REVIEW 2 major objections 3 minor 17 references
Mixed Segre zeta functions and their log-concavity
T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The mixed Segre zeta function of a sequence of homogeneous ideals is rational, and the homogenization of the numerator of $1-\zeta$ is denormalized Lorentzian.
desk verdict New mixed Segre zeta function with rationality and Lorentzian results; structurally sound, but the key cycle identity in Proposition 2.6 needs a more detailed proof before publication. 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 central object is the mixed Segre zeta function $\zeta_{I_1,\dots,I_m}(t_1,\dots,t_m)$, the generating function of the push-forwards of mixed Segre classes after extending the ideals to $\mathbb{P}^N$ for arbitrarily large $N$. The argument is carried by a joint blow-up construction: mixed Segre classes are expressed through the exceptional divisors $E_i$ of the joint blow-up, and the cycle identity $[P_X]=[P_Z]+[B]_n$ (Proposition 2.6) turns this into a blow-up formula (Theorem 2.7). A pull-back theorem for mixed Segre classes under projections (Theorem 3.8) explains how the classes change when the ambient projective space grows, which yields rationality. For log-concavity, the same blow-up expression identifies the numerator $Q$ with the push-forward of the total Chern class of globally generated quotient bundles; Proposition 5.2 shows such push-forwards are denormalized Lorentzian because volume polynomials are Lorentzian.
What would settle it
Compute the mixed Segre zeta function for two non-complete-intersection homogeneous ideals in $\mathbb{P}^5$, for instance $I_1=(x_0^2,x_0x_1,x_1^2)$ and $I_2=(x_2^3,x_2x_3,x_3^2)$, using the formula of Corollary 2.12; form $Q$ from $1-\zeta=Q\prod_{i,j}(1+d_{i,j}t_i)$ and check whether the homogenization of $Q$ is denormalized Lorentzian. A failure of nonnegative coefficients, M-convex support, or the one-positive-eigenvalue condition would refute Theorem B.
Extended reading notes
Core claim
The central claim is that mixed Segre zeta functions are rational with denominator $\prod_{i,j}(1+d_{i,j}t_i)$ determined by the generator degrees, and that the numerator of $1-\zeta_{I_1,\dots,I_m}$ is log-concave in a precise sense. Theorem A states that $\zeta_{I_1,\dots,I_m}(t_1,\dots,t_m)=P(t)/\prod_{i,j}(1+d_{i,j}t_i)$ for a polynomial $P$ with nonnegative integer coefficients. Theorem B states that, over an algebraically closed field, if $1-\zeta=Q\prod_{i,j}(1+d_{i,j}t_i)$, then the homogenization of $Q$ is denormalized Lorentzian: after normalizing coefficients by factorials it satisfies the Lorentzian conditions of nonnegative coefficients, M-convex support, and at most one positive eigenvalue in every relevant Hessian. The paper also shows the zeta function only depends on integral closure and gives a mixed formula expressing the Segre zeta function of the product ideal in terms of the mixed zeta function.
Load-bearing premise
The load-bearing step is the cycle identity $[P_X]=[P_Z]+[B]_n$ of Proposition 2.6, whose proof is compressed into an inclusion said to be checkable by expanding; if that inclusion fails for some ideals, the blow-up formula and both main theorems would not follow.
Editorial extensions
If this is right
- Every sequence of homogeneous ideals over an algebraically closed field yields a denormalized Lorentzian polynomial, so the class of Lorentzian polynomials contains a large, explicitly geometric family.
- The mixed Segre zeta function is rational and its numerator has nonnegative coefficients, so the function is determined by finitely many coefficients and inherits the pole structure of the single-ideal Segre zeta function.
- Replacing any ideal by its integral closure leaves the mixed Segre zeta function unchanged, so some generator degrees listed in the denominator may cancel and only integral-closure data is genuinely recorded.
- The mixed formula expresses the Segre zeta function of the product ideal $I_1\cdots I_m$ through the mixed zeta function, yielding explicit rational formulas for products of complete intersections in disjoint variables.
- Setting $m=1$ recovers the ordinary Segre zeta function, so the rationality and log-concavity results contain the single-ideal case as a special case.
Reading between the lines
- Inference: because $\zeta$ is invariant under integral closure, the polynomial $Q$ is an invariant of the integral closure of an ideal, and its coefficients may carry numerical information about the singularity that the paper does not extract.
- Inference: the mechanism suggests a general principle: joint blow-ups of ideal sequences give Lorentzian polynomials through Chern classes of globally generated bundles, a route that could be tested on multihomogeneous ideals, ideals in toric varieties, or flat families of ideals.
