{"id":"784c4a49-f6dc-42ff-aec9-6965ad0ca597","arxiv_id":"2412.04417","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For graded families of monomial or invariant ideals, the asymptotic resurgence is the dilation threshold at which one associated convex body fails to sit inside the other.","lead":"This paper shows that the asymptotic resurgence number of a pair of graded families of ideals can be computed from convex bodies attached to the families. A smart generalist may read it because it turns a hard algebraic containment problem into a geometric scaling calculation and extends the method beyond monomial ideals to determinantal and Pfaffian ideals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6's proof assumes that non-containment in b^rt forces a B-monomial term with value outside rt*Gamma, but Rees packages only describe integral closures; for b=(x^2,y^2), a*=m^{2*}, this inference is false, so the theorem as stated is not established.","rationale":"The reader's weakest assumption identified the same structural gap: convex-body and Rees-package data describe integral closures, while the theorems as stated concern ordinary ideal containments. My stress-test confirms this is not merely a missing overline or a stylistic imprecision. In Theorem 3.6, the proof's first direction asserts that non-membership in b_{rt} forces a B-monomial term with value outside rt*Gamma. Since the Rees package only spans the integral closure of b_{rt}, the argument conflates b_{rt} with its integral closure. The explicit example b=(x^2,y^2), a_i=m^{2i} shows the inference can fail: a_t is exactly the span of the allowed B-monomials, yet a_t is not contained in b^t. This means the central inequality for ordinary resurgence is not proved as written. The same example does not immediately disprove the theorem's numeric statement, since the convex-body value and the actual asymptotic resurgence may still coincide in that case, but it exposes that the proof needs a genuinely new argument or a hypothesis such as normality of the powers of b, or a restatement in terms of the integral-closure resurgence. The monomial theorem 2.7 has a closely related but more repairable issue: membership of an exponent vector in the Newton polyhedron gives a monomial in the integral closure, and a sufficiently high power of that monomial lies in the ideal, so the proof can be fixed without changing the statement. For Theorem 3.6, no such repair is evident. Therefore the paper should be accepted conditionally on supplying the missing hypothesis or a corrected proof for Theorem 3.6; the reader's verdict remains appropriate.","tokens_in":15101,"tokens_out":32573,"duration_ms":374903,"concrete_test":"Test the exact inference used in Theorem 3.6 on the example R=k[x,y], b=(x^2,y^2), Gamma=NP(b), B the monomial basis, and a_i=m^{2i}. For s=r=1 and t=1, verify that a_t is spanned by B-monomials with v-values in t*Gamma, yet a_t is not contained in b^t, so the proof's first step fails. Then compute both sides of the theorem for this family: rho-hat(a*,b*) and sup{lambda>0 : lambda*Gamma_R(a*) not subset Gamma}. If they agree, the theorem may still be true but needs a new proof or an extra hypothesis; if they disagree, the theorem as stated is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is in the proof of Theorem 3.6, first direction: from a_{st} not subset b_{rt} for t >> 0, the proof concludes that \"a_{st} is not spanned by the B-monomials whose v-values are in rt*Gamma\". This requires that b_{rt} itself is the B-span of {b in B : v(b) in rt*Gamma}. But Definition 3.2 only says that the integral closure of b^s is spanned by that set; b^s is contained in that span and can be strictly smaller. Hence an ideal a_{st} can be spanned by allowed B-monomials and still fail to be contained in b_{rt}. A concrete monomial counterexample to the inference is R=k[x,y], b=(x^2,y^2), Gamma=NP(b), and a_i=m^{2i} = overline{b^i}. For s=r=1, a_t = m^{2t} is spanned by monomials of degree at least 2t, exactly the B-monomials with v-values in t*Gamma, yet a_t is not contained in b^t, e.g. the monomial x^{t+1}y^{t-1} lies in m^{2t} but not in (x^2,y^2)^t. Thus the proof's first inequality rho-hat(a*,b*) <= sup{lambda : lambda*Gamma_R(a*) not subset Gamma} is not justified by the written argument. The analogous issue in Theorem 2.7 is more benign: for monomial ideals, if v is in NP(a_q), then x^v is in the integral closure of a_q, so some power x^{pv} lies in a_q^p; the proof can be patched. For Theorem 3.6, the false inference is not patchable by this observation, and the theorem as stated for ordinary (non-integral-closure) resurgence is unsupported unless one adds a normality hypothesis on the powers of b or explicitly invokes the equality between ordinary and integral-closure resurgence, e.g. via [21, Corollary 3.9].