{"id":"7137a259-16ba-48b6-8f8c-b670891944dd","arxiv_id":"2506.19663","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New local genus formulas and thickness bounds refine the computation of stable marked reduction of hyperelliptic curves in residue characteristic 2, enabling explicit cases up to genus 30.","lead":"This paper simplifies computation of the stable reduction of hyperelliptic curves when the prime 2 is wild. Its new local genus formulas make explicit examples possible up to genus 30.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorems depend on [10, Prop. 3.3.2] and [10, Prop. 4.5.1] as black boxes; a computational check of these propositions on the paper's own examples would settle the principal risk.","rationale":"The reader's verdict ACCEPT is reasonable. The paper's internal proofs are detailed and the only weak point is the reliance on prior work. The examples give computational evidence but are not a proof of the imported propositions. The proposed test is a direct verification that would settle whether the black-box assumption is safe. Since the concern is a risk rather than a known error, the verdict should remain unchanged.","tokens_in":29269,"tokens_out":28111,"duration_ms":263213,"concrete_test":"Recompute the normalization explicitly for the five worked curves X10, X11, X16, X17, X18 and for several random genus-2/3 examples using a computer algebra system (e.g., Magma's IntegralClosure or Sage), and compare the result with the claimed ring B of [10, Prop. 3.3.2]. Then compute the minimal semistable model (e.g., via the algorithm of [10] itself, or a separate blow-up computation) and check that the break points of w_{p,X} coincide with the components, as asserted by [10, Prop. 4.5.1]. If all tests pass, the black-box concern is resolved; if any fails, the central theorems require re-examination.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Theorems 4.2.2 and 4.3.1) are proved using two imported results: [10, Prop. 3.3.2], which gives the local equation B=R[x^{±1}][t]/(2^{1−γ/2}Ht+t²−G/2^γ) for the normalization, and [10, Prop. 4.5.1], which identifies break points of the square defect function w_{p,X} with irreducible components of ˆC0 above a double point. These are used throughout Sections 2.5, 4, and 5: the component structure over a smooth point is read off from [G/2^γ], the genus of a type-(d) component is computed via the Artin–Schreier equation derived from that proposition, and the correspondence of slopes to components is the backbone of Theorem 4.3.1. The present paper does not re-derive these propositions, and the worksheets [8] are not machine-checked. If either proposition has a missing hypothesis or a subtle error in characteristic 2, the local genus formulas would not be established. No internal inconsistency was found in the present paper; the risk is external dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops tools for computing the stable marked reduction of hyperelliptic curves over a characteristic-zero field with residue characteristic 2. Building on the author's earlier joint work with Pink [10], it defines a local genus for each point of the special fiber of the quotient curve and proves two central formulas: Theorem 4.2.2 expresses the local genus over a smooth unmarked point as half the multiplicity of a root of the derivative of the square-defect approximation, and Theorem 4.3.1 expresses the local genus over an even double point in terms of the largest and smallest slopes of the square defect function. It also proves approximation lemmas that allow replacing the Weierstrass polynomial by a simpler one, a thickness bound for components of type (d), a structure theorem for grounded double points, and leaf-component inequalities. The final section applies the methods to all genus-3 combinatorial cases with three grounded double points and to five explicit curves with many automorphisms, the largest of genus 30.","tokens_in":29485,"tokens_out":12695,"duration_ms":129154,"significance":"If correct, the local-genus formulas provide a genuinely efficient route to the stable reduction that avoids the large stability polynomial of [10], and the genus-30 example appears to be the largest explicit stable reduction computed in residue characteristic 2. The inductive proofs of Theorems 4.2.2 and 4.3.1 are detailed and checkable, and the paper is transparent about its assumptions: Conjecture 5.4.1 is explicitly labeled as conjectural, and the computational claims are backed by worksheets [8]. The main risk is external dependence: