{"id":"c9cf9f2e-f51c-4348-a7c6-1770302fec07","arxiv_id":"1909.00177","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under the moderate growth condition, Joris's power theorem holds for Denjoy-Carleman classes: coprime powers that are C_M force the function itself to be C_M.","lead":"The paper proves that if two coprime powers of a function germ are ultradifferentiable in a Denjoy-Carleman class with moderate growth, then the germ itself is ultradifferentiable. This extends H. Joris's classical smoothness theorem to a broad family of ultradifferentiable function classes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Fedja-style proof of Theorem 2.2.1 is internally coherent and the moderate-growth hypothesis is genuinely needed.","rationale":"The reader's weakest assumption correctly identifies moderate growth as the key hypothesis. The proof uses it in a controlled way, and the counterexamples separate necessity from sufficiency. The main theorem is new, the construction is complete, and my independent pass did not reveal a circular step or an omitted case. The minor constant and notation issues do not affect the central claim.","tokens_in":14588,"tokens_out":31991,"duration_ms":300578,"concrete_test":"Verify the necessity of moderate growth by recomputing Remark 2.2.3 for M^λ_j=exp(λ j^2/4): show that gλ^p ∈ C_{M^λ}(R,0) while gλ ∉ C_{M^λ}(R,0), so the conclusion of Theorem 2.2.1 fails if (4) is dropped.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central argument. Moderate growth is used exactly where the proof must convert fractional powers of h_M into h_M(Cε): in Lemma 3.2.4, in (9)/(31) of Proposition 3.3.2, and in the final step of §4.3. The reduction to f^m and f^{m+1}, the regularized quotient u_ε, the ∂̄-problem and the estimates in Lemmas 4.2.1–4.2.4 all line up; Remark 2.2.3 shows the hypothesis is not idle. The only issue I noted is notational: the displayed formula for u_ε in §4.2 needs a conjugate in the numerator for Lemma 4.2.3 to be consistent, and the constant a3 in the proof of Lemma 3.2.4 should be max(a1, sqrt(a1 L)) rather than max(sqrt a1, sqrt L). Neither affects the existence of the required constants or the logic of the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a Denjoy-Carleman analogue of Joris's theorem. For a weight sequence M satisfying the moderate-growth condition (4), if a germ f at the origin in R has two powers f^p and f^q, with gcd(p,q)=1, belonging to the Denjoy-Carleman class C_M, then f itself belongs to C_M. The proof follows the 2018 MathOverflow proof by \"fedja\" of the C^∞ case, and is built on a new approximation-theoretic characterization of C_M regularity on an interval (Proposition 3.3.2), together with estimates for solutions of the Cauchy-Riemann equation (Lemma 3.1.1), a subharmonicity lemma (Lemma 3.2.3), the Hadamard three-lines theorem (Lemma 3.2.4), and the moderate-growth inequality (8). The paper also contains a counterexample showing that the moderate-growth assumption is necessary for the two-power theorem (Remark 2.2.3), a single-power counterexample for strongly regular sequences (Proposition 2.1.1), and corollaries for functions of several variables under a non-quasianalyticity assumption (Corollary 2.2.5).","tokens_in":14807,"tokens_out":10378,"duration_ms":84826,"significance":"The main theorem is a natural and nontrivial extension of a classical result of Joris, and it answers a question raised in [24] about the role of moderate growth in the Denjoy-Carleman setting. The proof is self-contained in its main line, with all lemmas proved in detail; Proposition 3.3.2, an approximation-theoretic characterization of C_M regularity, is likely to be of independent interest. The counterexamples are well chosen and demonstrate that the hypotheses are not idle. The paper is clearly written and the central argument is internally coherent; the issues I found are typographical and