REVIEW 1 major objections 5 minor 6 references
Symbolic powers of monomial ideals
T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For monomial ideals generated in one degree with height at least 2, the second coefficient of the symbolic-power multiplicity quasi-polynomial is eventually constant.
desk verdict A plausible and genuinely new extension of the constancy result for Hilbert quasi-polynomial coefficients, but the written proof has two repairable gaps that a referee should ask to be fixed. 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 argument is carried by a homogeneous I-superficial element u, which exists because I is generated in a single degree, along with the Rees-module structures on L_F = ⊕_n A/I_{n+1}(J) and L_I = ⊕_n A/$I^{{n+1}}$. A short exact sequence 0 → W → L_I → L_F → 0 connects the two filtrations. Dividing by u produces a reduced filtration on A/(u) whose multiplicity g(n) satisfies the crucial relation g(n) = f(n) - f(n-1). This identity transfers the constancy of the top coefficient of g to the constancy of a_{c-1} for f.
What would settle it
Compute the full quasi-polynomial of e0(I_n(J)/I^n) for a concrete pair such as A = K[x,y], I = ($x^{2}$,$y^{2}$), J = (x,y); if the coefficient of $n^{{c-1}}$ is not constant across residue classes modulo the period for n large, the central claim is false.
Extended reading notes
Core claim
The central claim is Theorem 1.2: under the hypotheses that A is a standard graded polynomial ring over a field, I and J are monomial ideals, I is generated by some elements of the same degree, and height I >= 2, the coefficient a_{c-1}(n) in the quasi-polynomial expansion of f^I_J(n) = e0(I_n(J)/I^n) is constant for all sufficiently large n. The proof picks a homogeneous I-superficial element u, reduces modulo (u) to the ring A/(u), and compares the original filtration with the reduced one. Writing g(n) for the multiplicity of the reduced quotient, the proof obtains the identity g(n) = f(n) - f(n-1) for all large n. Since the leading coefficient of g is constant by the earlier theorem, the periodicity of a_{c-1} forces it to be constant as well.
Load-bearing premise
The proof relies on the unstated lemma that if the dimension of I_n(J)/I^n is strictly larger than the dimension of the same quotient modulo a homogeneous I-superficial element, then the quasi-polynomial degree c must be zero.
Editorial extensions
If this is right
- For monomial ideals with a single generator degree and height at least 2, the quasi-polynomial for e0(I_n(J)/I^n) has both its leading and next-to-leading coefficients independent of the residue class of n modulo the period.
- The same conclusion applies to any multiplicative filtration of homogeneous ideals satisfying the hypotheses of Theorem 2.2, since the proof is stated at the level of general filtrations.
- The coefficient a_{c-1} is determined by the leading coefficient of the reduced filtration on A/(u), so it can be computed from the same data used to compute the top term.
- If the theorem is correct, the periodic part of the quasi-polynomial starts only at degree c-2, mirroring the kind of rigidity seen in Ehrhart quasi-polynomials of rational polytopes.
- The proof suggests that homogeneous superficial elements are sufficient for coefficient-stability results of this kind, without needing the full strength of a general superficial element.
Reading between the lines
- The unproved dimension-drop lemma—asserting that a strict drop in dimension after reducing modulo a homogeneous superficial element forces the quasi-polynomial degree to be zero—is the one fragile step; if it fails, the conclusion r = s and hence l = c-1 would fail, so a_{c-1} could in principle vary periodically.
- A computational search over small monomial ideals with height 2 or 3, computing the full quasi-polynomial of e0(I_n(J)/I^n) for n up to a few periods, could test the theorem's scope and may reveal whether the single-degree assumption is necessary.
- The method likely extends to ideals generated in multiple degrees if one can find a homogeneous superficial element with respect to a suitable filtration, but the dimension-comparison step would need a separate proof in that setting.
- The result fits the broader pattern that multiplicity functions of symbolic powers are eventually polynomial-like to a high order, suggesting that the Ehrhart-grade phenomenon for rational polytopes has an algebraic analogue for monomial filtrations.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. Let A = K[X_1,...,X_d] and let I,J be monomial ideals. The paper studies f^I_J(n) = e_0(I_n(J)/I^n), where I_n(J) = I^n : J^∞ is the nth symbolic power with respect to J. Earlier work [4] established that this function is of quasi-polynomial type and that its leading coefficient a_c(n) is eventually constant, and also that dim I_n(J)/I^n is eventually constant. The present paper proves (Theorem 1.2) that if I is generated by elements of a single degree and height I ≥ 2, then the next coefficient a_{c-1}(n) is also eventually constant. The proof chooses a homogeneous I-superficial element u, reduces modulo u, and uses a short exact sequence relating I_n(J)/I^n, I_{n-1}(J)/I^{n-1}, and the corresponding quotient. The main steps are: an injectivity claim for multiplication by u on symbolic powers, an application of the earlier theorem to the quotient filtration, and a comparison of leading coefficients of f(n) and f(n)-f(n-1). The paper is short and the strategy is natural, but the written proof contains an unsupported assertion that is load-bearing for the central claim.
