{"id":"348049e2-e25b-4657-9521-fd3e78eaefe7","arxiv_id":"2505.15608","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that astab(I)=1 for every graded ideal in dimension two with vstab(I) arbitrary, and constructs ideals realizing every pair (astab(I), vstab(I))=(a,b) in higher dimension.","lead":"This paper compares two stability indices of powers of graded ideals, astab and vstab. It shows the two indices can be chosen independently in high dimension, while in two variables astab is always 1 and vstab can be any positive integer.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unverified hypotheses of [6, Cor. 2.3 and 4.2] are the load-bearing risk for vstab in Theorem 2.1; a direct check for (a,b)=(3,2) would settle it.","rationale":"I traced the main computation and found no internal contradiction: Theorem 1.1's staircase calculation checks out, the astab induction gives a, and the arithmetic after equations (5)-(7) gives vstab b. The only point where the central claim could fail is the validity of [6, Cor. 2.3 and 4.2] for the specific ideals L and H. Since these are from the authors' unpublished preprint and no formal verification is supplied, this is the most load-bearing concern. A small direct computation, or an independent reading of the cited corollaries, would resolve it. I keep the reader's ACCEPT because the cited results are plausible and the rest of the proof is coherent; the concern is about verification, not a demonstrated flaw.","tokens_in":5544,"tokens_out":46596,"duration_ms":400483,"concrete_test":"Run a direct monomial-ideal computation for the smallest non-trivial case (a=3, b=2) in Macaulay2: build I = x0^3(J1+J2) + (x^5, x^2y^3, y^5), and compute Ass(I^k) and v(I^k) for k=1,...,4 from the definition via monomial colon ideals. Check that astab(I)=3 and that v(I^k) equals 9,14,18,23,... (i.e., 5k+4 for k<2 and 5k+3 for k≥2). Also read the statements of [6, Cor. 2.3 and 4.2] and confirm that L and H satisfy their stated hypotheses; if either check fails, equations (5) and (7) are invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem's vstab half rests on two imported formulas: equation (5) via [6, Cor. 2.3] for v(L^k), and equation (7) via [6, Cor. 4.2] for v((L+H)^k). The manuscript does not state the hypotheses of these corollaries or verify them for L=x0^(2b-1)J and H. The dangerous point is (7): [6, Cor. 4.2] must allow replacing the true v-number of the sum by a minimum over ℓ<k of v(L^(k-ℓ)) + v(H^(ℓ+1)), where J^k is not equigenerated for k≥2 and H has exactly one associated prime while L has many. If the corollary requires extra hypotheses (e.g., equigenerated summands or v-stability of the summands), the computed value of v(I^k) and hence the conclusion vstab(I)=b is unsupported. The astab half is less exposed: it uses the published [11, Thm 4.1(3)] and the induction is explicit. Apart from this import, the exponent arithmetic and the use of Theorem 1.1 are internally consistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the relationship between two asymptotic invariants of powers of a graded ideal: the index of ass-stability astab(I) and the v-stability index vstab(I). In Theorem 1.1, for a two-dimensional polynomial ring, the authors prove that astab(I)=1 for every graded ideal and exhibit ideals I=(x^{2b+1}, x^2 y^{2b-1}, y^{2b+1}) for which vstab(I)=b, with explicit formulas for v(I^k). In Theorem 2.1, for any positive integers a,b, they construct a monomial ideal in a 3(a+1)-dimensional polynomial ring such that (astab(I), vstab(I))=(a,b), thereby showing that the two indices are independent in general. The construction combines a sum of triangle edge ideals with the two-dimensional example, and the proof relies on imported formulas for v-numbers and associated primes of products and sums of ideals in disjoint variables.","tokens_in":5800,"tokens_out":30256,"duration_ms":239041,"significance":"If correct, Theorem 2.1 settles a natural independence question: the indices astab and vstab can be prescribed arbitrarily, so in general they are not comparable. The two-dimensional part is an attractive, explicit family of examples with astab=1 and unbounded vstab. The construction in Theorem 2.1 is explicit and the overall question is well motivated. The main strength is the concrete nature of the computations: Theorem 1.1 is proved with a self-contained monomial order argument. However, the