REVIEW 5 minor 28 references
The multiplicity sequence of monomial ideals
T0 review · 0 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read The multiplicity sequence of any monomial ideal is a signed sum of mixed volumes of pyramids built from the facets of its Newton polyhedron.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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.
minor comments (5)
- [Section 4, after Eq. (4)] 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.
- [Remark 3.4] 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.
- [References] Reference [6] (Bhattacharya) appears in the bibliography but is not cited in the text; it should either be cited where relevant or removed.
- [Proof of Proposition 4.10, second inequality] 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.
- [Remarks 3.4 and 4.12] 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.
Circularity Check
No significant circularity: the multiplicity-sequence formula is derived from Newton-polyhedron lattice-point counts, and the known j-multiplicity formula is recovered as a special case rather than assumed.
full rationale
The derivation is self-contained. The invariants c_i(I) are defined in Definition 2.1 as coefficients of the Hilbert polynomial of the bigraded algebra gr_m(gr_I(R)), independently of any Newton-polyhedron data. Lemma 2.5 reduces their computation to the Hilbert function λ_R(I^u/(m^v I^u+I^{u+1})), using standard integral-closure stability results; this is an algebraic input, not an assumption of the desired formula. Section 4 then expresses this length as a lattice-point count in uΓ and decomposes uΓ\(u+1)Γ into facet cones (Lemma 4.6), with Proposition 4.10 giving λ = Σ_F L_F(u,v) + o(d−1). Proposition 4.3 and Theorem/Definition 3.2 (McMullen's mixed-volume theorem) convert the lattice-point counts into mixed-volume polynomials, and Theorem 4.11 extracts the coefficients. No parameter is fitted, no normalization forces the conclusion, and the known j-multiplicity formula of Jeffries-Montaño is recovered afterward (Remark 4.12) as a consistency check, not used as a premise. The delicate error-term cancellation in Proposition 3.10 is governed by explicit stated hypotheses (P_2 spanned by coordinate vectors, P_1 with lattice vertices), so it is a technical condition rather than a circular reduction. The counterexample to the Achilles-Manaresi conjecture is computed independently from the new formula and compared with an external conjecture. Self-citations appear only for background and benchmarking and do not carry the argument.
Assumptions & free parameters
assumptions (5)
- standard math The lattice point enumerator E_Q(t_1,...,t_r) of rational polytopes is a quasi-polynomial whose degree d leading homogeneous part has constant coefficients equal to the mixed volume expansion (McMullen's theorem).
- standard math Integral closures of powers of an ideal stabilize: I^{n+l} = I^n I^l for n >> 0 (Rees; Huneke-Swanson, [19, Cor. 9.2.1]).
- standard math Multiplicity sequences are additive on short exact sequences and can be defined for modules.
- standard math The saturated multi-Rees algebra associated to integral closures of powers is a finitely generated module over the Rees algebra (Das-Roy [13], Lemma 3.1).
- domain assumption For a monomial ideal, the monomials in I^u correspond exactly to lattice points in the u-th dilation of its Newton polyhedron (Huneke-Swanson, [19, Prop. 1.4.2 and 1.4.6]).
Cite this review
Pith. "Pith review of The multiplicity sequence of monomial ideals." pith.science (2026). https://pith.science/paper/NRT3UB5B
@misc{pith2026260812731,
author = {Pith},
title = {Pith review of: The multiplicity sequence of monomial ideals},
year = {2026},
howpublished = {\url{https://pith.science/paper/NRT3UB5B}},
note = {Machine review of arXiv:2608.12731}
}
read the original abstract
We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.
Figures
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