REVIEW 1 major objections 4 minor 13 references
Heights of complete intersections in toric varieties
T0 review · 1 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read For two Laurent polynomials on a toric variety, the height of the intersection of their torsion-twisted hypersurfaces converges to an adelic sum of mixed integrals of roof functions and duals of Ronkin functions.
desk verdict The proof of Theorem B has a real gap at inequality (5.5), but the strategy and local results are solid; the paper needs revision before acceptance. 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 auxiliary function $$F_v(t) = \int_{$X_v^{{\mathrm{an}}$}} \log |t^*g|_v \, c_1(\overline{D}_{0,v}) \wedge \dots \wedge c_1(\overline{D}_{n-2,v}) \wedge \delta_{$Z_v^{{\mathrm{an}}$}},$$ where $Z$ is the hypersurface defined by $f$. Theorem 3.4 shows that $F_v$ extends to the whole analytic torus with at most logarithmic singularities along the closed subset where $t^*g$ is proportional to $f$: a function with at most logarithmic singularities is continuous outside that subset and bounded below near it by a constant times the logarithm of the maximum of the defining equations. At Archimedean places this extension uses a continuity theorem for fiber integrals, while at non-Archimedean places it uses a formal-model description of the Monge\,--Amp\`ere measure as a weighted sum over irreducible components of the special fiber. Combined with the Archimedean logarithmic equidistribution theorem for torsion points and the non-Archimedean theorem on linear forms in roots of unity, this makes each local error term tend to zero; a separate adelic argument using the Poisson formula for the sparse resultant bounds the sum over all but finitely many places.
What would settle it
For a fixed number field and a pair of Laurent polynomials $f,g$, compute the Galois-orbit average of the local function $I_v$ along a strict sequence of torsion points at a non-Archimedean place where the reduction of $f$ or $g$ is reducible; if that average does not tend to 0, or the explicit norm identity in Lemma 5.4 fails, then Theorem 4.4 and hence Theorem B would be false.
Extended reading notes
Core claim
The central claim (Theorem 6.2) is that for nonzero Laurent polynomials $f,g \in K[M]$ and any quasi-strict sequence of torsion points $(\omega_\ell)_\ell$ in $T(K)^2$, the height satisfies $$\lim_{\ell\to\infty} h_{D_0,\dots,D_{n-2}}\bigl(Z_T(\omega_{\ell,1}^* f,\omega_{\ell,2}^* g)\bigr) = \sum_{v\in M} n_v \, \mathrm{MI}_M\bigl(\vartheta_{D_0,v},\dots,\vartheta_{D_{n-2},v},\rho_{f,v}^\vee,\rho_{g,v}^\vee\bigr).$$ The left side is the height of the intersection cycle of two translated hypersurfaces inside a complete toric variety, and the right side is an adelic sum of mixed integrals of the $v$-adic roof functions of the metrized divisors and the convex duals of the $v$-adic Ronkin functions of $f$ and $g$. The theorem is proved by reducing to smooth projective toric data with very ample divisors and algebraic or smooth metrics, applying the arithmetic B\'ezout theorem to separate a hypersurface height, and then proving that the remaining local integrals vanish both at each place, by logarithmic equidistribution, and away from a finite set of bad places, by an adelic argument.
Load-bearing premise
The argument stands on the claim that, at every place, the local integral function $F_v$ extends from torsion points to the whole analytic torus with only logarithmic singularities; if that extension broke down at even one place, the local error terms would not vanish and the limit formula would collapse.
Editorial extensions
If this is right
- For any two nonzero Laurent polynomials over a number field, the typical height of the intersection of their torsion-twisted hypersurfaces in a complete toric variety is explicitly computable from the Newton polytopes and the Ronkin data of $f$ and $g$.
- The result settles the previously open case $k=2$ of Conjecture A and, for bivariate Fermat polynomials of arbitrary degrees, confirms the authors' earlier conjecture on limit heights.
- Because the proof passes through reduction steps that allow arbitrary semipositive toric metrics to be approximated by smooth or algebraic ones, the formula holds for the full class of semipositive toric metrized divisors.