the proofs repeatedly invoke [10, Prop. 3.3.2] and [10, Prop. 4.5.1] as black boxes, so the validity of the central theorems is conditional on those results. I found no internal inconsistency in the present manuscript, and the cited dependence is legitimate rather than circular, but a separate verification of the quoted propositions is needed for full confidence.","major_comments":[],"minor_comments":[{"comment":"In the case α > 4 the break points of w_{p,X} should be λ = 2 and λ = α − 2; the text currently writes λ1 := 2 − α and λ2 := α − 2, and 2 − α is negative for α > 4. In the same proposition, the second sentence of part (b) says that T2 intersects the proper transform of X, but it should say the proper transform of Y.","section":"§5.2, Proposition 5.2.4"},{"comment":"The text says that over each grounded double point there is a unique component of type (b) and a unique component of type (c); for thickness 1, Proposition 5.2.4(a)(i) gives a component of type (b) and a component of type (d), so the example should refer to type (d).","section":"§5.2, Example 5.2.8"},{"comment":"In the induction step with a break point, the sentence 'Equations 4.2.4 and 4.2.3 yield' appears to mis-cite the line numbers: the value of s2 is defined in equation (4.2.5), so the reference should be to equations (4.2.4) and (4.2.5).","section":"§4.2, proof of Theorem 4.2.2"},{"comment":"The final display writes a sum over X1 in the sentence 'for each 1 ≤ i ≤ 3, we have ∑_{q∈X1} g_q = 0'; this should be a sum over Xi, and the claimed vanishing should be stated uniformly for i = 1, 2, 3.","section":"§5.3, Example 5.3.2(b)"},{"comment":"The proof says that for a component of type (b) the stable reduction of C and of C̃ over the closed points of the component is isomorphic, but the proposition statement only asserts isomorphisms of the preimages of the components of types (b), (c), and (d). A brief clarification that the isomorphism is of the full preimage of the open component minus the double points would make the logical structure of the proof easier to follow.","section":"§3.3, Proposition 3.3.4"}],"recommendation":"minor_revision","confidential_remarks":"The central results are sound as far as I can verify from the present manuscript, but they rest on two key propositions from the unpublished preprint [10]. I recommend that the editor either obtain a separate report on [10] or ask the authors to include precise statements of [10, Prop. 3.3.2 and 4.5.1] and a short verification of their hypotheses in the main applications, since these propositions carry the full weight of Theorems 4.2.2 and 4.3.1. The remaining issues are local typos in §5.2 and §5.3 that can be fixed without affecting the mathematics."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nHere is my take on Gehrunger's paper. The new results are real: Theorems 4.2.2 and 4.3.1 give explicit local-genus formulas at smooth unmarked points and even double points, and they are proven with induction arguments that I could follow. The thickness bound in 5.1.1 and the grounded-double-point criteria in 5.2 are useful computational tools, and the genus-30 example is a concrete demonstration that the method scales. This is not a repackaging: those formulas do not appear in [10] and they genuinely reduce the need to compute the stability polynomial.\n\nWhere the paper is soft is not in its own proofs but in its dependence on the prior square-defect machinery. The main theorems lean on [10, Prop. 3.3.2] (local equations of the normalization) and [10, Prop. 4.5.1] (break points correspond to components) as black boxes. That dependence is acknowledged, not hidden, and the cited work ships worksheets. But it does mean the present paper is only as strong as those two propositions. A referee should check at least one application of each in the current examples, or ask the author to state the needed hypotheses in full. This is a real risk, not a fatal one.\n\nOther caveats are minor and properly labeled. Conjecture 5.4.1 is labeled a conjecture; the genus-3 count is explicitly not exhaustive; the high-genus computations in Section 6.2 are deferred to worksheets that are not machine-checked. That is fine for a computational paper as long as the worksheets are accessible. The citation pattern is reasonable: the paper builds directly on the author's joint work with Pink, and it says so.