{"id":"849d3f14-2c20-4434-b2dd-54ddcd309a13","arxiv_id":"1908.08430","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Skew rational functions admit Taylor expansions and residues, with a residue formula and a change-of-variables rule, extending classical residue calculus to Ore polynomial rings.","lead":"This paper builds a theory of residues for skew rational functions, which are fractions of polynomials with a twisted multiplication rule. It proves a residue formula and a change-of-variables formula that extend classical calculus-like identities to this noncommutative setting.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The residue formula is not valid over imperfect fields: the proof sums classical residues only over the separable closure, omitting purely inseparable poles that can carry nonzero residue.","rationale":"The reader identified the simple-pole hypothesis in Theorem 3.2.2 as the weakest assumption. That is a real limitation, but the concern raised here is more serious: Theorem 3.2.1 itself, which requires no simple-pole hypothesis and is the foundation of the advertised residue formula, can fail over imperfect fields in characteristic p. The proof's use of the classical residue formula over F^s silently assumes that the separable closure contains all poles, which is false for purely inseparable closed points. The explicit r=1 example shows the claimed identity can fail by an amount equal to the residue of the omitted inseparable point. This is not a stylistic or scope issue; it is a correctness gap under the stated hypotheses. The fix is straightforward for the intended applications: require F perfect, or assume characteristic 0, or sum over all closed points with trace maps. Since the construction and change-of-variables theorems appear otherwise coherent, the appropriate verdict is CONDITIONAL rather than outright rejection: accept the paper's framework once the residue theorem is restricted to settings where all poles are separable, or the field is perfect. The coding-theory applications over finite fields are unaffected, since finite fields are perfect.","tokens_in":26542,"tokens_out":51791,"duration_ms":541411,"concrete_test":"Specialize to the commutative case r=1, theta=id, K=F=F_p(t), and set f=Y^{p-1}/(Y^p-t). Compute sres_{z,0}(f) for z in F^s directly from Definition 3.1.1 and sres_infinity,0(f) from Definition 3.1.3. The finite contributions vanish because Y^p-t has no root in F^s, while the infinity contribution is -1, so the left side of Theorem 3.2.1 equals -1, not 0. Then repeat the same computation over a perfect field such as F_p(t) replaced by its perfection, or over a finite field, and verify that the formula becomes 0; this isolates the missing inseparable-pole contribution.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 3.2.1 reduces sres_{z,0}(f) to the classical residue rho_z(sigma_0(f)dY) and then invokes the classical residue formula with z ranging over the separable closure F^s. This step is legitimate only if F^s is algebraically closed (e.g., F perfect) or if all poles are separable. Over an imperfect field, a rational differential can have a pole at a purely inseparable closed point, which admits no F^s-rational point yet contributes to the classical residue sum. Concretely, specialize to r=1, theta=id, K=F=F_p(t), and take f=Y^{p-1}/(Y^p-t). The denominator has no root in F^s, so every finite sres_{z,0}(f) is 0. At infinity, TS_infinity(f) = \\tilde X + t\\tilde X^{p+1} + ..., so Definition 3.1.3 gives sres_infinity,0(f) = -1. Over the algebraic closure, the unique root alpha of Y^p-t has residue 1 for f dY, and the residue at infinity is -1, so the omitted inseparable residue is exactly the term needed to make the total sum zero. Thus Theorem 3.2.1, and Theorem 3.2.2 insofar as it relies on the same classical summation, is false as stated for arbitrary fields; it requires an explicit perfectness, characteristic-zero, or all-poles-separable hypothesis.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26794,"tokens_out":8905,"duration_ms":92529,"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":[{"comment":"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.","section":"§3.2, Theorem 3.2.1"},{"comment":"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.","section":"§3.2, Theorem 3.2.2"}],"minor_comments":[{"comment":"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.","section":"§3.1, Definition 3.1.1"},{"comment":"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.","section":"§3.2, proof of Theorem 3.2.2"},{"comment":"The characteristic p is used before it is defined; the paper should state at the first use that p denotes the characteristic of K.","section":"§1.4.2"},{"comment":"The statement says that γ(f) has a single pole at f; it should say that γ(f) has a single pole at z.","section":"§3.3.1, Theorem 3.3.2(ii)"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is real and is reflected in my two major comments. The counterexample is elementary and shows that Theorems 3.2.1 and 3.2.2 cannot stand as stated. I recommend major revision rather than rejection because the fix appears local: add a perfectness hypothesis or replace the F^s-summation by summation over all closed points with suitably generalized residues. The rest of the paper, including the change-of-variables results, seems structurally sound, but the authors should also re-check every later statement that relies on the residue formula. I did not attempt to verify the announced coding-theoretic application in the forthcoming reference [5]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":27317,"tokens_out":5771,"would_cite":false,"duration_ms":59569,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16S36"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["skew polynomials","Ore rings","skew rational functions","Taylor expansions","residue formula","change of variables","section operators","linearised Reed-Solomon codes"],"falsifier":"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.","tokens_in":26324,"feed_emoji":"🧮","tokens_out":14251,"duration_ms":131211,"temperature":0.7,"pith_summary":"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.","feed_headline":"Skew rational functions now have a residue calculus","feed_subtitle":"Noncommutative multiplication still yields Taylor expansions, and the degree -1 coefficient obeys residue and substitution laws.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the noncommutative polynomial rings and Euclidean division that define the paper's object of study.","marker":"[23]"},{"why":"Gives the fraction-field existence condition used to construct Frac(A).","marker":"[22]"},{"why":"Supplies the division-algebra and norm-one cocycle background behind the change-of-variables conjugation.","marker":"[14]"},{"why":"Supplies the p-curvature identity used to prove the characteristic obstruction to a canonical derivation.","marker":"[24]"},{"why":"Provides the evaluation-theoretic background for Ore polynomials cited in the foundational section.","marker":"[15]"}],"fun_headline_variants":["Skew rational residues: zero sum and change of variables","Residue formula and change of variables for skew rationals","Noncommutative residues: Taylor expansions and zero sums","A skew residue calculus with vanishing total residue","Skew rational function residues obey residue and substitution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Skew rational residues: zero sum and change of variables","Residue formula and change of variables for skew rationals","Noncommutative residues: Taylor expansions and zero sums","A skew residue calculus with vanishing total residue","Skew rational function residues obey residue and substitution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000422,"raw_usage":{"total_tokens":2178,"prompt_tokens":964,"completion_tokens":1214,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":1137}},"tokens_in":580,"tokens_out":1214,"duration_ms":11431,"temperature":1.0,"reasoning_tokens":1137,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:39:25.602659+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ore, Theory of non-commutative polynomials , Ann","cited_arxiv_id":null,"evidence_quote":"Introduces the noncommutative polynomial rings and Euclidean division that define the paper's object of study."},{"cited_title":"Ore, Linear equations in non-commutative fields , Ann","cited_arxiv_id":null,"evidence_quote":"Gives the fraction-field existence condition used to construct Frac(A)."},{"cited_title":"Jacobson, Finite-Dimensional Division Algebras Over Fields , Grundlehren der Mathema- tischen Wissenschaften Series (1996), Springer","cited_arxiv_id":null,"evidence_quote":"Supplies the division-algebra and norm-one cocycle background behind the change-of-variables conjugation."},{"cited_title":"Van der Put, Differential equations in characteristic p, Compositio Math","cited_arxiv_id":null,"evidence_quote":"Supplies the p-curvature identity used to prove the characteristic obstruction to a canonical derivation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the evaluation-theoretic background for Ore polynomials cited in the foundational section."}],"review_version":1}