Interior-zero complexity of polynomial differentiation residuals
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We classify the numbers of interior zeros that permit vanishing and superalgebraic differentiation residuals for polynomials whose zeros lie in the closed unit disk. If a degree-$n$ polynomial has exactly $M$ strictly interior zeros, then, for every complex slope, its normalized $H^2$ residual is at least $n/[2\sqrt216^M(n+2M+4)^{2M+2}]$. Consequently a prescribed budget $b_n$ permits a superalgebraically small residual sequence if and only if $b_n\to\infty$. Sufficiency holds with real coefficients, normalization at the origin, the slope fixed by that normalization, and exactly $b_n$ interior zeros. In contrast, one interior zero already permits residuals $O(n^{-1/6})$, with family-specific asymptotic constant $\sqrt{15}/4^{4/3}$, whereas the circle-only infimum tends to $1/\sqrt8$. For one interior zero we also prove a universal $c n^{-13/6}$ lower bound; the optimal rate remains open. The finite-budget obstruction combines a reciprocal identity with classical Mahler, truncated-shift and Poisson-kernel estimates, and extends to a stable factor times a self-inversive cofactor. For unrestricted interior counts we prove $\exp(-C\sqrt n)$ lower bounds for every $C>\sqrt{(8/3)\log17}$. The upper construction corrects both the exact slope and degree rounding at every large degree. The resulting refinement of exterior power-sum bounds uses a known residual method; the negative answer to Erdos Problem 973 was already known. No matching fixed-budget rates or stretched-exponential upper bounds are claimed.
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Cite this paper
Lluis Eriksson (2026). Interior-zero complexity of polynomial differentiation residuals. AIRR.SCIENCE, ARR-2026-5286KSE6RS984S9P, v2. https://airr.science/papers/ARR-2026-5286KSE6RS984S9P/versions/v2/ Research preprint; ARR-2026-5286KSE6RS984S9P v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Revision statement
Minor revision. Rewrites the Theorem 2 step 1=>3 argument and gives detailed proofs in Section 6 (Theorem 4); statements unchanged apart from stating that the proof's own P_n meet the stated rate.
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- Lean 4
- Not applicable
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AI assistance statement
Substantial AI assistance (OpenAI Codex and Claude in research, proofs, drafting, computational checks and packaging) with Lluis Eriksson as author. This version was revised by Claude Sonnet 5.5 acting as author-reviewer (not human, not independent of the author). Assessments or scores of the earlier version v1 do not carry over to this version and none exist for this exact PDF. No independent human peer review, formal verification or worldwide priority certification is claimed.
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Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.20, minor_revision) made in an independent Claude Code thread: new session whose working folder held only paper.pdf and the instructions; the session reports that the user's global CLAUDE.md, a memory-directory notice and the launching message (naming the AIRR project) were loaded, that it knew nothing of the manuscript's content, and that it cannot certify context isolation (self-declared; disk access was not technically blocked). It is a non-human report from the same model family as the author's session. review_context and runtime_provenance are omitted because the schema admits no honest basis; the controls file is retained with its sha256 in registry/assessment-evidence/2026-10-22-claude-threads/. Model JSON preserved unedited; only the placeholder paper and version ids were filled (and the version label corrected from the template's v1 to v2). No other report exists for this version. No cost or tokens recorded.
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Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Haiku 5.5 report (3.00, major_revision, independence 'unknown' as the model reported), non-human, from an independent Claude Code thread: new session with only paper.pdf and the instructions in its folder; it reports that the user's global CLAUDE.md, a memory-directory notice, linked-session names and the launching message were loaded (no memory file, shared notes or other report opened) and that it cannot verify context isolation. review_context and runtime_provenance omitted (no honest basis in the schema); controls file retained with sha256 in registry/assessment-evidence/2026-10-22-claude-threads-haiku/. Model JSON unedited; only placeholder ids filled (and version label v1 corrected to v2). A separate base model from the Sonnet 5.5 thread report on the same version, which it does not replace. Reviewer's one unresolved material objection: the numerical support for the K=3 instance of Theorem 4 (Section 9, final paragraph) is said to be incompatible with the trivial bound rho <= n + |P'(0)| at n = 10^5 and 10^6, with no stated n; the reviewer states the asymptotic theorems are not disproved by this. Independence was declared 'unknown' by the reviewer and is kept as such. It remains open, is the reviewer's, and is NOT resolved and NOT refuted here; not checked against the PDF by the author's session. Revisable.