- Averaging over strict sequences of finite sets of torsion points, the same limit describes the typical height and shows that almost all twists have height within any positive tolerance of the limit.
Reading between the lines
- The local extension theorem for the auxiliary function $F_v$ is the main technical bottleneck; the same strategy could in principle prove Conjecture A for $k>2$ once such an extension is available for $k$ polynomials.
- Quantitative versions of the logarithmic equidistribution theorems used in the proof should yield explicit convergence rates for the height of twists of bounded torsion order, extending the special-case estimates the authors cite.
- The limit formula suggests a convex-geometric picture of arithmetic complexity for complete intersections, analogous to the way sparse-resultant geometry computes typical degrees by mixed volumes; testing it on families with more than two polynomials or on non-split tori would be a natural next step.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves Theorem B: for two nonzero Laurent polynomials f and g over a number field K, a complete toric variety X compactifying a split torus T, and a quasi-strict sequence of torsion points (ωℓ) in T(K)^2, the height of the intersection cycle ZT(ωℓ,1^* f, ωℓ,2^* g) converges to an adelic sum of mixed integrals of roof functions and Legendre–Fenchel duals of Ronkin functions. The proof combines toric intersection theory, arithmetic Bézout, an extension of an auxiliary function to the analytic torus with logarithmic singularities (Section 3), local logarithmic equidistribution theorems (Section 4), and an adelic vanishing statement (Section 5). The paper also contains a detailed reduction appendix and applications to average heights over torsion sets.
Significance. If the proof is completed, the result establishes a nontrivial two-codimensional case of the authors' conjecture and gives an arithmetic analogue of the Bernstein–Kushnirenko–Khovanskii theorem for typical intersections. The paper is well structured and makes use of deep external results (Stoll, Tate–Voloch, Dimitrov–Habegger) in a modular way, with most reduction steps moved to the appendix. The main technical concern is a specific false inequality in Section 5 that is used to prove the adelic vanishing; this gap appears fixable, but it is load-bearing for the proof of Theorem B.
major comments (1)
- [Section 5.B, Eq. (5.5)] The inequality (5.5) is false when the differences m−m' for m,m' in supp(f) generate a proper sublattice of M. For example, take K=Q, T=G_m^3, and f=g=1+x1+x2, which is absolutely irreducible and non-binomial. Let pℓ and qℓ be distinct primes with qℓ >> pℓ, and set ωℓ=(ζ_{pℓ}, ζ_{pℓ}^{ℓ}, ζ_{qℓ}). This sequence is strict: for every nonzero character (a,b,c), the value ζ_{pℓ}^{a+bℓ} ζ_{qℓ}^{c} is eventually different from 1, since for c≠0 the qℓ-component cannot cancel the pℓ-component once qℓ>|c|, while for c=0 the relation a+bℓ=0 holds for at most one ℓ. However dℓ=ord(ωℓ)=pℓ qℓ, whereas every character χ^{m−m'} with m,m'∈supp(f) involves only x1 and x2, so max_{m,m'} φ(ord(χ^{m−m'}(ωℓ))) = φ(pℓ) ≈ pℓ. This contradicts (5.5) for any c1≥1. Consequently the bound c2 log dℓ / dℓ^{c1} used to prove that the first summand in Theorem 5.2 tends to zero does not follow. The proof needs to replace dℓ by a quantity that measures the order of the projection of ωℓ to the subtorus generated by supp(f)−supp(f), or equivalently the lcm of the finitely many orders ord(χ^{m−m'}(ωℓ)), rather than the full order in T(K).
minor comments (4)
- [Introduction, p. 4] In the sentence 'we need to show prove that there exists a finite subset S ⊂ M', the word 'prove' appears to be a typographical error and should be deleted.
- [Section 4.A, proof of Theorem 4.1] In the phrase 'for all ℓ such that the strictness degree δ(ωℓ) si sufficiently large', 'si' should read 'is'.