vstab half of Theorem 2.1 depends on several imported formulas from [6] whose hypotheses are neither stated nor verified, and at least one of these formulas is not universally true. This is a load-bearing correctness risk that must be fixed before the paper can be accepted.","major_comments":[{"comment":"Equation (7) is obtained by applying [6, Corollary 4.2] to L and H, but neither the hypotheses of that corollary nor their verification for L=x0^{2b-1}J and H are provided. This matters because the displayed min-formula is not true in general: for L=(x,y)^2 and H=(z^2) in disjoint variables one has v(L)=v(H)=1, while v(L+H)=3 for the sum ideal (x^2,xy,y^2,z^2). Please state the exact hypotheses of [6, Corollary 4.2] and check them for the pair (L,H), or give a direct proof of (7). Until then, the conclusion vstab(I)=b is unsupported.","section":"Section 2, Eq. (7)"},{"comment":"Equation (5) for v(L^k) is imported from [6, Corollary 2.3], and the input v(J^k) in equation (4) is imported from [6, Theorem 5.2]. The hypotheses of these results are not stated, nor is it verified that L=x0^{2b-1}J and J=sum Ji satisfy them. Since the subsequent vstab computation feeds directly on (5), the authors should quote the precise statements and confirm their applicability, or prove (5) directly in the paper.","section":"Section 2, Eq. (5)"},{"comment":"The proof that astab(J)=a for a>2 is compressed into a single sentence ('Applying again this result ... proceeding iteratively'). Please spell out the induction: at each step, identify the first power at which the new maximal prime m1+...+mr appears and verify that no further associated primes appear afterwards. This is needed to justify the claim astab(I)=a in equation (6).","section":"Section 2, astab(J)=a"}],"minor_comments":[{"comment":"In the displayed minimum, the index range '0≤j≤i+1' is not well-defined because u_{i+1,i+2} does not exist; the intended range is likely 0≤j≤i.","section":"Section 1, proof of Theorem 1.1(a)"},{"comment":"The definition of vstab(I) says 'the least integer vstab(I) for which v(I^k)=α(I)k+b', but the constant b is not introduced before this point; please define b as the constant from the eventual linear form.","section":"Introduction"},{"comment":"The indexing S=K[x0,x1,...,x_{3a},x,y] gives 3a+3 variables, which equals 3(a+1); this is correct, but the notation may be clearer if the total number of variables is stated explicitly.","section":"Section 2, Theorem 2.1"},{"comment":"A short comment explaining how the ordered chain of monomials matches the hypotheses of [7, Proposition 5.5] would improve readability.","section":"Section 1, proof of Theorem 1.1(a)"}],"recommendation":"major_revision","confidential_remarks":"The central concern is that the proof of Theorem 2.1 leans heavily on two preprints from the same group ([6]) for v-number formulas, and the missing hypotheses make the proof non-reproducible. The likely fix is local: state the exact theorems and verify the hypotheses for the specific ideals L and H. A small direct check for a concrete case (e.g., a=b=3) would substantially increase confidence. I would not reject on this basis, but the current version is not yet suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main news is that in a two-dimensional polynomial ring, astab is always 1 while vstab can be any positive integer, and that in higher dimension the two indices are independent: the authors construct, for any a,b>=1, an ideal with (astab,vstab)=(a,b). If the second theorem holds, it settles a natural comparability question. The dimension-two result is proved by a clean monomial computation, and the construction in Theorem 2.1 is a plausible and clever use of ideals in disjoint variables. I checked the exponent arithmetic and the use of Theorem 1.1; they are internally consistent. The paper is honestly written and does not oversell. The soft spot is exactly where the stress-test note points: equation (5) and especially equation (7) rely on [6, Cor. 2.3 and 4.2] for v-numbers of products and sums of monomial ideals in disjoint variables. The manuscript does not state the hypotheses of those corollaries or verify them for L=x0^(2b-1)J and H. If those corollaries impose conditions that L and H do not satisfy, the computed v(I^k) and the conclusion vstab(I)=b are unsupported. This is not a manufactured worry; the formulas are