local, not load-bearing. Once the notational slips described below are corrected, the paper will be a solid contribution to the literature on ultradifferentiable functions.","major_comments":[],"minor_comments":[{"comment":"The displayed definition of u_epsilon appears to be missing a conjugate on g_epsilon: to have u_epsilon = h_epsilon/g_epsilon on the set {|g_epsilon| > r_epsilon}, as used in Lemma 4.2.2, the numerator should be \\bar{g}_epsilon h_epsilon rather than g_epsilon h_epsilon. Accordingly, in formula (41) of Lemma 4.2.3 the factor g'_epsilon should be \\bar{g'}_epsilon. The subsequent estimates use only |g'_epsilon|, so the argument is unaffected, but the displayed formulas should be corrected for consistency.","section":"Section 4.2, display after (35) and Lemma 4.2.3"},{"comment":"The stated value of the constant a3, namely a3 = max(a1^{1/2}, L^{1/2}), is not consistent with the immediately preceding estimate |g(w)| ≤ a1 (K h_M(a2 epsilon))^{1/2} with K = max(1, L/a1). The correct value is a3 = max(a1, (a1 L)^{1/2}). This is a typographical error in an explicit constant and does not affect the existence of the required a3 and a4.","section":"Section 3.2, proof of Lemma 3.2.4"},{"comment":"The final step of the proof is somewhat compressed: after bounding the error by c12 delta_{2epsilon}^{1/s}, the paper uses the moderate-growth property to conclude |f - f_epsilon| ≤ c'_1 h_M(c'_2 epsilon). A one-sentence explanation of how the polynomial and logarithmic factors from Lemma 4.2.4 are absorbed by the fast decay of delta_epsilon (using h_M(t) = O(t^N) for every N) would improve readability.","section":"Section 4.3, end of proof"},{"comment":"There is a minor typo: \"it is not difficult the check\" should read \"it is not difficult to check\".","section":"Section 2.1, paragraph before (12)"}],"recommendation":"minor_revision","confidential_remarks":"No conflicts of interest. The paper is well within the scope of the journal; the remaining issues are typographical and local in nature, and I do not see a need for a second full round of review once they are corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is exactly what it says: a Joris-type theorem for Denjoy-Carleman classes, proved by adapting fedja's holomorphic-approximation argument. The main result, Theorem 2.2.1, is new and is proved from scratch, not by importing the author's earlier work. The approximation-theoretic characterization of C_M regularity (Proposition 3.3.2) is a genuinely useful byproduct and is likely to be cited independently. I checked the logic carefully. The moderate-growth hypothesis is used exactly where you'd expect—turning fractional powers of h_M into h_M(Cε) through condition (8)—and Remark 2.2.3 shows it's not idle. The proof is detailed, lemmas are proved, and the structure is honest: counterexamples (one-variable and several-variable) are cleanly separated from the positive result. The paper deserves a serious referee and, in my view, publication after fixing a few notational slips. On those slips: the displayed formula for u_ε in §4.2 is missing a conjugate in the numerator; as written it isn't consistent with Lemma 4.2.3. The constant a3 in Lemma 3.2.4 should be max(a1^{1/2}, sqrt(a1 L)) rather than max(sqrt a1, sqrt L). Both are minor and do not affect the existence of constants or the logic. One further point: the proof of Lemma 4.2.4 is compressed at the end, where the claim that δ_ε = o(ε^j) absorbs all the epsilons; this is fine given condition (3), but a referee should ask for one clarifying sentence. The citation pattern looks fair: Joris, fedja's MathOverflow post, Dynkin, and the Denjoy-Carleman background literature are all credited appropriately. The author's own earlier results are used only in the ancillary counterexample, not in the central proof, so there's no circularity. In short: the math is sound, the novelty is real, the main claim holds up. Who