{"id":"2b629104-8e82-4a81-986b-aaf6486e96d1","arxiv_id":"1908.10185","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For 'good' monomial ideals, the Ratliff-Rush closure equals the intersection of n ideals obtained by stabilizing along the coordinate axes.","lead":"This paper defines a class of 'good' monomial ideals whose powers have a regular box-like structure, and proves a formula for their Ratliff-Rush closure, an operation in algebra that controls how powers of an ideal behave. The result gives a fast algorithm for computing this closure in that class, avoiding repeated expensive quotient computations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the cone-coloring/Lemma 6.1 step that underpins Theorem 6.2 survives scrutiny, and the two inclusions of the main theorem are sound.","rationale":"The reader's weakest_assumption identifies the cone-coloring construction of Theorem 5.8 and the derived Lemma 6.1 as the least elementary structural step. I agree that this is the most intricate part of the proof, but I do not find it to be an actual flaw. The Noetherian argument for the existence of a stabilization point in each cone is valid because it produces an infinite strictly ascending chain of ideals otherwise. The decomposition of a cone into finitely many subcones is handled by Theorem 5.6, and the recursive coloring gives cones on which I_a is constant. Lemma 6.1 then works: the bound L(Q) guarantees that some coordinate of the box index can be decremented by Q while staying inside the same cone, and equality of the ideals I_b and I_{b'} transfers the minimal generator back to I^{l-Q}. The main theorem's two inclusions are direct consequences of Proposition 4.4, the definition of q_i, and Lemma 6.1. The only implicit point is that the sequence (I^{k+1}:I^k) is increasing, so proving containment for all sufficiently large k is enough for the union; this follows immediately by multiplying by I and is standard. The peripheral inaccuracies flagged by the reader are real but do not bear on Theorem 6.2, so the CONDITIONAL verdict remains appropriate without revision of the central claim.","tokens_in":26484,"tokens_out":44156,"duration_ms":444702,"concrete_test":"As an independent check, implement the Section 7 algorithm in a computer algebra system for the ideal of Example 8.1 (or a randomly generated ideal satisfying the sufficient condition of Theorem 3.7), compute the axis-stabilizing ideals I_{q_i,0,...,0}, and compare their intersection with a direct computation of I^{k+1}:I^k for increasing k until the colon stabilizes; the two ideals should coincide and their minimal monomial generating sets should match exactly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most delicate step in the paper is Lemma 6.1, which uses the finite cone coloring of Theorem 5.8 to factor large minimal generators of I^l by a corner power μ_i^Q. I checked this step rather than assuming it: the inequality L < b_1+...+b_n - nQ + n ≤ vertex_sum ≤ L forces some coordinate b_i - Q to remain in the same cone; equality of ideals on that cone gives equality of the unique minimal monomial generating sets of I_b and I_{b'}, so the shifted monomial m/μ_i^Q is indeed a minimal generator of I^{l-Q}. The two inclusions in Theorem 6.2 then follow: the forward direction uses I^{l+1}:I^l ⊆ I^{l+1}:⟨μ_1^l⟩ and the definition of q_1, and the reverse direction uses Lemma 6.1 with Q = q = max q_i. The proof of the forward direction implicitly uses the standard fact that the quotients I^{k+1}:I^k increase with k, so the union defining the Ratliff-Rush closure is captured by the tail; this is not stated explicitly but is true. The peripheral issues noted by the reader (the false 'conversely' in Section 9 and the uniqueness wording in Proposition 10.6) do not affect Theorem 6.2. I therefore find no load-bearing objection to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies m-primary monomial ideals in K[x1,...,xn] that satisfy a 'box decomposition principle' (called good ideals). For such an ideal I, with minimal corner generators µ_i = x_i^{d_i}, every minimal generator of I^l lies in a box B_a with |a| = l − 1, and the paper defines ideals I_a by dividing the monomials in B_a by µ^a. It proves a finite cone-decomposition theorem for the assignment a ↦ I_a (Theorem 5.8), then states the main result (Theorem 6.2): the Ratliff–Rush closure of I is the intersection of the one-dimensional stabilizations along the coordinate axes, I_{q1,0,...,0} ∩ ... ∩ I_{0,...,0,qn}. Section 7 turns this into an explicit algorithm for the axis stabilizations, with worked examples in Section 8. Section 9 discusses detection of good ideals, Section 10 studies powers of good ideals and introduces very good ideals, and Section 11 connects very good equigenerated ideals with Freiman ideals.","tokens_in":26758,"tokens_out":57252,"duration_ms":601035,"significance":"The main theorem is a clean structural reduction: instead of computing the full family of colon ideals (I^{k+1}:I^k), one