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to compute the asymptotic resurgence number of a pair of graded families of ideals from convex bodies: Newton-Okounkov bodies for monomial families and Rees-package bodies for invariant ideals. The main results are Theorem 2.7 for monomial families, Theorems 3.6 and 3.8 for ideals with Rees packages, and Theorem 4.3 on truncations. The central claim is that the asymptotic resurgence equals the dilation threshold of one convex body inside another.","tokens_in":15476,"tokens_out":10066,"duration_ms":97616,"significance":"The idea of reducing an infinite family of ideal containments to a convex-geometric computation is attractive and would be valuable if correct. The paper also contains a useful-looking truncation result (Theorem 4.3) and several worked examples. However, the equivalence between ideal non-containment and convex-body non-containment is not valid for non-normal ideals, and the main theorems fail on a simple monomial example. The truncation part of the paper appears independent and may be correct, but the advertised reduction of resurgence to convex bodies is not established.","major_comments":[{"comment":"The claimed equality is false. Let R=k[x,y], b=(x^2,y^2), and a_i=m^{2i}. Then a• and b• are graded families of monomial ideals and R(b•) is Noetherian. For every integer S≥1, a_{St}=m^{2St} is not contained in b^t=(x^2,y^2)^t for any t, since x^{2St}∈a_{St} has total degree 2St>2t and therefore cannot lie in b^t. Hence \\hatρ(a•,b•)=∞. On the other hand, ∆(a•)=∆(b•)={u∈R^2_{\\ge0}:u_1+u_2≥2}, so {λ>0:λ∆(a•)⊄∆(b•)}=(0,1) and its supremum is 1. The proof's step \"u_t∈NP(a_{st})\\setminus NP(b_{rt})\" from x^{u_t}∈a_{st}\\setminus b_{rt} is invalid: membership in the Newton polyhedron describes integral closure, not membership in the ideal. For example, u=(t+1,t-1) lies in NP((x^2,y^2)^t) while x^u is not in that ideal.","section":"Theorem 2.7(1), proof, p.6"},{"comment":"The load-bearing inference \"a_{st}\\not\\subseteq b^{rt} implies a_{st} is not spanned by the B-monomials whose v-values are in rtΓ\" is unjustified. Definition 3.2 only says that the integral closure \\overline{b^{rt}} is spanned by those B-monomials; the ideal b^{rt} itself can be strictly smaller. The same counterexample applies: for b=(x^2,y^2), Γ=NP(b), and a_i=m^{2i}, each a_t=m^{2t} is spanned by monomials with v-values in tΓ, yet a_t\\not\\subseteq b^t. Consequently the first inequality in the proof fails and the theorem's RHS is 1 while \\hatρ(a•,b•)=∞. This is a central gap, not a local typo.","section":"Theorem 3.6, proof, p.9"},{"comment":"Because Theorems 2.7 and 3.6 are false as stated, the derived results that rely on them are unsupported. In particular, Theorem 3.8 uses Theorem 3.6 directly to compute \\hatρ(a•,b•) for b-equivalent families, and Corollary 3.10 extends this to determinantal, Pfaffian, and Hankel ideals. These applications therefore do not follow from the manuscript's arguments unless additional normality hypotheses are introduced or the integral-closure resurgence is used instead.","section":"Corollaries 2.11, 2.12; Theorems 3.7, 3.8; Corollary 3.10"}],"minor_comments":[{"comment":"The phrase \"for any integers s,r ∈ 1\" should read \"s,r ∈ N\".","section":"Proof of Theorem 2.7, p.6"},{"comment":"The displayed sentence \"the integral closure I s is spanned\" would be clearer with an overline or explicit \\overline{I^s}, since the distinction between I^s and its integral closure is the source of the main gap.","section":"Definition 3.2, p.8"},{"comment":"The argument asserts that the ascending union \\bigcup_n ∆(a_{n,•}) is a closed set; an arbitrary ascending union of closed sets need not be closed. This step needs justification or a different argument, though the statement of Theorem 4.2(2) may still be true.","section":"Proof of Theorem 4.2(2), p.12"},{"comment":"The notation \\hatρ(a•,\\overline{b•}) is used without prior definition; please define the integral-closure variant explicitly if it is needed.","section":"Example 4.4, p.14"}],"recommendation":"reject","confidential_remarks":"The monomial counterexample with b=(x^2,y^2) and a_i=m^{2i} appears to decisively falsify Theorems 2.7 and 3.6 as stated. The authors may be able to repair the results by replacing ordinary ideal containments with integral-closure containments or by imposing normality conditions, but as written the main results are not correct. The truncation section seems more solid and could form the basis of a revised manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper has a good idea and one genuinely useful theorem, but the main results as written conflate integral closures with the ideals themselves, and that is not a cosmetic issue.