Significance. If the theorem is correct, it is a genuine refinement of [4]: it shows that not only the top Hilbert coefficient but also the second coefficient of the symbolic-power multiplicity function stabilizes under a mild hypothesis on the generators of I. The result fits naturally in the literature on Hilbert coefficients of powers and symbolic powers and should be of interest to commutative algebraists. The proof is concise and mostly self-contained, and the reduction to the quotient by a superficial element is elegant. No code or machine-checked material is included; this is a traditional mathematical proof.
major comments (1)
- [Section 3, proof of Theorem 1.2] The step "If r > s then notice c = 0 which is a contradiction" is asserted without proof. This step is load-bearing because it is used to conclude r = s, which in turn justifies l = c-1 and the coefficient comparison f(n)-f(n-1)=g(n). The missing justification is that in the exact sequence 0 → M_{n-1} → M_n → N_n → 0 with dim M_{n-1} = dim M_n = r and dim N_n = s < r, the Hilbert series of N_n, when rewritten with denominator (1-z)^r, has numerator divisible by (1-z)^{r-s}; hence its contribution to e_0 computed with respect to dimension r vanishes, and e_0(M_n) = e_0(M_{n-1}) for all n >> 0. Therefore f(n) is eventually constant, contradicting the standing assumption c > 0. This argument should be stated explicitly in the proof.
minor comments (5)
- [Section 3, coefficient comparison] The displayed identity "a_c + a_{c-1}(n) - a_{c-1}(n-1) = b" is missing the factor c multiplying a_c. The correct leading coefficient of f(n)-f(n-1) is c a_c + a_{c-1}(n) - a_{c-1}(n-1). The subsequent periodicity argument is unaffected because c a_c is also a constant, but the displayed equation as written is incorrect.
- [Section 3, grade statement] The sentence "Notice grade I = height I ≥ 2. So grade I ≥ 1" should refer to the image ideal \bar I = I/(u) in Rbar = A/(u), not to I itself. Please clarify the notation.
- [Section 3, notation] The symbol R is used both for the Rees algebra A[It] and later for the quotient ring A/(u) within the same proof, which is confusing. Consider using a different symbol, such as \mathcal R for the Rees algebra and S or \bar A for the quotient.
- [Section 3, exact sequence passage] The passage from the sequence (**) to the displayed short exact sequence of quotients is very compressed; a brief explanation of the index shift and of the isomorphism between (I_n(J),u)/(I^n,u) and \bar I_n(J)/\bar I^n would greatly help the reader.
- [Introduction, typos] The name Ehrhart is misspelled as "Erhart" in two places in the introduction; there are also several OCR artifacts (e.g., "affine", "di m", "coinc ides") that should be corrected in the final version.
Circularity Check
No significant circularity: the paper legitimately reuses its own earlier theorem for the leading coefficient and dimension constancy, and the new a_{c-1} claim is derived by an independent exact-sequence argument.
full rationale
The proof of Theorem 1.2 does not reduce its conclusion to its inputs by construction. It invokes the author's prior theorem [4, 2.4] (restated as Theorem 2.2) for two facts: dim(I_n(J)/I^n) is eventually constant and the leading coefficient a_c is constant. That cited result is a published theorem whose assumptions do not include the target statement that a_{c-1} is constant, so it supplies independent support rather than a circular premise. The new content is the exact sequence 0 -> I_{n-1}(J)/I^{n-1} -> I_n(J)/I^n -> \bar{I}_n(J)/\bar{I}^n -> 0, the conclusion r = s, and the consequent relation f(n) - f(n-1) = g(n). Comparing coefficients in this relation and using periodicity of a_{c-1} is a genuine derivation, not a renamed assumption. One sentence in the proof, 'If r > s then notice c = 0 which is a contradiction,' is asserted without justification; it is a gap in exposition, repairable by multiplicity additivity along the exact sequence, and it is not an instance of the proof assuming what it proves. The displayed coefficient comparison also omits a harmless factor c, but this does not affect the periodicity conclusion. No fitted parameter is relabelled as a prediction, and no uniqueness theorem is imported from the authors' prior work.
Assumptions & free parameters
assumptions (5)
- standard math The polynomial ring A = K[X1,...,Xd] is standard graded and Cohen-Macaulay, so grade equals height for homogeneous ideals.
- domain assumption The symbolic Rees algebra \bigoplus_{n>=0} I_n(J) is finitely generated over A for monomial ideals I and J.
- domain assumption Theorem 2.2 of the paper (constancy of dimension and of the leading coefficient for any filtration with finitely generated Rees algebra) is accepted as proved in [4].
- domain assumption A homogeneous I-superficial element u exists when I is generated in a single degree over an infinite field.
- standard math If f(n) is a quasi-polynomial of degree c with leading coefficient a_c, then f(n)-f(n-1) has degree c-1 with leading coefficient c a_c.
Cite this review
Pith. "Pith review of Symbolic powers of monomial ideals." pith.science (2026). https://pith.science/paper/HNREMDYQ
@misc{pith2026190802085,
author = {Pith},
title = {Pith review of: Symbolic powers of monomial ideals},
year = {2026},
howpublished = {\url{https://pith.science/paper/HNREMDYQ}},
note = {Machine review of arXiv:1908.02085}
}
abstract
Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n \gg 0$ say \[ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + \text{lower terms}, \] where for $i = 0, \ldots, c$, $a_i \colon \mathbb{N} \rt \mathbb{Z}$ are periodic functions of period $g$ and $a_c \neq 0$. In an earlier paper we (together with Herzog and Verma) proved that $\dim I_n(J)/I^n$ is constant for $n \gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I \geq 2$ then $a_{c-1}(-)$ is also a constant.
Reference graph
Works this paper leans on
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Reviewed August 14, 2026 · model on record in the stance chip above.
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