{"id":"400ac824-f4e5-4318-803e-58a69d9824a2","arxiv_id":"2608.12731","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The multiplicity sequence of every monomial ideal equals a sum of mixed volumes of polytopes constructed from its Newton polyhedron, a formula that also produces a counterexample to the Achilles-Manaresi conjecture.","lead":"This paper reveals that the entire sequence of numeric invariants of a monomial ideal can be read off from volumes of geometric shapes made from the ideal. It also shows that an earlier proposed way to compute one of those invariants was wrong, and that a related class of invariants has a similar volume formula.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"deepseek-v4-flash","summary":"This paper gives a complete convex-geometric formula for the multiplicity sequence of a monomial ideal in a polynomial ring localized at the maximal ideal. The main result (Theorem 4.11) expresses each coefficient c_i(I) as a signed sum of mixed volumes of pyramids over the non-coordinate facets of the Newton polyhedron of I. The authors also disprove a conjecture of Achilles and Manaresi in dimension three (Theorem 5.2) and provide a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals (Theorem 6.1). The proofs use lattice-point enumeration, integral closure of ideals, and convex-geometric decompositions of the Newton polyhedron.","tokens_in":47,"tokens_out":42564,"duration_ms":1186115,"significance":"If the results are correct, they provide a definitive convex-geometric description of the multiplicity sequence, generalizing the known j-multiplicity formula of Jeffries–Montaño and offering a formula for mixed multiplicities that was previously missing. The counterexample to the Achilles–Manaresi conjecture is a useful correction to the literature. The proofs are self-contained and rely on standard lattice-point theory (McMullen) and integral closure facts; no fitted parameters or ad hoc normalizations are introduced. The paper is clearly written and the computations in the example are internally consistent.","major_comments":[],"minor_comments":[{"comment":"The authors should explicitly state that the unbounded directions of every face of the Newton polyhedron Γ are coordinate directions and that all vertices of bounded faces are lattice points; this follows from Γ = conv(v_i) + R_{\\ge0}^d, but it is the key hypothesis that makes Proposition 3.10 applicable to the facets.","section":"Section 4, after Eq. (4)"},{"comment":"The convention for the empty set should specify that it applies when the multiplicity n_r is positive; as written, 'for every n1,...,nr' would incorrectly force MV_d(Q[n]) = 0 when n = 0, contradicting standard mixed-volume notation.","section":"Remark 3.4"},{"comment":"Reference [6] (Bhattacharya) appears in the bibliography but is not cited in the text; it should either be cited where relevant or removed.","section":"References"},{"comment":"The justification of the second inequality is terse; adding a sentence explaining why a point lying in more than one cone_u(F) and not in any coordinate plane must belong to some cone_u(G) with dim G ≤ d-2 would improve readability.","section":"Proof of Proposition 4.10, second inequality"},{"comment":"The symbol 'H' is used for the empty set; replacing it with the standard empty-set symbol '∅' would avoid confusion, especially for readers not familiar with the notation.","section":"Remarks 3.4 and 4.12"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical content is sound; the central theorem and the counterexample are correct as far as I can verify. The paper is appropriate for the journal and I recommend acceptance after minor revisions. The main proofs are dense but valid; the minor comments above address clarity and presentation only."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a good paper and the main results look right. Theorem 4.11 is genuinely new — it gives the first full mixed-volume formula for every entry of the multiplicity sequence, not just the j-multiplicity. The counterexample to the Achilles–Manaresi conjecture in dimension three is concrete and the numbers check out: c1=6 versus the predicted 4. Theorem 6.1 is a useful bonus, giving a mixed-volume formula for mixed multiplicities of arbitrary monomial ideals. The overall derivation is coherent: lattice-point counts of the layers uΓ\\(u+1)Γ are matched to lengths via the integral closure reduction, and the leading coefficients are extracted after discarding controlled lower-order terms. I did not find any fitted parameters, circular reasoning, or invented entities.\n\nThe soft spots are real but minor. The passage from quotient lengths to lattice-point counts is written too quickly in Proposition 4.10 and Theorem 6.1. Those equalities are only true after passing to integral closures, or up to o(d-1) error terms. Lemma 2.5 and Lemma 2.7 already do the heavy lifting, but the proofs as printed sometimes omit the error term and state exact equalities. A referee should ask the authors to make the reduction explicit. Relatedly, the text occasionally loses the overline on integral closures, which confuses the reading.