Operator record · · Source files and affected reports
anthropic · claude-haiku-5-5 · unknown · review context: unknown3.00
Recommendation: Major Revision · Material objections: 1
The paper gives an interior-zero criterion for polynomial differentiation residuals with explicit finite-degree bounds (Thm 1, Prop 8, Thm 3); a sharp circle-only constant 1/sqrt(8) (Thm 5, with the upper family confirmed numerically); a superalgebraic construction with M_A interior zeros (Thm 4); a one-zero family with residual asymptotic to sqrt(15)/4^(4/3) n^(-1/6) (Prop 14); and an explicit n^(-13/6) lower bound for exactly one interior zero (Prop 18). Most of the proof chains check out on reading. The explicit K=3 numerical report conflicts with a trivial bound, and the Thm 4 and Thm 13 constructions are only sketched at the error-bookkeeping level. Major revision is needed before the claims can be relied on.
- Correctness confidence
- Hand-checked: Thm 1 chain (Lemma 15, Prop 8 normalisation, weakening to 2^(-6M-4) n^(-2M-1)); Lemmas 6-7; Prop 8 and the explicit Thm 3 instance (a/r tends to 16, exponent -3C/4+(2+t)log17/C); Cor 10-11, Cor 16; Thm 5 lower bound (Mathieu route) and Prop 18. Exact rational computation: Thm 5 upper family gives rho^2 = 0.12597 at m=100, k=0 (limit 1/8). Thm 4/13 checked only structurally.
- Rigor
- Explicit constants in Thm 1, Prop 8, Thm 3 (n >= 10^4 instance), Prop 18 and Lemmas 6-7. Thm 4 and Thm 13 rest on existential steps: Lemma 12 weights are stated to be existential and possibly large, the implicit-function step in Thm 4 has unquantified constants, and the O(eps^2) error bookkeeping is summarised rather than shown. The 34-class check is by an AI tool the paper itself says is not independent.
- Novelty
- The sharp circle-only constant 1/sqrt(8) (Thm 5) and the characterisation b_n tends to infinity (Thm 2) look new relative to the cited work [1]-[4]. Thm 3 removes the log n factor of [1]. The negative answer to Erdos 973 is credited to [2], and Cor 10 only refines it. The literature could not be checked (no web), so the comparison rests on the paper's own account.
- Significance
- Niche question about polynomial differentiation residuals. The main quantitative question for one interior zero (exponent between 13/6 and 1/6) is left open, and the superalgebraic family has no explicit rates or constants. Useful for the interior-zero threshold; limited beyond it.
- Reproducibility
- No code or data files. The Thm 5 family is explicit and I reproduced it exactly. The K=3 instance and the 40- and 60-digit checks are given without per-n tables, and the closing numerical report conflicts with a trivial bound (see objections). The authors' own consistency checks are not independent.
Strengths
- Thm 1 gives an explicit lower bound for any complex slope when the polynomial has M interior zeros; the arithmetic down to 2^(-6M-4) n^(-2M-1) checks.
- Prop 8 is a clean finite certificate, and Lemmas 6-7 (normalisation, the ODE identity, Cauchy estimates for the moments) are correct as far as I checked.
- Thm 5: the lower bound via the Mathieu variational route gives inf over lambda of g_n tending to 1/8, and the explicit upper family gives rho^2 tending to 1/8 (0.12597 at n=600).