load-bearing. But it is also not fatal on its face: the ideals in question are simple enough that a referee can check the hypotheses directly, and the authors' own preprints are likely to contain exactly what is needed. A concrete check for (a,b)=(3,2) would settle it either way. The astab half is less exposed, since it uses the published result [11, Thm 4.1(3)] and the induction is explicit. The paper is short and targeted. Readers who work on v-numbers, stability indices, or associated primes of powers will want this result, and it gives a nice answer to a small natural question. I would send it to peer review, with instructions to verify the imported formulas in Section 2. A referee should not desk-reject it over the reliance on the authors' own preprints, because the cited results appear to be real and the construction is transparent enough to check. If the formulas check out, this is a solid accept.","headline":"Short, useful paper with a real new result, but the vstab half of the main theorem leans on imported formulas whose hypotheses are not checked in the text.","tokens_in":645,"tokens_out":640,"would_cite":true,"duration_ms":16256,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F20","13F55","05C70","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any a,b, one ideal has ass-stability a and v-stability b","keywords":["v-number","ass-stability","v-stability","associated primes","monomial ideals","powers of ideals","stability index"],"falsifier":"For the constructed ideal with $a=2$ and $b=3$, compute $v(I^k)$ directly for $k=1,2,3$ from the definition of the v-number using the explicit generators. The claimed values are $(2b+1)k+2b$, i.e. $7k+6$, for $k=1,2$, and $(2b+1)k+2b-1$, i.e. $7k+5$, for $k=3$. A direct computation that yields any other values would disprove the formulas on which the $\\mathrm{vstab}$ half rests.","tokens_in":5358,"feed_emoji":"🧮","tokens_out":6624,"duration_ms":53094,"temperature":0.7,"pith_summary":"This paper asks how two asymptotic invariants of powers of a graded ideal compare: the ass-stability index $\\mathrm{astab}(I)$, where the set of associated primes stops changing, and the v-stability index $\\mathrm{vstab}(I)$, where the v-number becomes linear. The main result is that the two indices are independent: given any positive integers $a$ and $b$, there is a monomial ideal with $\\mathrm{astab}(I)=a$ and $\\mathrm{vstab}(I)=b$. In two-dimensional polynomial rings the situation is rigid on one side—$\\mathrm{astab}(I)=1$ for every graded ideal—but $\\mathrm{vstab}(I)$ can be any positive integer. A reader should care because the two indices measure different forms of stabilization, and knowing that no universal inequality relates them is a structural fact about powers of ideals.","feed_headline":"For any a,b, one ideal has both stability indices you choose","feed_subtitle":"The two indices are not comparable: in low dimension one is fixed, in high dimension both can be chosen freely.","key_machinery":"The construction separates the two indices: a sum $J$ of edge ideals on disjoint triples of variables produces $\\mathrm{astab}(J)=a$ with $\\mathrm{vstab}(J)=1$, and then multiplying $J$ by $x_0^{2b-1}$ and adding the planar ideal $H=(x^{2b+1},x^2y^{2b-1},y^{2b+1})$ shifts the v-number growth so that $\\mathrm{vstab}(I)=b$ without changing $\\mathrm{astab}(I)$. The proof uses imported formulas for the v-number of products and sums of ideals in disjoint variables, together with a formula for the associated primes of powers of sums of ideals.","core_discovery":"The paper establishes that for every pair of positive integers $(a,b)$, there exists a monomial ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\\mathrm{astab}(I),\\mathrm{vstab}(I))=(a,b)$. It also proves that in a two-dimensional polynomial ring every graded ideal satisfies $\\mathrm{astab}(I)=1\\le \\mathrm{vstab}(I)$, and exhibits ideals where $\\mathrm{vstab}(I)$ is any prescribed positive integer. Consequently, the two stability indices are not comparable in general.","pith_inferences":["The same disjoint-variable technique could likely be adapted to prescribe the v-stability indices of individual associated primes, not only the global $\\mathrm{vstab}(I)$.","A direct computer algebra check for the smallest new cases, $a=2$ with $b=2$ or $b=3$, would test whether the sharp switch in $v(I^k)$ at $k=b$ actually occurs, thereby