- Inference: a concrete next test is to compute $Q$ for ideals defining non-reduced schemes, such as fat points, and compare the resulting polynomials with known log-concave families; the paper leaves this comparison open.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces the mixed Segre zeta function of a sequence of homogeneous ideals in a polynomial ring, a power series that packages the push-forwards of Kleiman–Thorup mixed Segre classes after extending the ideals to projective spaces of arbitrarily large dimension. The main results are: rationality of this zeta function with a denominator built from the generator degrees and a numerator with nonnegative coefficients (Theorem A); invariance of the zeta function under integral closure of the ideals; a mixed formula relating the Segre zeta function of a product ideal to the mixed Segre zeta function; and a Lorentzian/log-concavity statement asserting that the homogenization of the numerator of 1 minus the mixed Segre zeta function is denormalized Lorentzian (Theorem B). The technical core consists of a blow-up formula for mixed Segre classes, a projection/join formula for how these classes pull back under projections, and an application of Brändén–Huh volume-polynomial theory.
Significance. If the main results are correct, the paper supplies a large and systematic family of Lorentzian polynomials associated with arbitrary homogeneous ideals, genuinely generalizing and unifying work of Kleiman–Thorup and Aluffi. The Lorentzian conclusion is particularly notable because covolume polynomials, the analogous objects in Aluffi's multihomogeneous framework, can fail to be Lorentzian even after normalization. The paper is honest about its technical debts: the arguments are built on Fulton's intersection theory, Kleiman–Thorup mixed Segre classes, and the Branden–Huh theorem, rather than on fitted parameters or assumed conclusions. The inclusion of a Macaulay2 implementation for mixed Segre classes and worked examples is a concrete strength that helps verify the formulas.
major comments (2)
- [Proposition 2.6] The cycle identity [P_X] = [P_Z] + [B]_n is load-bearing for the blow-up formula (Theorem 2.7) and therefore for the rationality and Lorentzian theorems. Its proof is compressed into the sentence that the inclusion K_1^{n1+1}...K_m^{nm+1} ∩ tO_bX ⊂ tK_1^{n1}...K_m^{nm} 'can be checked by expanding.' The inclusion is plausible and appears to hold, but the entire central claim depends on it; the manuscript should provide the expansion as a short lemma or at least an explicit demonstration for the multi-ideal case, including the edge cases where some J_i is zero or the unit ideal.
- [Lemma 4.6] The displayed equality at the start of the proof of Lemma 4.6 is too terse: it moves from the expression in Remark 2.8, which has denominators 1 + E_i t_i, to an expression involving c_t(O(-e_i)) and the quotient bundles Q_i without spelling out how the sign conventions for O_B(-e_i), O_{P(E)}(-e_i), and the exceptional divisors E_i interact. Since this lemma is the source of the nonnegativity of the numerator in Theorem 4.3 and is also used in the proof of Theorem 5.3, the author should expand this computation so that a reader can check the equality line by line.
minor comments (3)
- [Example 6.4] In the display 'ζI(I) = 1 − 3/(1+4t) + 2/(1+5t)', the notation ζI(I) appears to be a typo for ζ_I(t).
- [Introduction] In the last paragraph of the introduction, 'It should mentioned that' should read 'It should be mentioned that.'
- [Example 6.6] The Macaulay2 output is given only up to total degree 5; it would help the reader to state explicitly that this is the truncation predicted by the closed formula, since the equality is with the power series and not with a finite polynomial.
Circularity Check
No circularity: the rationality and Lorentzian theorems are derived from independent inputs; the compressed verification in Proposition 2.6 is a proof gap, not a circular step.
full rationale
The derivation chain is self-contained in the relevant sense. The mixed Segre zeta function is defined by push-forwards of Kleiman–Thorup mixed Segre classes, and Theorem A derives rationality from the blow-up formula (Theorem 2.7), which is proved via the cycle identity in Proposition 2.6 rather than assumed. Theorem B defines Q by the identity 1 − ζ = Q ∏(1 + d_{i,j}t_i) and then identifies Q with the push-forward of the total Chern class of globally generated quotient bundles on the joint blow-up; the Lorentzian conclusion follows from Proposition 5.2, whose proof invokes the independent Brändén–Huh theorem that volume polynomials are Lorentzian. No parameter is fitted to the target conclusion, and no step assumes that Q is Lorentzian. The one self-citation, [7], is used for the joint blow-up construction of more than two subschemes, but Setup 2.1 explicitly constructs the same multi-Rees algebra, so the citation is not load-bearing. The only fragile point is the inclusion in Proposition 2.6, which the paper says “can be checked by expanding”; this is an omitted verification and therefore a correctness risk, but it is not a circular reduction of the theorem to its own conclusion.