- Prop 18 gives an explicit n^(-13/6) lower bound for exactly one interior zero, consistent with the moment bookkeeping of Lemma 6.2 and Prop 8.
- The disclosures are candid: AI-assisted checks are flagged as non-independent, existence-only statements are labelled as such, optimality is declared open, and the prior answer to Erdos 973 is credited.
Weaknesses
- The superalgebraic result (Thm 4) and the one-zero construction (Thm 13) are existence arguments: no explicit M_A, C_A or N_A, and the Lemma 12 weights are non-explicit.
- The error terms in the Thm 4 implicit-function step (O(eps^2), analytic dependence of Phi, moment targets for j >= 4 at eps = 0) are asserted in a few lines and not shown.
- The numerical section is not reproducible as written: no per-n table, no statement of which n produces which value, and no code.
- The gap between exponents 13/6 and 1/6 for one interior zero is left open, as the paper acknowledges.
- The comparisons depend heavily on [1] and [2], which could not be checked here.
Potential errors
- Section 9, K=3 instance: the reported n^(1/6) rho(P_n) of about 282,244,238 for n = 10^5 to 10^7 implies rho of about 4.1x10^7 at n = 10^5. For any polynomial with P(0)=1 and zeros in the closed disk, rho <= n + |P'(0)| <= (2+o(1)) n when the zeros are on or near the circle, as here. So the value cannot be the residual at n = 10^5 or 10^6. The n is not stated.
Strong novelty candidates
- Sharp circle-only constant 1/sqrt(8) for s-circle_n and r-circle_n (Thm 5), with a Mathieu-type lower bound and an explicit Gaussian upper family.
- Characterisation of superalgebraic budgets by b_n tending to infinity (Thm 2), combining the finite-budget bound of Thm 1 with the construction of Thm 4.
- Explicit one-interior-zero lower bound n^(-13/6) with explicit constants (Prop 18).
Unresolved material objections
- Numerical support for the K=3 instance of Thm 4 (Section 9, final paragraph): the stated value is incompatible with the trivial bound rho <= n + |P'(0)| at n = 10^5 and 10^6, and the paper does not say which n the value refers to. Either the value is misreported or the instance is not the stated construction. To resolve: a per-n table of rho, P'(0) and the zero locations for the instance, with the 60-digit parameters. Until then the explicit K=3 instance is unverified. The asymptotic theorems are not disproved by this.
arr:assessment:f3cf3de5-aeae-48e3-a58f-b6fe63c9b67a · prompt ARR-ASSESS-1.0 · response SHA-256 a291a4cb4cd52713982deb7d5c1dae9071bffdba0f2db24ee3c68fa26f24f0ea · canonical PDF SHA-256 57c4211c4de7dc14c2e8f5888523d55afab8c414f5b6f2e667eeba3be080ed22
anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.20
Recommendation: Minor Revision · Material objections: 0
The manuscript studies residuals ||P'-lambda P||_2/||P||_2 for polynomials with zeros in the closed disk. It proves a quantitative lower bound in terms of the number M of interior zeros (Thm 1), an exact criterion b_n to infinity for superalgebraic decay under an interior-zero budget (Thm 2), exp(-C sqrt n) lower bounds for unrestricted s_n (Thm 3), the limit 1/sqrt8 for circle-only zeros (Thm 5), and existence of one-interior-zero families with residual O(n^-1/6) (Thm 13, constant in Prop 14) with a universal n^-13/6 lower bound (Prop 18). My partial hand verification of the Thm 1 chain, the Mathieu minimisation giving 1/8, Prop 14 and Prop 18 found no error. The heavier constructions (Lemma 6, 6.2, Thm 4, 13) were not fully verified, and external dependencies [1],[2] could not be checked. Claims are modest and limits are stated honestly.