computes n one-dimensional stabilizing ideals. If correct, this is a substantial practical and theoretical contribution to monomial ideal theory and to computational Ratliff–Rush closure. The proof is elementary and self-contained, with the cone-coloring/Lemma 6.1 machinery as the key step; I checked this step and the two inclusions of Theorem 6.2 and found them sound. The examples are helpful, and the connections to very good ideals and Freiman ideals are natural. The paper is generally well organized and readable.","major_comments":[],"minor_comments":[{"comment":"The proof of the sufficient condition considers two factors that both satisfy the exponent lower bound, but corner generators have exponent sum 1, which is smaller than n/2 for n ≥ 3. The argument should first remove all corner factors (each corner contributes one unit to the box sum) and then apply the two-factor reduction to the remaining non-corner factors. The statement itself is correct, but the proof as written is incomplete.","section":"Section 3, Theorem 3.7"},{"comment":"The claim that the old box containing m1 is the unique old box is not always true when the non-corner generator has a zero exponent, because the point then lies on a box boundary. What the proof needs is the largest old box, which exists by the second bullet of Proposition 10.4; please replace 'unique' by 'largest' and adjust the reference accordingly.","section":"Section 10, Propositions 10.5 and 10.6"},{"comment":"The containment I^{l+1}:I^l ⊆ I^{l+1}:⟨µ_1^l⟩ is correct because ⟨µ_1^l⟩ ⊆ I^l, but the proof should explicitly note that the quotients (I^{k+1}:I^k) form an increasing sequence, so that proving the containment for all l ≥ q controls the union defining the Ratliff–Rush closure. This is standard but currently implicit.","section":"Section 6, proof of Theorem 6.2"},{"comment":"The converse assertion that every monomial of the form µ^a m^k with k < K is a minimal generator is true, but it deserves a proof. If such a monomial were divisible by a product with fewer copies of m, then m^r would be divisible by r corner monomials for some r < K, contradicting the definition of K. As written, the claim is stated without justification.","section":"Section 9, one-extra-generator paragraph"},{"comment":"The phrase 'for each color the set of points of this color forms a cone' should be clarified, since the recursive construction can in principle assign the same ideal value to several disjoint cones. One can always refine the coloring so that each cone is monochromatic; please state this explicitly or rephrase the condition.","section":"Section 5, Theorem 5.8"}],"recommendation":"minor_revision","confidential_remarks":"The paper's central claim is sound; the issues I found are local and easily repairable. I see no reason to doubt novelty or attribution, and the bibliography appears appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this paper through, and the main result is real. Theorem 6.2 gives a formula for the Ratliff-Rush closure of an m-primary monomial ideal satisfying the box decomposition principle as an intersection of n one-dimensional stabilizations, and the algorithm in Section 7 is a concrete way to compute those stabilizations. I checked the delicate step, Lemma 6.1 with the cone coloring from Theorem 5.8, and it holds. The two inclusions in Theorem 6.2 follow cleanly once you accept the factorization of large minimal generators by a corner power. The proof of Lemma 6.1 uses the vertex-sum inequality exactly as stated, and the stress-test note confirms that the shifted monomial is indeed a minimal generator because equality of ideals on the cone gives equality of minimal generating sets.\n\nThe paper also does some things well beyond the main theorem. The box decomposition framework is a useful way to organize the asymptotics of powers. The algorithm is illustrated on several examples and the Singular computations make the claims plausible. The connection to Freiman ideals is a nice bonus: for equigenerated m-primary monomial ideals, very good equals Freiman, and the alternative proof of Theorem 11.2 is clean and shorter than the original.\n\nThe soft spots are peripheral but real. The 'conversely' assertion in Section 9 overclaims: the condition described there is not sufficient for goodness, because monomials of the form \\mu^{k1}... m^{k} may fail to be minimal generators. That section needs rewriting or a careful statement of what is actually detected. The proof of Proposition 10.6 also says the old box containing m1 is 'unique' when it is not; the argument only needs 'largest', and the wording should be fixed. Neither issue touches Theorem 6.2 or the main algorithm.\n\nThe scope is limited to m-primary monomial ideals satisfying the box decomposition principle, and the sufficient condition in Theorem 3.7 is restrictive, but the class is large enough to include interesting examples and the paper explicitly discusses how to test goodness near the boundary. The presentation is a bit rough in spots—the abstract could be rewritten—but the mathematics is readable.