\n\nThe other place to scrutinize is Proposition 3.10. It is the load-bearing estimate behind the coefficient extraction, and the error term is delicate. The argument is plausible and uses standard ideas, but it is not machine-checked and it deserves careful referee attention. This is not a fatal flaw, just the spot where I would focus.\n\nOne small thing: reference [6] appears in the bibliography but is never cited in the text.\n\nThe paper is aimed at commutative algebraists and convex geometers working on multiplicities, monomial ideals, and mixed volumes. It deserves a serious referee and, assuming the authors tighten the proofs around the integral-closure reduction, I would accept it. Recommend sending to peer review.","headline":"Solid paper: a new convex-geometric formula for the full multiplicity sequence, a credible counterexample to a published conjecture, and a mixed-multiplicity formula; needs minor tightening around integral-closure reductions.","tokens_in":16331,"tokens_out":12980,"would_cite":true,"duration_ms":154676,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13H15","13F55","52B20","52A39","05E40","13A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The multiplicity sequence of any monomial ideal is a signed sum of mixed volumes of pyramids built from the facets of its Newton polyhedron.","keywords":["multiplicity sequence","monomial ideal","Newton polyhedron","mixed volume","j-multiplicity","mixed multiplicities","Achilles-Manaresi conjecture","lattice point counting"],"falsifier":"Compute the multiplicity sequence of the ideal $I=(x_1^2 x_3, x_2^2 x_3, x_1 x_2)$ directly from the Hilbert function of the double associated graded ring and compare it with the mixed-volume sum; if the result differs from $(0,6,0,0)$, Theorem 4.11 is false.","tokens_in":15671,"feed_emoji":"📐","tokens_out":12370,"duration_ms":105753,"temperature":0.7,"pith_summary":"This paper establishes a convex-geometric formula for the entire multiplicity sequence of a monomial ideal, the sequence of coefficients appearing in the Hilbert polynomial of the double associated graded ring. The formula expresses each coefficient as a signed sum of mixed volumes of pyramids over the non-coordinate facets of the Newton polyhedron. The paper also constructs a three-dimensional monomial ideal for which a volume-based conjecture stated in [3] fails, and it derives a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals. A sympathetic reader would care because the multiplicity sequence characterizes integral dependence, and the new formula turns this algebraic invariant into a purely geometric computation.","feed_headline":"Mixed volumes compute the multiplicity sequence of monomial ideals","feed_subtitle":"A signed sum of pyramid mixed volumes over Newton-polyhedron facets gives every coefficient; a rival volume conjecture fails.","key_machinery":"The mechanism is the Newton polyhedron $\\Gamma$ of the monomial ideal, together with two polytopes associated to each non-coordinate facet $F$: the pyramid $\\mathrm{pyr}(F) = \\mathrm{conv}(0, \\mathrm{conv}(\\text{vertices of }F))$ and the simplex $\\Delta_F = \\mathrm{conv}(\\text{unbounded direction vectors of }F)$, with $\\Delta_F^{\\leq} = \\mathrm{conv}(0, \\Delta_F)$. The load-bearing step is Proposition 4.10, which shows that the algebraic length $\\lambda_R(I^u/(m^v I^{u+1} + I^{u+1}))$ equals, up to lower-order error, the sum over facets of the lattice-point counts of the slice $\\mathrm{cone}_u(F)$ between successive dilates of $F$. Theorem/Definition 3.2 then converts these lattice-point counts into mixed volumes of the associated pyramids and simplices, giving the exact formula after error terms are discarded.","core_discovery":"On the paper's own terms, the central result is Theorem 4.11: for every $0 \\leq i \\leq d$, the $i$-th coefficient $c_i(I)$ of the multiplicity sequence of a monomial ideal $I$ equals $\\sum_{F \\in \\mathcal{F}_{d-1}(\\Gamma)} d! \\left( \\mathrm{MV}_d(\\mathrm{pyr}(F)[d-i], \\Delta_F^{\\leq}[i]) - \\mathrm{MV}_d(\\mathrm{pyr}(F)[d-i], \\Delta_F[i]) \\right)$, where $\\Gamma$ is the Newton polyhedron of $I$, $\\mathrm{pyr}(F)$ is the pyramid over the convex hull of the vertices of $F$, $\\Delta_F$ is the simplex spanned by the unbounded direction vectors of $F$, and $\\Delta_F^{\\leq} = \\mathrm{conv}(0, \\Delta_F)$. The paper proves the formula by relating the length of the quotient $I^u/(m^v I^{u+1} + I^{u+1})$ to lattice-point counts of slices of $\\Gamma$, then extracting coefficients. It also proves Theorem 5.2, giving a counterexample to the conjecture of [3], and Theorem 6.1, a mixed-volume formula for mixed multiplicities.","pith_inferences":["If the formula holds, the multiplicity sequence of a monomial ideal is a piecewise-linear function of the exponent vectors of its generators, so one expects the sequence to be constant on chambers of the Newton polyhedron's normal fan.","The failure of the conjecture in [3] suggests that a correct discrete-volume description must weight each bounded face by something like a mixed volume of the face and the cone of unbounded directions, and that the minimum operation appearing in that conjecture should be replaced by a sum.","The error-term cancellation in Proposition 3.10 may transfer to other counting problems where the lattice points of a half-open polyhedron are compared with their Minkowski