\n\nWho should read this: anyone working on stable reduction of hyperelliptic curves in the wild case, and people who need concrete high-genus reductions. I don't see a load-bearing flaw. The central argument is coherent and the new formulas are checkable. I would send it to a serious referee, with the specific instruction to scrutinize the two imported propositions and to spot-check the worksheets. Accept after revision.","headline":"A genuine continuation of the square-defect program with checkable local-genus formulas; black-box dependence on two prior propositions is the main risk, not the paper's own proofs.","tokens_in":30079,"tokens_out":2043,"would_cite":true,"duration_ms":19967,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H30","14H10","11G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves two closed-form formulas for the local genus of the stable reduction of a hyperelliptic curve in residue characteristic 2, and shows how they make explicit computation practical up to genus 30.","keywords":["hyperelliptic curves","stable reduction","residue characteristic 2","square defect","local genus","semistable models","marked curves","toric rank"],"falsifier":"Recompute the stable marked reduction of $X_{18}$ (genus 30, $z^2 = r_5 s_5 t_5$) by the full normalization algorithm of [10] without the shortcuts, and compare the resulting components and genera with the paper's Figure 11; a mismatch in any local genus over a smooth point or even double point refutes Theorems 4.2.2 and 4.3.1. A smaller check: for a curve with known $F$, compute the root multiplicity of $[dG/dx/2^{\\gamma}]$ directly and compare it with the genus of the Artin-Schreier component obtained from [10, Prop. 3.3.2].","tokens_in":29027,"feed_emoji":"🧮","tokens_out":8547,"duration_ms":84672,"temperature":0.7,"pith_summary":"The paper aims to make the stable reduction of hyperelliptic curves in residue characteristic 2 computable without running the full, heavy normalization algorithm at every step. It proves that two elementary local quantities determine the local genus: half the multiplicity of a certain derivative's root at a smooth unmarked point, and a combination of the extreme slopes of a piecewise-linear function at an even double point. These formulas let one read off the combinatorial structure of the stable marked model from the dual graph and thicknesses of the base model, often without solving a high-degree stability polynomial. The payoff is demonstrated by explicit computations up to genus 30, including curves of genus 9, 12, 20, 24, and 30.","feed_headline":"Two formulas compute stable hyperelliptic reduction at genus 30","feed_subtitle":"Root multiplicities and square-defect slopes replace the costly stability polynomial.","key_machinery":"The central object is the square defect $w(F)$, defined as the minimum of $2$ and the largest valuation of $F - H^2$ over all Laurent polynomials $H$; it measures how closely $F$ can be approximated by a square. At a double point this becomes the piecewise-linear square defect function $w_{\\bar p,X}(\\lambda) = w(F(2^{\\lambda}u))$, whose break points correspond to components of the base model and whose slopes encode genera. The argument runs through optimal decompositions $F = H^2 + G$ and the explicit local equations of the normalization from [10, Prop. 3.3.2], which is what turns the two formulas into actual computations.","core_discovery":"The central discovery is that the hard local information in the stable reduction is carried by two formulas. If $F = H^2 + G$ is an optimal decomposition of a Weierstrass equation and $\\gamma := w(F) < 2$, then the local genus over a smooth unmarked point $\\bar p$ is half the multiplicity of the root of $[dG/dx/2^{\\gamma}]$ at $\\bar p$ (Theorem 4.2.2). Over an even double point, writing $s_1$ and $s_2$ for the largest and smallest slopes of the square defect function $w_{\\bar p,X}$, the local genus is $(s_1 - s_2 + \\delta_{0s_1} + \\delta_{0s_2})/2$ (Theorem 4.3.1). These formulas, together with the grounded-double-point criteria and leaf-component bounds, convert a substantial part of stable reduction into elementary arithmetic on a single Laurent polynomial and its derivative.","pith_inferences":["A natural algorithmic consequence, not pursued in the paper, is to factor $[dG/dx/2^{\\gamma}]$ in $k[x^2]$ early: the total local genus over a smooth component is bounded by half its degree, so a partial root count could terminate the normalization computation before all components are built.","The leaf-component inequalities in Propositions 5.3.1 and 5.3.3 are necessary conditions on the dual graph that could be turned into an automated search for all possible dual graphs of the stable reduction for a