction field construction yields the unique field Frac(A) with the stated universal property.
Used in Theorem 1.2.1 to represent Frac(A) as Frac(Z) tensor_Z A; this is a classical theorem in noncommutative algebra (Ore, Lam).
standard math Hilbert's Theorem 90 (additive and multiplicative forms) for cyclic Galois extensions.
Used in Propositions 1.3.3, 1.3.4, 1.4.2, and Theorem 3.3.7 to characterize C-linear endomorphisms and derivations and to conjugate by U.
standard math Hensel's lemma and Newton approximation in complete rings.
The proof of Theorem 2.2.1 constructs a root of N in the N-adic completion by successive approximation, relying on separability of N.
standard math Classical residue theorem for rational differential forms on P^1 over an algebraically closed field.
Used in the proofs of Theorems 3.2.1 and 3.2.2 to conclude sums of commutative residues vanish.
domain assumption The fixed field F of a finite automorphism group gives a Galois extension K/F of degree r.
Throughout Section 1, K is equipped with a finite-order automorphism theta of order r, forcing K/F to be cyclic Galois; all norms and traces are taken relative to this.
domain assumption The polynomial N is separable and has nonzero constant term.
Section 2.2 fixes a separable N in Z+ with nonzero constant term to define the Taylor isomorphism tau_N; separability is essential for the Hensel construction and for unicity of tau^C_z.
domain assumption For the canonical theory, the characteristic p does not divide r.
Definition 1.4.3 and Theorem 2.2.5 assume p does not divide r; Proposition 1.4.4 shows that when p divides r no analogous canonical derivation with vanishing p-th power exists.

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Cite this review

Pith. "Pith review of Residues of skew rational functions." pith.science (2026). https://pith.science/paper/HUDVIMB2

@misc{pith2026190808430,
  author       = {Pith},
  title        = {Pith review of: Residues of skew rational functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUDVIMB2}},
  note         = {Machine review of arXiv:1908.08430}
}

read the original abstract

This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Syndrome-Based Error-Erasure Decoding of Interleaved Linearized Reed-Solomon Codes
cs.IT 2024-11 conditional novelty 6.0 of 10

New error-erasure decoders for interleaved linearized Reed-Solomon codes correct full errors, row erasures, and column erasures up to a probabilistic radius of s/(s+1)(n-k) with O(s n^2) average complexity.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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