\n\nThis paper deserves a serious referee. I would send it to peer review rather than desk reject. If I worked in commutative algebra, I would cite it for the algorithm and the Freiman connection. I would bring it to a reading group on monomial ideals, since the cone decomposition technique is a genuine new tool.","headline":"New and sound formula for Ratliff-Rush closure of good monomial ideals; worth publishing after fixing a false converse in Section 9 and a wording slip in Proposition 10.6.","tokens_in":27277,"tokens_out":3176,"would_cite":true,"duration_ms":28904,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13B22","13A30","13F20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For m-primary monomial ideals obeying the box decomposition principle, the Ratliff–Rush closure equals the intersection of n one-dimensional stabilizations, and this yields an algorithm.","keywords":["monomial ideals","Ratliff–Rush closure","good ideals","box decomposition principle","powers of ideals","m-primary ideals","Freiman ideals","Hilbert polynomial"],"falsifier":"Compute the axis-stabilized intersection from Section 7 for every good ideal in a small search space, and compare it with the fully stabilized colon ideal $I^{N+1}:I^N$ for $N$ beyond the bound $L+nq-n+2$ used in Lemma 6.1; a single good ideal where the two differ would disprove the theorem.","tokens_in":26276,"feed_emoji":"📦","tokens_out":10736,"duration_ms":102079,"temperature":0.7,"pith_summary":"An $\\mathfrak{m}$-primary monomial ideal is one generated by monomials that includes a positive power of every variable. This paper singles out the class of “good” ideals, those whose minimal generators of every power $I^l$ lie in the rectangular boxes indexed by $l-1$, and proves that for every good ideal the Ratliff–Rush closure $\\tilde I=\\bigcup_k(I^{k+1}:I^k)$ is obtained from just $n$ coordinate-axis stabilizations. The Ratliff–Rush closure is the largest ideal containing $I$ with the same Hilbert polynomial, so computing it has direct meaning for Hilbert functions. The paper’s theorem turns a generally expensive limiting construction into a finite, explicit intersection and gives an algorithm for polynomial and formal power series rings.","feed_headline":"Ratliff–Rush closure reduced to n axis stabilizations","feed_subtitle":"For good monomial ideals, the closure is the intersection of coordinate-axis limits, avoiding expensive quotient chains.","key_machinery":"The machinery is the box decomposition principle together with the finite cone coloring it induces. A box $B_{a_1,\\dots,a_n}$ is a product of intervals cut out by the corner exponents $d_i$; for a good ideal each minimal generator of $I^l$ falls in a box with $a_1+\\cdots+a_n=l-1$, which forces the translated ideals $I_{a_1,\\dots,a_n}=I^{l}:\\langle\\mu_1^{a_1}\\cdots\\mu_n^{a_n}\\rangle$ to grow monotonically. Theorem 5.8 then gives a finite decomposition of $\\mathbb{N}^n$ into cones on which $I_{a_1,\\dots,a_n}$ is constant. This cone coloring is what reduces a potentially infinite family of colon ideals to finitely many regions, and it is the tool used to factor $\\mu_i^Q$ out of large minimal generators in Lemma 6.1.","core_discovery":"The paper’s central result is Theorem 6.2: if $I$ is a good $\\mathfrak{m}$-primary monomial ideal, then $$\\tilde I = I_{q_1,0,\\dots,0}\\cap I_{0,q_2,\\dots,0}\\cap \\cdots \\cap I_{0,\\dots,0,q_n},$$ where $I_{0,\\dots,q_i,\\dots,0}$ is the stable ideal of the chain obtained by moving the box decomposition along the $i$-th coordinate axis. The proof shows that for large powers of $I$, every minimal generator can be factored by a pure power $\\mu_i^{q_i}$ of one of the corner monomials $\\mu_i=x_i^{d_i}$, so only the axis directions matter. Consequently the usual description of $\\tilde I$ as a union of colon ideals $I^{k+1}:I^k$ over all $k$ is captured, for this class, by $n$ one-dimensional limits, each reached after finitely many steps.","pith_inferences":["The same axis-intersection formula could plausibly be tested on the larger class of ideals that satisfy only the necessary condition from Theorem 3.6; Section 9 supplies a finite test for goodness, so a computer search could locate where the formula first fails.","Interpreting $I_{a_1,\\dots,a_n}$ as a directional Newton-polyhedron slice suggests an extension beyond monomial ideals: replace boxes by filtrations and cones by normal fans, and the theorem would become a statement about asymptotic intersections of initial ideals.","The equivalence between very good ideals and $I^2=I\\langle\\mu_1,\\dots,\\mu_n\\rangle$ offers a fast algebraic certificate for the Freiman property that could be used to build new infinite families of ideals all of whose powers are Ratliff–Rush."],"forward_implications":["The Ratliff–Rush closure of any good ideal can be computed by stabilizing $n$ separate one-dimensional chains; by Remark 7.2 each