\n\nThe genuinely new content is extending the convex-body description of asymptotic resurgence from a single squarefree monomial ideal ([34]) to arbitrary graded families of monomial ideals (Theorem 2.7), and to families of arbitrary ideals via Rees packages (Theorems 3.6 and 3.8). The truncation theorem (4.3) is clean and correct, and the worked examples line up with known values. The Rees package framework is a reasonable tool for this. So the paper is worth engaging with.\n\nThe problem: in Theorem 3.6, the proof's first direction says that if a_st is not contained in b_rt, then for t >> 0, a_st is not spanned by the B-monomials whose v-values lie in rtΓ. But Definition 3.2 only guarantees that the integral closure of b_rt is spanned by those monomials, not b_rt itself. The two can differ. Concretely, R=k[x,y], b=(x^2,y^2), a_i=m^{2i}. For s=r=1, a_t is not contained in b^t, yet every monomial in a_t has v-value in tΓ. So the inference fails. The same conflation appears in Theorem 2.7: v ∈ NP(a_q) does not imply x^v ∈ a_q. There the gap is patchable—integral closure gives x^v in the closure of a_q, hence some power lies in a_q^p—but the written proof doesn't say this. Theorem 3.6 is not patchable that way; you need an explicit normality hypothesis on the powers of b, or a restatement using integral-closure resurgence, e.g. via [21, Corollary 3.9].\n\nBottom line: the central claim is plausible and likely fixable, but as stated it is not established. This paper deserves a serious referee and a major revision, not a desk reject. If the authors add the missing hypotheses and fix the proof's first direction, the result would be solid and useful. I'd send it out.","headline":"A promising but under-supported main theorem: the proofs conflate integral closures with the ideals themselves, and Theorem 3.6 needs a normality hypothesis or a restatement in terms of integral-closure resurgence.","tokens_in":16112,"tokens_out":2905,"would_cite":true,"duration_ms":28342,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A18","13F20","13A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Asymptotic resurgence of graded families of ideals is a convex-body dilation threshold, proved for monomial and Rees-package ideals.","keywords":["asymptotic resurgence","graded families of ideals","Newton-Okounkov body","monomial ideals","Rees package","determinantal ideals","integral closure","convex bodies"],"falsifier":"Take a non-normal monomial ideal $I$ and compare the true asymptotic resurgence $\\hat{\\rho}(I^{(\\bullet)},I_\\bullet)$ with the threshold $\\sup\\{\\lambda>0 : \\lambda\\cdot\\operatorname{SP}(I)\\not\\subseteq\\operatorname{NP}(I)\\}$; because $\\operatorname{NP}(I)=\\operatorname{NP}(\\bar{I})$, any discrepancy shows the convex formula computes the integral-closure resurgence rather than the actual one, so Theorem 2.7 would fail without a normality-type hypothesis.","tokens_in":14779,"feed_emoji":"📐","tokens_out":16056,"duration_ms":135811,"temperature":0.7,"pith_summary":"This paper seeks to show that the asymptotic resurgence number of a pair of graded families of ideals—the supremum of $s/r$ for which $a_{st}$ fails to be contained in $b_{rt}$ for all large $t$—is encoded in convex geometry. For monomial ideals, the paper proves that, when the Rees algebra of the second family is Noetherian, $\\hat{\\rho}(a_\\bullet,b_\\bullet)$ equals the largest $\\lambda>0$ such that $\\lambda\\cdot\\Delta(a_\\bullet)$ is not contained in $\\Delta(b_\\bullet)$, where $\\Delta$ is the Newton-Okounkov body of a family. For ideals admitting a Rees package, the same statement holds with $\\Delta$ replaced by the closed convex set $\\Gamma_R(a_\\bullet)$ built from valuation vectors, covering determinantal and Pfaffian ideals among others. If the identification is correct, an infinite family of ideal-containment questions collapses into a single convex-geometric computation.","feed_headline":"Convex bodies compute asymptotic resurgence","feed_subtitle":"For monomial and invariant ideals, containment failures reduce to scaling one polytope inside another.","key_machinery":"The