differences, possibly yielding sharp estimates for the discrepancy.","It is natural to ask whether the mixed-multiplicity formula extends to non-monomial ideals by replacing each ideal by its integral closure and using a suitable Newton polyhedron, though no evidence for such an extension appears in the paper."],"forward_implications":["The multiplicity sequence of a monomial ideal, which detects whether two ideals have the same integral closure, becomes computable directly from the geometry of the Newton polyhedron rather than from Hilbert functions.","The $i=0$ case of the formula recovers the known $j$-multiplicity formula for monomial ideals, so the new theorem contains that earlier result as a special case.","Since the formula is explicit in the facet data, it yields a finite algorithm: compute the Newton polyhedron, decompose it into facets, and compute a finite list of mixed volumes.","The counterexample to the conjecture of [3] shows that any correct volume formula for the multiplicity sequence must take the full facet geometry into account, not just the counts of unbounded directions and bounded faces.","For mixed multiplicities, the formula extends the known realization of mixed volumes as mixed multiplicities from ideals generated in a single degree to all monomial ideals, giving a combinatorial description in full generality."],"supporting_citations":[{"why":"defines the multiplicity sequence as the multidegrees of the double associated graded ring and supplies its Hilbert polynomial.","marker":"[1]"},{"why":"states a conjectured volume formula for the multiplicity sequence that Theorem 5.2 disproves.","marker":"[3]"},{"why":"gives the j-multiplicity formula for monomial ideals that Theorem 4.11 recovers as the i=0 case.","marker":"[20]"},{"why":"provides the quasi-polynomial and mixed-volume expansion for lattice-point counts of dilated polytopes used in Theorem/Definition 3.2.","marker":"[22]"},{"why":"shows that mixed volumes are realizable as mixed multiplicities of certain monomial ideals, which Theorem 6.1 extends to arbitrary monomial ideals.","marker":"[27]"},{"why":"allows replacement of powers of an ideal by their integral closures, a step in Lemma 2.5 that underlies the coefficient extraction.","marker":"[19]"}],"fun_headline_variants":["Mixed volumes crack monomial multiplicity sequence","New mixed-volume formula for monomial multiplicity sequence","Multiplicity sequence via Newton-polyhedron mixed volumes","Monomial ideals: mixed volumes yield multiplicity sequence","Counterexample to Achilles-Manaresi: mixed volumes win"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula holds only if the lower-order error terms in the lattice-point counts of the Newton polyhedron slices can be discarded; this requires each facet's unbounded directions to be coordinate axes and all facet vertices to be lattice points, as they are for monomial ideals.","fun_headline_variants_meta":{"raw":{"variants":["Mixed volumes crack monomial multiplicity sequence","New mixed-volume formula for monomial multiplicity sequence","Multiplicity sequence via Newton-polyhedron mixed volumes","Monomial ideals: mixed volumes yield multiplicity sequence","Counterexample to Achilles-Manaresi: mixed volumes win"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001381,"raw_usage":{"total_tokens":5548,"prompt_tokens":856,"completion_tokens":4692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":4619}},"tokens_in":472,"tokens_out":4692,"duration_ms":32865,"temperature":1.0,"reasoning_tokens":4619,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:29:22.575182+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the multiplicity sequence of the ideal $I=(x_1^2 x_3, x_2^2 x_3, x_1 x_2)$ directly from the Hilbert function of the double associated graded ring and compare it with the mixed-volume sum; if the result differs from $(0,6,0,0)$, Theorem 4.11 is false.","supporting_citations":[{"cited_title":"Achilles and M","cited_arxiv_id":null,"evidence_quote":"defines the multiplicity sequence as the multidegrees of the double associated graded ring and supplies its Hilbert polynomial."},{"cited_title":"Generalized Samuel multiplicities of monomial ideals and volumes.Exp","cited_arxiv_id":null,"evidence_quote":"states a conjectured volume formula for the multiplicity sequence that Theorem 5.2 disproves."},{"cited_title":"Thej-multiplicity of monomial ideals.Math","cited_arxiv_id":null,"evidence_quote":"gives the j-multiplicity formula for monomial ideals that Theorem 4.11 recovers as the i=0 case."},{"cited_title":"McMullen","cited_arxiv_id":null,"evidence_quote":"provides the quasi-polynomial and mixed-volume expansion for lattice-point counts of dilated polytopes used in Theorem/Definition 3.2."},{"cited_title":"Mixed multiplicities of ideals versus mixed volumes of polytopes.Trans","cited_arxiv_id":null,"evidence_quote":"shows that mixed volumes are realizable as mixed multiplicities of certain monomial ideals, which Theorem 6.1 extends to arbitrary monomial ideals."},{"cited_title":"Cambridge University Press, Cambridge, 2006","cited_arxiv_id":null,"evidence_quote":"allows replacement of powers of an ideal by their integral closures, a step in Lemma 2.5 that underlies the coefficient extraction."}],"review_version":1}