is this for? Anyone working on ultradifferentiable regularity and Joris-type phenomena; it's a meaningful extension of a classic result within that theory, not a paradigm shift. I'd take it to reading group and would cite the approximation characterization. Recommendation: send it to peer review — it will survive, and a careful referee will just polish the corners.","headline":"A solid, careful extension of Joris's theorem to Denjoy-Carleman classes under moderate growth, with a clean approximation characterization; the main result is new and the proof is essentially complete.","tokens_in":15264,"tokens_out":615,"would_cite":true,"duration_ms":6474,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26E10","46E25","30E10","32W05"],"pacs":[],"model":"deepseek-v4-flash","headline":"If two coprime powers of a function belong to a moderate Denjoy–Carleman class, then the function itself does.","keywords":["Denjoy-Carleman classes","ultradifferentiable functions","moderate growth","weight sequences","holomorphic approximation","non-quasianalytic classes","function germs","coprime powers"],"falsifier":"To test the role of the hypothesis, compute $h_M(t)=\\inf_{j\\ge0}t^jM_j$ for $M^\\lambda_j=\\exp(\\lambda j^2/4)$ and check inequality (8) with $s=2$: it fails. With $g_\\lambda(x)=\\exp(-(\\ln x)^2/\\lambda)$ for $x>0$ and $g_\\lambda(x)=0$ for $x\\le0$, the germ $f=g_\\lambda^{1/p}$ satisfies $f^p\\in\\mathcal{C}_{M^\\lambda}(\\mathbb{R},0)$ and $f^q\\in\\mathcal{C}_{M^\\lambda}(\\mathbb{R},0)$ for $p<q$, yet $f\\notin\\mathcal{C}_{M^\\lambda}(\\mathbb{R},0)$, as shown in Remark 2.2.3. A counterexample with a sequence that does satisfy (8) would falsify the theorem.","tokens_in":14431,"feed_emoji":"∞","tokens_out":21840,"duration_ms":161159,"temperature":0.7,"pith_summary":"The paper proves an ultradifferentiable analogue of the classical theorem on powers: if a germ $f$ at the origin in $\\mathbb{R}$ has two powers $f^p$ and $f^q$, with $\\gcd(p,q)=1$, belonging to a Denjoy–Carleman class $\\mathcal{C}_M$, and if the weight sequence $M$ satisfies the moderate-growth condition, then $f$ itself belongs to $\\mathcal{C}_M$. This carries a result known for $\\mathcal{C}^\\infty$ functions into finer scales of ultradifferentiability, where a single power no longer suffices. The proof adapts a holomorphic-approximation strategy: it approximates $f^m$ and $f^{m+1}$ by holomorphic functions in narrow ellipses around the real interval, builds a quotient-like approximant for $f$, solves a $\\bar\\partial$-problem to make it holomorphic, and uses the moderate-growth inequality to control the error. A corollary extends the conclusion to several variables when the class is also non-quasianalytic.","feed_headline":"Two coprime powers put a function in the ultradifferentiable class","feed_subtitle":"If $f^p$ and $f^q$ lie in a moderate Denjoy–Carleman class with $\\gcd(p,q)=1$, then $f$ lies in it too.","key_machinery":"The key machinery is a characterization of $\\mathcal{C}_M$ regularity by holomorphic approximation (Proposition 3.3.2): $f$ belongs to $\\mathcal{C}_M$ on $[-1,1]$ exactly when, for each small $\\varepsilon>0$, there is a function $f_\\varepsilon$ holomorphic in the ellipse $\\Omega_\\varepsilon$ (the image of a strip under the sine map), uniformly bounded by a constant $K$, approximating $f$ on $[-1,1]$ with error $c_1h_M(c_2\\varepsilon)$, where $h_M(t)=\\inf_{j\\ge0}t^jM_j$. To prove the theorem, one starts from holomorphic approximants $g_\\varepsilon$ and $h_\\varepsilon$ of $f^m$ and $f^{m+1}$, forms the quotient-like approximant $u_\\varepsilon=\\chi_\\varepsilon g_\\varepsilon h_\\varepsilon/\\max(|g_\\varepsilon|,r_\\varepsilon)^2$ with $r_\\varepsilon=\\delta_\\varepsilon^{1/(m+1)}$, and controls its $\\bar\\partial$-derivative through an integral estimate on $|g'_\\varepsilon|^2$ in the region where $|g_\\varepsilon|<r_\\varepsilon$. The moderate-growth