Newton-Okounkov body of a graded family of monomial ideals, $\\Delta(a_\\bullet)=\\bigcup_{k\\ge1}(1/k)\\operatorname{NP}(a_k)$, is the key object: dilating it by $\\lambda$ and asking whether the dilated set leaves $\\Delta(b_\\bullet)$ mirrors whether $a_{st}$ can escape $b_{rt}$. The polar set $C^\\circ=\\{a\\in\\mathbb{R}^n : \\langle a,b\\rangle\\ge1 \\text{ for all } b\\in C\\}$, together with the bipolar lemma for sets absorbing $\\mathbb{R}^n_{\\ge0}$, converts this dilation threshold into the reciprocal of a minimum pairing $\\langle a,b\\rangle$. For non-monomial ideals, the Rees package $(B,v,\\Gamma)$ substitutes a vector of $B$-monomial valuations for monomial exponents, and $\\Gamma_R(a_\\bullet)=\\bigcup_{k\\ge1}(1/k)\\operatorname{conv}\\{v(b) : b \\text{ is a } B\\text{-monomial appearing in some } f\\in a_k\\}$ plays the role of $\\Delta(a_\\bullet)$. The mechanism throughout is to translate ideal containments into containments of convex sets and then to read off the threshold where one body pokes out of the other.","core_discovery":"The central claim is that asymptotic resurgence is a dilation-threshold invariant of convex bodies. Theorem 2.7 states that for graded families $a_\\bullet$, $b_\\bullet$ of monomial ideals in a polynomial ring, if $R(b_\\bullet)$ is Noetherian then $\\hat{\\rho}(a_\\bullet,b_\\bullet)=\\sup\\{\\lambda>0 : \\lambda\\cdot\\Delta(a_\\bullet)\\not\\subseteq\\Delta(b_\\bullet)\\}$, and when this set is nonempty the supremum equals $1/\\inf\\{\\langle a,b\\rangle : a\\in\\Delta(a_\\bullet), b\\in\\Delta(b_\\bullet)^\\circ\\}$. The same shape is proved in Theorem 3.6 for arbitrary ideals in a universally Japanese ring once $b$ carries a Rees package $(B,v,\\Gamma)$: $\\hat{\\rho}(a_\\bullet,b_\\bullet)=\\sup\\{\\lambda>0 : \\lambda\\cdot\\Gamma_R(a_\\bullet)\\not\\subseteq\\Gamma\\}$, with $b_\\bullet$ the ordinary powers of $b$, and Theorem 3.8 extends this to $b$-equivalent families when $a_\\bullet$ is a filtration. The paper also shows (Theorem 4.3) that truncating the first family gives a sequence whose supremum and limit are the true resurgence numbers, while truncating the second family need not.","pith_inferences":["If the threshold formula survives without normality assumptions, it would compute the integral-closure resurgence rather than the ordinary one; a direct check on non-normal monomial ideals would separate the two notions.","The same dilation-threshold picture likely applies to other asymptotic invariants of pairs, such as skew initial-degree or valuation-growth invariants, by replacing the convex body with a sublevel set of a valuation.","Question 2.10 asks whether the Noetherian hypothesis on $R(b_\\bullet)$ can be dropped; a natural test is to compare the convex threshold with the limit over truncations of $a_\\bullet$, since truncations approximate $a_\\bullet$ but not $b_\\bullet$.","The Rees-package formulation suggests a numerical recipe: approximate $\\Gamma_R(a_\\bullet)$ by finite unions of convex hulls of valuation vectors, then solve the dilation containment problem to bound the asymptotic resurgence from below."],"forward_implications":["For monomial families with Noetherian second family, asymptotic resurgence is computable from two Newton-Okounkov bodies; when both are rational polyhedra the computation is a linear program.","The formula recovers and generalizes the known edge-ideal duality: $\\hat{\\rho}(a^{(\\bullet)},b_\\bullet)=1/\\min\\{\\langle a,b\\rangle : a\\in\\operatorname{SP}(a), b\\in\\operatorname{SP}(b^\\vee)\\}$ for squarefree monomial ideals.","For ideals with Rees packages—determinantal ideals, symmetric determinantal ideals, Pfaffian ideals, and products of Hankel determinantal ideals—the same threshold formula applies when $b_\\bullet$ is the family of ordinary powers, so the asymptotic resurgence is again a single convex computation.","In the $b$-equivalent filtration case the convex formula forces the equalities $\\hat{\\rho}(a_\\bullet,b_\\bullet)=\\hat{\\rho}(a_\\bullet,\\overline{b_\\bullet})=\\rho(a_\\bullet,b_\\bullet)$, recovering the example value $10/9$ for symbolic-versus-ordinary powers of a specific monomial ideal.","Truncating the first family approximates the true resurgence from below, with the supremum and limit agreeing with $\\rho$ and $\\hat{\\rho}$; Example 4.4 shows the analogous statement for the second family is false."],"supporting_citations":[{"why":"Defines resurgence and asymptotic resurgence for graded families of ideals and supplies equalities between them under integral-closure hypotheses