Assumptions & free parameters
assumptions (5)
- standard math Kleiman-Thorup theory of mixed Segre classes and joint blow-ups
- standard math Fulton's intersection theory, including the projective bundle formula, projection formula, specialization, and Chow group exact sequences
- standard math Branden-Huh theorem that volume polynomials are Lorentzian
- domain assumption Theorem B assumes k is algebraically closed, while Theorem A works over an arbitrary field
- domain assumption Each ideal is homogeneous with a chosen finite set of homogeneous generators
Cite this review
Pith. "Pith review of Mixed Segre zeta functions and their log-concavity." pith.science (2026). https://pith.science/paper/N5GGYBUG
@misc{pith2026250706424,
author = {Pith},
title = {Pith review of: Mixed Segre zeta functions and their log-concavity},
year = {2026},
howpublished = {\url{https://pith.science/paper/N5GGYBUG}},
note = {Machine review of arXiv:2507.06424}
}
read the original abstract
We introduce and study the mixed Segre zeta function of a sequence of homogeneous ideals in a polynomial ring. This function is a power series encoding information about the mixed Segre classes obtained by extending the ideals to projective spaces of arbitrarily large dimension. Our work generalizes and unifies results by Kleiman and Thorup on mixed Segre classes and by Aluffi on Segre zeta functions. We prove that this power series is rational, with poles corresponding to the degrees of the generators of the ideals. We also show that the mixed Segre zeta function only depends on the integral closure of the ideals. Finally, we prove that the homogenization of the numerator of a modification of the mixed Segre zeta function is denormalized Lorentzian in the sense of Br\"and\'en and Huh.
Reference graph
Works this paper leans on
-
[1]
Aluffi, Computing characteristic classes of projective schemes, J
P. Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput. 35 (2003), no. 1, 3–19
work page 2003
-
[2]
, The Segre zeta function of an ideal, Adv. Math. 320 (2017), 1201–1226
work page 2017
-
[3]
, Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones , Adv. Math. 437 (2024), Paper No. 109440, 37
work page 2024
-
[4]
P. Aluffi and E. Faber, Chern classes of splayed intersections, Canad. J. Math. 67 (2015), no. 6, 1201–1218
work page 2015
- [5]
-
[6]
P. Br ¨and´en and J. Huh, Lorentzian polynomials, Ann. of Math. (2) 192 (2020), no. 3, 821–891
work page 2020
-
[7]
Y . Cid-Ruiz,Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities, arXiv:2311.15105 (2023)
arXiv 2023
-
[8]
D. Eisenbud and J. Harris, 3264 and all that—a second course in algebraic geometry , Cambridge University Press, Cam- bridge, 2016
work page 2016
Show all 17 references
-
[9]
Eklund, C
D. Eklund, C. Jost, and C. Peterson, A method to compute Segre classes of subschemes of projective space, J. Algebra Appl. 12 (2013), no. 2, 1250142, 15
2013
-
[10]
Fulton, Intersection theory, Second, Ergebnisse der Mathematik und ihrer Grenzgebiete
W. Fulton, Intersection theory, Second, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathemat- ics], vol. 2, Springer-Verlag, Berlin, 1998
1998
-
[11]
I. M. Gessel and I. Kar, Binomial convolutions for rational power series, J. Integer Seq. 27 (2024), no. 1, Paper No. 24.1.3, 27
2024
-
[12]
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http: //www.math.uiuc.edu/Macaulay2/. MIXED SEGRE ZETA FUNCTIONS AND THEIR LOG-CONCA VITY 27
-
[13]
Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977
R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52
1977
-
[14]
Helmer, Algorithms to compute the topological Euler characteristic, Chern-Schwartz-MacPherson class and Segre class of projective varieties, J
M. Helmer, Algorithms to compute the topological Euler characteristic, Chern-Schwartz-MacPherson class and Segre class of projective varieties, J. Symbolic Comput. 73 (2016), 120–138
2016
-
[15]
Helmer and C
M. Helmer and C. Harris, SegreClasses: A Macaulay2 package. Version 1.03
-
[16]
Kleiman and A
S. Kleiman and A. Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), no. 1, 168–231
1994
-
[17]
, Mixed Buchsbaum-Rim multiplicities, Amer. J. Math. 118 (1996), no. 3, 529–569. DEPARTMENT OF MATHEMATICS , NORTH CAROLINA STATE UNIVERSITY , RALEIGH , NC, USA Email address: ycidrui@ncsu.edu
1996
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