condition enters as the inequality $h_M(t)\\le h_M(\\kappa_st)^s$, which converts the smallness $\\delta_\\varepsilon=c_4h_M(c_5\\varepsilon)$ into approximation of $f$ with decay $h_M(c_2\\varepsilon)$.","core_discovery":"The central result, Theorem 2.2.1, states that for a weight sequence $M=(M_j)_{j\\ge0}$ with $M_0=1$, increasing, logarithmically convex, and satisfying the moderate-growth condition $M_{j+k}\\le A^{j+k}M_jM_k$, the following holds for a complex-valued germ $f$ at the origin in $\\mathbb{R}$: if $f^p$ and $f^q$ belong to the Denjoy–Carleman class $\\mathcal{C}_M(\\mathbb{R},0)$ for coprime positive integers $p$ and $q$, then $f$ belongs to $\\mathcal{C}_M(\\mathbb{R},0)$. The statement is local, so it applies to functions on any open interval. The paper also shows the hypothesis on two coprime powers is not an artifact: with a single power the conclusion fails even for strongly regular sequences, and without moderate growth it fails for the sequence $M^\\lambda_j=\\exp(\\lambda j^2/4)$.","pith_inferences":["Editorial inference: the approximation characterization of Proposition 3.3.2 is likely a reusable tool for local questions; any regularity statement that can be phrased as uniform holomorphic approximation in the ellipses $\\Omega_\\varepsilon$ with $h_M$-error could be attacked the same way, for instance stability of $\\mathcal{C}_M$ under other nonlinear maps.","Editorial inference: tracking the constants through Lemmas 4.2.2 and 4.2.4 should yield a quantitative form of the theorem; the $\\mathcal{C}_M$ constants of $f^p$ and $f^q$ determine the $\\mathcal{C}_M$ constant of $f$ through the moderate-growth constant $\\kappa_s$ and the integer $m$, and a reader needing such bounds could extract them from the proof.","Editorial inference: the quasianalytic case is not reached by the curve-reduction argument; a natural next question is whether the quasianalytic analogue of the power theorem follows from the algebraic power-series route instead, since the paper notes that the quasianalytic case needs only derivation stability, not moderate growth."],"forward_implications":["On any open subset of $\\mathbb{R}$, the local conclusion gives: if $f^p$ and $f^q$ are in $\\mathcal{C}_M$ with $\\gcd(p,q)=1$, then $f$ is in $\\mathcal{C}_M$.","The proof shows a stronger one-pair reduction: it is enough that $f^m$ and $f^{m+1}$ belong to $\\mathcal{C}_M$ for the single integer $m$ such that every integer $j\\ge m$ is a nonnegative combination of $p$ and $q$.","In the non-quasianalytic case, the same conclusion holds for germs in $\\mathbb{R}^n$ (Corollary 2.2.5), by reducing the problem to one variable along suitable curves.","The theorem applies to the standard factorial-type strongly regular weight sequences $M_j=(j!)^\\alpha$, so for those sequences coprime powers in the class force the function into the same class.","The moderate-growth and non-quasianalytic assumptions cannot simply be dropped: a single power fails even for strongly regular sequences, and the exponential-sequence example shows the two-power theorem fails without moderate growth."],"supporting_citations":[{"why":"Supplies the holomorphic-approximation strategy and the uniform Cauchy–Riemann estimates (Lemmas 3.1.1, 3.2.3, 3.2.4) that the proof follows.","marker":"[6]"},{"why":"Provides the pseudoanalytic extension theorem used to establish the approximation characterization (Proposition 3.3.2).","marker":"[5]"},{"why":"Establishes the equivalence between moderate growth and the inequality $h_M(t)\\le h_M(\\kappa_st)^s$, which the proof uses to control error terms.","marker":"[16]"},{"why":"The original $\\mathcal{C}^\\infty$ theorem on powers, whose ultradifferentiable analogue is proved here.","marker":"[12]"},{"why":"Supplies the curve-reduction results used to pass from the one-dimensional theorem to several variables and in the construction of the strongly regular