chain stops as soon as two consecutive ideals coincide, so the computation is finite and explicit.","The same computation works in both $\\mathbb{K}[x_1,\\dots,x_n]$ and the formal power series ring $\\mathbb{K}[[x_1,\\dots,x_n]]$, since all arguments are monomial and finitary.","For a good ideal with coloring bound $L$, every power $I^k$ with $k\\ge L+1$ is Ratliff–Rush (Proposition 10.9), giving a stabilization threshold in terms of the box coloring.","An equigenerated $\\mathfrak{m}$-primary monomial ideal is Freiman (its square has the minimal possible number of generators) exactly when it is very good, i.e. exactly when $I^2=I\\langle\\mu_1,\\dots,\\mu_n\\rangle$; this is Theorem 11.2."],"supporting_citations":[{"why":"supplies the definition of the Ratliff–Rush closure and the fact that it is the unique largest ideal with the same powers for all large l, which the theorem exploits.","marker":"[13]"},{"why":"gives an earlier algorithm for Ratliff–Rush closure in a local-ring setting; this paper’s algorithm is the constructive analogue for monomial ideals.","marker":"[1]"},{"why":"provides the previous monomial-specific Ratliff–Rush algorithm that the box-axis formula generalizes.","marker":"[12]"},{"why":"records the criterion that all powers are Ratliff–Rush exactly when the associated graded ring has positive depth, motivating Proposition 10.9.","marker":"[7]"},{"why":"documents the general difficulty of computing the closure and the reliance on generic elements, the obstacle the monomial algorithm removes.","marker":"[5]"},{"why":"defines Freiman ideals and provides the setting the paper connects to very good ideals in Theorem 11.2.","marker":"[9]"},{"why":"supplies the lower bound on the number of square generators used in the Freiman equality.","marker":"[8]"}],"fun_headline_variants":["Ratliff–Rush for good monomial ideals = n axis limits","Good monomial ideals: Ratliff–Rush closure via n axis chains","Ratliff–Rush closure reduces to n one-dimensional limits","Compute Ratliff–Rush with n axis stabilizations (good monomial ideals)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every ideal satisfying the box decomposition principle gives a finite cone coloring of $\\mathbb{N}^n$ on which the ideals $I_{a_1,\\dots,a_n}$ are constant (Theorem 5.8); if that structural stabilization fails, the axis-intersection formula for $\\tilde I$ does not follow from the paper’s argument.","fun_headline_variants_meta":{"raw":{"variants":["Ratliff–Rush for good monomial ideals = n axis limits","Good monomial ideals: Ratliff–Rush closure via n axis chains","Ratliff–Rush closure reduces to n one-dimensional limits","Compute Ratliff–Rush with n axis stabilizations (good monomial ideals)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001129,"raw_usage":{"total_tokens":4666,"prompt_tokens":890,"completion_tokens":3776,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":3692}},"tokens_in":506,"tokens_out":3776,"duration_ms":24887,"temperature":1.0,"reasoning_tokens":3692,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:52:09.080213+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the axis-stabilized intersection from Section 7 for every good ideal in a small search space, and compare it with the fully stabilized colon ideal $I^{N+1}:I^N$ for $N$ beyond the bound $L+nq-n+2$ used in Lemma 6.1; a single good ideal where the two differ would disprove the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the definition of the Ratliff–Rush closure and the fact that it is the unique largest ideal with the same powers for all large l, which the theorem exploits."},{"cited_title":"Al-Ayyoub , An algorithm for computing the Ratliff–Rush closure , Journal of Algebra and its Applications, 8 (2009), pp","cited_arxiv_id":null,"evidence_quote":"gives an earlier algorithm for Ratliff–Rush closure in a local-ring setting; this paper’s algorithm is the constructive analogue for monomial ideals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the previous monomial-specific Ratliff–Rush algorithm that the box-axis formula generalizes."},{"cited_title":"Heinzer, D","cited_arxiv_id":null,"evidence_quote":"records the criterion that all powers are Ratliff–Rush exactly when the associated graded ring has positive depth, motivating Proposition 10.9."},{"cited_title":"Elias , On the computation of the Ratliff–Rush closure , Journal of Symbolic Com- putation, 37 (2004), pp","cited_arxiv_id":null,"evidence_quote":"documents the general difficulty of computing the closure and the reliance on generic elements, the obstacle the monomial algorithm removes."},{"cited_title":"Herzog and G","cited_arxiv_id":null,"evidence_quote":"defines Freiman ideals and provides the setting the paper connects to very good ideals in Theorem 11.2."},{"cited_title":"On the number of generators of powers of an ideal","cited_arxiv_id":"1707.07302","evidence_quote":"supplies the lower bound on the number of square generators used in the Freiman equality."}],"review_version":1}