used throughout.","marker":"[21]"},{"why":"Constructs Newton-Okounkov bodies for graded families of monomial ideals and shows that Noetherianity gives a stable description of the body; also supplies the worked example's polyhedra.","marker":"[22]"},{"why":"Introduces Rees packages and proves their existence for monomial and classical invariant ideals, the technical foundation for Section 3.","marker":"[1]"},{"why":"The ic-resurgence duality for edge ideals that motivated Theorem 2.7 and that the paper recovers as a corollary.","marker":"[34]"},{"why":"Gives equality of resurgence and asymptotic resurgence via integral closures, used to recover the edge-ideal formula.","marker":"[11]"},{"why":"Provides polar-set and duality facts for convex bodies associated to graded families, used in the proof of Theorem 2.7(2) and Corollary 2.11.","marker":"[14]"},{"why":"Supplies the known resurgence value 10/9 used to illustrate equality of all resurgence variants in Example 2.9.","marker":"[23]"},{"why":"Computes asymptotic resurgence of classical varieties, providing the comparison for the symmetric determinantal example.","marker":"[28]"}],"fun_headline_variants":["Resurgence as a scaling threshold of convex bodies","Convex body scaling finds asymptotic resurgence","Ideal resurgence from dilation of bodies","Scaling one body inside another reveals resurgence","Polytope scaling tests ideal containment"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof treats a point in the Newton polyhedron or in $\\Gamma_R(a_\\bullet)$ as a certificate that some element of the ideal really exists there, but these convex sets describe integral closures; the equality therefore rests on the assumption that convex non-containment matches ideal non-containment, which normality or a suitable integral-closure hypothesis would guarantee.","fun_headline_variants_meta":{"raw":{"variants":["Resurgence as a scaling threshold of convex bodies","Convex body scaling finds asymptotic resurgence","Ideal resurgence from dilation of bodies","Scaling one body inside another reveals resurgence","Polytope scaling tests ideal containment"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000583,"raw_usage":{"total_tokens":2703,"prompt_tokens":868,"completion_tokens":1835,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":1771}},"tokens_in":484,"tokens_out":1835,"duration_ms":16553,"temperature":1.0,"reasoning_tokens":1771,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:25:07.630207+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-normal monomial ideal $I$ and compare the true asymptotic resurgence $\\hat{\\rho}(I^{(\\bullet)},I_\\bullet)$ with the threshold $\\sup\\{\\lambda>0 : \\lambda\\cdot\\operatorname{SP}(I)\\not\\subseteq\\operatorname{NP}(I)\\}$; because $\\operatorname{NP}(I)=\\operatorname{NP}(\\bar{I})$, any discrepancy shows the convex formula computes the integral-closure resurgence rather than the actual one, so Theorem 2.7 would fail without a normality-type hypothesis.","supporting_citations":[{"cited_title":"Resurgence number of graded families of ideals","cited_arxiv_id":"2308.16410","evidence_quote":"Defines resurgence and asymptotic resurgence for graded families of ideals and supplies equalities between them under integral-closure hypotheses used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs Newton-Okounkov bodies for graded families of monomial ideals and shows that Noetherianity gives a stable description of the body; also supplies the worked example's polyhedra."},{"cited_title":"Villarreal, A duality theorem for the ic-resurgence of edge ideals","cited_arxiv_id":null,"evidence_quote":"The ic-resurgence duality for edge ideals that motivated Theorem 2.7 and that the paper recovers as a corollary."},{"cited_title":"Dipasquale, C.A","cited_arxiv_id":null,"evidence_quote":"Gives equality of resurgence and asymptotic resurgence via integral closures, used to recover the edge-ideal formula."},{"cited_title":"DiPasquale, T.T","cited_arxiv_id":null,"evidence_quote":"Provides polar-set and duality facts for convex bodies associated to graded families, used in the proof of Theorem 2.7(2) and Corollary 2.11."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the known resurgence value 10/9 used to illustrate equality of all resurgence variants in Example 2.9."},{"cited_title":"Symbolic Powers of Classical Varieties","cited_arxiv_id":"2402.18693","evidence_quote":"Computes asymptotic resurgence of classical varieties, providing the comparison for the symmetric determinantal example."}],"review_version":1}