counterexample.","marker":"[14]"},{"why":"Provides the counterexample showing the failure without moderate growth and poses the question answered by the theorem.","marker":"[24]"},{"why":"Supplies the flat element with the lower bound $h_M(b|t|)$ used in proving that one power does not suffice for strongly regular sequences.","marker":"[22]"}],"fun_headline_variants":["Coprime powers pin down the ultradifferentiable class","Two coprime powers guarantee the same smoothness for f","Moderate growth: coprime powers decide the Denjoy-Carleman class","If f^p and f^q are ultradifferentiable, so is f","Coprime exponents lock in ultradifferentiable regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the moderate-growth condition (4) on the weight sequence $M$; every decay estimate in the proof passes through the equivalent inequality (8), and the paper shows the conclusion is false without it.","fun_headline_variants_meta":{"raw":{"variants":["Coprime powers pin down the ultradifferentiable class","Two coprime powers guarantee the same smoothness for f","Moderate growth: coprime powers decide the Denjoy-Carleman class","If f^p and f^q are ultradifferentiable, so is f","Coprime exponents lock in ultradifferentiable regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000321,"raw_usage":{"total_tokens":1802,"prompt_tokens":934,"completion_tokens":868,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":780}},"tokens_in":550,"tokens_out":868,"duration_ms":510010,"temperature":1.0,"reasoning_tokens":780,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:59:34.057734+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the role of the hypothesis, compute $h_M(t)=\\inf_{j\\ge0}t^jM_j$ for $M^\\lambda_j=\\exp(\\lambda j^2/4)$ and check inequality (8) with $s=2$: it fails. With $g_\\lambda(x)=\\exp(-(\\ln x)^2/\\lambda)$ for $x>0$ and $g_\\lambda(x)=0$ for $x\\le0$, the germ $f=g_\\lambda^{1/p}$ satisfies $f^p\\in\\mathcal{C}_{M^\\lambda}(\\mathbb{R},0)$ and $f^q\\in\\mathcal{C}_{M^\\lambda}(\\mathbb{R},0)$ for $p<q$, yet $f\\notin\\mathcal{C}_{M^\\lambda}(\\mathbb{R},0)$, as shown in Remark 2.2.3. A counterexample with a sequence that does satisfy (8) would falsify the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the holomorphic-approximation strategy and the uniform Cauchy–Riemann estimates (Lemmas 3.1.1, 3.2.3, 3.2.4) that the proof follows."},{"cited_title":", Pseudoanalytic extension of smooth functions","cited_arxiv_id":null,"evidence_quote":"Provides the pseudoanalytic extension theorem used to establish the approximation characterization (Proposition 3.3.2)."},{"cited_title":"Structure theorems and a character ization, J","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between moderate growth and the inequality $h_M(t)\\le h_M(\\kappa_st)^s$, which the proof uses to control error terms."},{"cited_title":", Une C ∞-application non immersive qui possède la propriété univer selle des immersions, Arch","cited_arxiv_id":null,"evidence_quote":"The original $\\mathcal{C}^\\infty$ theorem on powers, whose ultradifferentiable analogue is proved here."},{"cited_title":", The convenient setting for non-quasianalytic Denjoy-Carleman Differentiable Mappings , J","cited_arxiv_id":null,"evidence_quote":"Supplies the curve-reduction results used to pass from the one-dimensional theorem to several variables and in the construction of the strongly regular counterexample."},{"cited_title":", Smooth solutions of quasianalytic or ultraholomorphic equ ations, Monatsh","cited_arxiv_id":null,"evidence_quote":"Provides the counterexample showing the failure without moderate growth and poses the question answered by the theorem."},{"cited_title":", On closed ideals in smooth classes , Math","cited_arxiv_id":null,"evidence_quote":"Supplies the flat element with the lower bound $h_M(b|t|)$ used in proving that one power does not suffice for strongly regular sequences."}],"review_version":1}