stress-testing the imported formulas.","The independence of the two indices may be stable under small deformations of the constructed monomial ideals, though the paper does not address generic perturbations."],"forward_implications":["Neither $\\mathrm{astab}(I)\\le \\mathrm{vstab}(I)$ nor $\\mathrm{vstab}(I)\\le \\mathrm{astab}(I)$ holds for all graded ideals in dimension at least three.","In dimension two, $\\mathrm{astab}(I)=1$ is forced while $\\mathrm{vstab}(I)$ can be any positive integer, so the low-dimensional regime is completely described.","The constructed ideals are monomial, so the independence phenomenon already appears in combinatorial commutative algebra.","The open problem of realizing arbitrary quadruples $(\\mathrm{astab},\\mathrm{dstab},\\mathrm{rstab},\\mathrm{vstab})$ remains; the present work fixes two of the four coordinates in a uniform construction."],"supporting_citations":[{"why":"Supplies the result used in Theorem 1.1 that $\\mathrm{astab}(I)=1$ for every graded ideal in a two-dimensional polynomial ring.","marker":"[9]"},{"why":"Supplies the formula $v_{\\mathfrak{m}}(J)=\\min\\{a_i+b_{i+1}-2\\}$ for $\\mathfrak{m}$-primary monomial ideals and the asymptotic linearity results underlying both theorems.","marker":"[7]"},{"why":"Supplies the v-number formulas for products and sums of ideals in disjoint variables used in equations (5) and (7) to compute $\\mathrm{vstab}(I)$.","marker":"[6]"},{"why":"Supplies the formula for associated primes of powers of sums of ideals, used to compute $\\operatorname{Ass}(I^k)$ and to show $\\mathrm{astab}(J)=a$.","marker":"[11]"},{"why":"Supplies the value $v(J^k)=2k-1$ for edge ideals with linear resolution and the lower bound $v(I^k)\\ge \\alpha(I)k-1$.","marker":"[5]"},{"why":"Supplies the persistence property for associated primes of edge ideals, used to determine $\\operatorname{Ass}(J_i^k)$ for all $k\\ge 2$.","marker":"[10]"}],"fun_headline_variants":["For any pair (a,b), some ideal has astab=a, vstab=b","Stability indices not comparable: any (a,b) achievable","In 2D astab fixed; in high dimension both indices free","Pick any a,b: a graded ideal exists with those stability numbers","Astab and vstab: any pair realizable in some polynomial ring"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that $\\mathrm{vstab}(I)=b$ depends on imported formulas for the v-number of sums and products of ideals in disjoint variables; if those formulas require additional hypotheses that the particular constructed ideals fail, the equalities giving $\\mathrm{vstab}(I)=b$ would not follow.","fun_headline_variants_meta":{"raw":{"variants":["For any pair (a,b), some ideal has astab=a, vstab=b","Stability indices not comparable: any (a,b) achievable","In 2D astab fixed; in high dimension both indices free","Pick any a,b: a graded ideal exists with those stability numbers","Astab and vstab: any pair realizable in some polynomial ring"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000962,"raw_usage":{"total_tokens":4015,"prompt_tokens":780,"completion_tokens":3235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":3139}},"tokens_in":396,"tokens_out":3235,"duration_ms":19342,"temperature":1.0,"reasoning_tokens":3139,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:14:54.291813+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the constructed ideal with $a=2$ and $b=3$, compute $v(I^k)$ directly for $k=1,2,3$ from the definition of the v-number using the explicit generators. The claimed values are $(2b+1)k+2b$, i.e. $7k+6$, for $k=1,2$, and $(2b+1)k+2b-1$, i.e. $7k+5$, for $k=3$. A direct computation that yields any other values would disprove the formulas on which the $\\mathrm{vstab}$ half rests.","supporting_citations":[{"cited_title":"Herzog, A","cited_arxiv_id":null,"evidence_quote":"Supplies the result used in Theorem 1.1 that $\\mathrm{astab}(I)=1$ for every graded ideal in a two-dimensional polynomial ring."},{"cited_title":"Nguyen, Q.H","cited_arxiv_id":null,"evidence_quote":"Supplies the formula for associated primes of powers of sums of ideals, used to compute $\\operatorname{Ass}(I^k)$ and to show $\\mathrm{astab}(J)=a$."},{"cited_title":"Mart ´ ınez-Bernal, S","cited_arxiv_id":null,"evidence_quote":"Supplies the persistence property for associated primes of edge ideals, used to determine $\\operatorname{Ass}(J_i^k)$ for all $k\\ge 2$."}],"review_version":1}