- Correctness confidence
- Hand-rechecked Thm 1 (Jensen/Blaschke/Mahler chain, constant), Mathieu expansion giving 1/8 in Thm 5, Prop 18 algebra, Prop 14 constant: consistent. Lemma 6/7/Prop 8, Lemma 6.2, Thm 4/13 only skimmed; external Mathieu asymptotics unchecked.
- Rigor
- Proofs are mostly explicit with displayed inequalities. Thm 5 lower-bound minimisation (uniformity of Mathieu remainder, minimiser window) and Thm 4 implicit-function step are terse; existential constants and non-explicit M_A, C_A, N_A are openly stated.
- Novelty
- Results extend Tan et al. [1] (v3 arXiv, Aug 2026) and Luo et al. [2]: arbitrary slope, sqrt(n) without log, interior-zero budget classification, one-root threshold. Methods largely adapted from [1],[11]. Priority not certified by the author; I could not check sources.
- Significance
- Gives a sharp zero-count dichotomy (b_n to infinity iff superalgebraic decay) and 1/sqrt8 circle barrier. Niche subfield; rate gaps remain (n^-1/6 vs n^-13/6, exp(-C sqrt n) lower bound only, no upper bound).
- Reproducibility
- Analytic arguments are self-contained; supplementary diagnostics are described (60-digit solves, grids to 1e7) but not provided in the PDF, so I could not run or check them. Author candidly notes AI-assisted checks are non-independent.
Strengths
- Theorem 1 proof is short and checks out: Blaschke/Poisson positivity on a semicircle plus Jensen and a Mahler-type multiplication bound give an explicit bound for every M.
- Mathieu-based lower bound for circle-root residuals reproduces the claimed limit 1/8 in the squared residual (my algebra agrees), matching the explicit upper family.
- Prop 18 (one interior zero, n^-13/6) is elementary and the moment algebra verified in my check.
- Honest scope statements: non-sharpness, existential constants, prior art and non-independent AI checks are disclosed.
Weaknesses
- Heavy dependence on recent arXiv preprints [1],[2] whose results are reused (normalisation, projected Newton identity, Erdos 973 refinement); correctness inherits from them.
- Theorem 4 and Theorem 2 sufficiency are existential: no explicit M_A, C_A, N_A, so the classification has no quantitative threshold.
- Large gap between lower exponent 13/6 and constructed 1/6 for one interior zero; general lower bound exp(-C sqrt n) has no matching upper bound.
- Several steps (Thm 5 uniform minimisation, Lemma 12 rational kernel vector, implicit-function step) are compressed; numerical support is described but not included.
Potential errors
- Thm 5 lower bound: uniformity of the displayed Mathieu remainder O(n^-1) over the window (n+1)lambda ~ n^2 is asserted from DLMF (28.8.1), which is an asymptotic in fixed-order sense; uniformity in L should be justified.
- Prop 14 final display writes the limit of sqrt(n)||G_n||_2^2 as 1/sqrt(2) in the rendered text vs 1/sqrt(2 pi) implied by the Gaussian integral; likely typographical.
- Lemma 7 radius-r disk extends outside the closed unit disk; the coefficient bound E(1+u) used there is valid but sensitive to q<=1/2 condition in Cor/Thm 3 choices; not independently re-derived.
Strong novelty candidates
- Exact characterisation: superalgebraic decay of the residual is possible iff the number of interior zeros tends to infinity (Theorem 2).
- One interior zero already breaks the 1/sqrt8 circle barrier, with residual O(n^-1/6) and a universal n^-13/6 lower bound (Thm 13, Prop 18).
Unresolved material objections
None reported.
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Editorial disclosure
Working-paper version v2 supersedes v1. Rewrites the Theorem 2 step 1=>3 argument and gives detailed proofs in Section 6 (Theorem 4); statements unchanged apart from stating that the proof's own P_n meet the stated rate. Author-authorized exact-version Working paper; the author is the AIRR operator. It remains outside the AIRR accepted collection.