fixed genus.","The approximation results suggest a divide-and-conquer strategy for very high genus: approximate the Weierstrass polynomial locally at each point of the base model, compute local genera independently, and assemble the reduction; a natural stress test would be random genus-10 polynomials compared against the full algorithm of [10].","If the local-genus formula extends to other tamely ramified double covers, derivative-root multiplicity might replace full normalization in a broader class of wild stable-reduction problems."],"forward_implications":["Because $F$ can be replaced by any Laurent polynomial $\\tilde F$ that is sufficiently close in valuation, the semistable normalization above a smooth point or even double point is unchanged; this lets one discard high-degree pieces of $F$ and avoid the high-degree stability polynomial.","If the base curve has $g$ grounded double points, every one of them has local genus $1$, and its entire reduction is determined by the thicknesses, with the explicit structure described in Proposition 5.2.4.","The toric rank of the reduction of the Jacobian equals the number of grounded double points of thickness strictly greater than $4$ (Corollary 5.2.7).","For genus 3 curves with three grounded double points, 115 combinatorial types of the stable reduction are obtained, each depending only on the thicknesses of the double points.","The curves $X_{10}, X_{11}, X_{16}, X_{17}, X_{18}$ all have toric rank $0$ in their reductions over $2$, and $X_{18}$ is the highest-genus hyperelliptic curve whose stable reduction over $2$ has been explicitly computed."],"supporting_citations":[{"why":"Supplies the square-defect machinery, local normalization equations, and the base algorithm that this paper refines.","marker":"[10]"},{"why":"Gives the five many-automorphism curves $X_{10}, X_{11}, X_{16}, X_{17}, X_{18}$ whose stable reductions are computed in Section 6.2.","marker":"[15]"},{"why":"Records the computer-algebra worksheets underlying the genus-3 and high-genus examples.","marker":"[8]"},{"why":"Used to enumerate the 32 possible dual graphs of the marked base curve in genus 3.","marker":"[4]"},{"why":"Supplies worksheet computations used to read off thicknesses in the genus-2 example of Section 5.1.","marker":"[11]"}],"fun_headline_variants":["Two formulas crack stable reduction in characteristic 2","Local genus via root multiplicity and slope difference","Stable reduction: two elegant formulas replace heavy polynomial","Compute char-2 stable reduction with root and slope data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the square-defect machinery of [10], above all the explicit local equations for the normalization and the statement that break points of the square defect function correspond to components of the base model; if either of those results were false, the two local-genus theorems would not be established.","fun_headline_variants_meta":{"raw":{"variants":["Two formulas crack stable reduction in characteristic 2","Local genus via root multiplicity and slope difference","Stable reduction: two elegant formulas replace heavy polynomial","Compute char-2 stable reduction with root and slope data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1450,"prompt_tokens":813,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":576}},"tokens_in":429,"tokens_out":637,"duration_ms":6488,"temperature":1.0,"reasoning_tokens":576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:28:15.825050+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the stable marked reduction of $X_{18}$ (genus 30, $z^2 = r_5 s_5 t_5$) by the full normalization algorithm of [10] without the shortcuts, and compare the resulting components and genera with the paper's Figure 11; a mismatch in any local genus over a smooth point or even double point refutes Theorems 4.2.2 and 4.3.1. A smaller check: for a curve with known $F$, compute the root multiplicity of $[dG/dx/2^{\\gamma}]$ directly and compare it with the genus of the Artin-Schreier component obtained from [10, Prop. 3.3.2].","supporting_citations":[{"cited_title":"Hyperelliptic curves with many automorphisms","cited_arxiv_id":null,"evidence_quote":"Gives the five many-automorphism curves $X_{10}, X_{11}, X_{16}, X_{17}, X_{18}$ whose stable reductions are computed in Section 6.2."},{"cited_title":"ReductionofHyperellipticCurves in Residue Characteristic 2","cited_arxiv_id":null,"evidence_quote":"Supplies worksheet computations used to read off thicknesses in the genus-2 example of Section 5.1."}],"review_version":1}