- [Section 5.A, proof of Proposition 5.3] The final sentence 'the statement follows from Lemma 5.1' appears to cite the wrong lemma: Lemma 5.1 only establishes finiteness of S, whereas the displayed norm formula is proved in Lemma 5.4. The reference should be corrected.
- [Example 6.3] The notation 'MIZ2(0∆, ρ∨_{f,v}, ρ∨_{g,v})' is not defined in the paper; if it denotes the mixed integral for n=2 with the indicator of the standard simplex, this should be stated explicitly.
Circularity Check
No significant circularity: Theorem B is derived from prior k=0,1 height formulas plus new local and adelic vanishing arguments, and the target formula is never assumed.
full rationale
The paper's derivation chain is self-contained rather than circular. The proof of Theorem B (Theorem 6.2) follows the strategy in Section 2.B: the arithmetic B\'ezout formula gives the exact recursive identity in Proposition 2.5, separating the height into the hypersurface height h_{D0,...,D_{n-2},D_Ron_g}(Z_T(f)) plus a sum of local error terms. The first term is evaluated using Theorem 2.2 from [Gua18b], the previously established k=1 case whose hypotheses do not include the k=2 target. The remaining error terms are shown to vanish by two genuinely new inputs: Theorem 4.4, which combines the auxiliary-function regularity of Theorem 3.4 with the logarithmic equidistribution theorems of Dimitrov-Habegger and Tate-Voloch, and Theorem 5.2, the adelic vanishing argument. No equation defining the limit is inserted as an assumption, and no fitted quantity is renamed as a prediction. The self-citations to [BPS14] and [Gua18b] are legitimate prior published results used as base cases and technical tools; they do not presuppose Conjecture A or Theorem B. The challenged inequality (5.5) in Section 5.B is a potential correctness gap in the adelic vanishing proof, but a false estimate is not circularity: even if the bound fails, the target formula is not being used as its own input. Thus the score is 0.
Assumptions & free parameters
assumptions (6)
- standard math Arithmetic Bezout formula for heights [BPS14, Chapter 1]
- standard math Stoll's theorem on the continuity of fiber integrals [Sto67]
- standard math Tate-Voloch theorem on linear forms in p-adic roots of unity [TV96]
- standard math Dimitrov-Habegger logarithmic equidistribution of torsion points [DH24]
- standard math Laurent's theorem (toric Manin-Mumford) [BG06, Theorem 7.4.7]
- standard math Poisson formula for the sparse resultant [DS15]
Cite this review
Pith. "Pith review of Heights of complete intersections in toric varieties." pith.science (2026). https://pith.science/paper/2UM3KQWS
@misc{pith2026241216308,
author = {Pith},
title = {Pith review of: Heights of complete intersections in toric varieties},
year = {2026},
howpublished = {\url{https://pith.science/paper/2UM3KQWS}},
note = {Machine review of arXiv:2412.16308}
}
abstract
The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions. Here we extend these results to the $2$-codimensional situation by presenting a limit formula predicting the typical height of the intersection of two hypersurfaces on a toric variety. More precisely, we prove that the height of the intersection cycle of two effective divisors translated by a strict sequence of torsion points converges to an adelic sum of mixed integrals of roof and duals of Ronkin functions. This partially confirms a previous conjecture of the authors about the average height of families of complete intersections in toric varieties.
Reference graph
Works this paper leans on
-
[1]
[BE21] S. Boucksom and D. Eriksson, Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry , Adv. Math. 378 (2021), Paper No. 107501, 124 pp. [Ber75] D. N. Bernstein, The number of roots of a system of equations , Funkcional. Anal. i Priloˇ zen. 9 (1975), no. 3, 1–4. [Ber90] V. G. Berkovich, Spectral theory and analytic g...
work page 2021
-
[10]
[Mai00] V. Maillot, G´ eom´ etrie d’Arakelov des vari´ et´ es toriques et fibr´ es en droites int´ egrables, M´ em. Soc. Math. Fr. (N.S.) (2000), no. 80, vi+129. [MS19] C. Mart ´ ınez and M Sombra, An arithmetic Bernˇ stein-Kuˇ snirenko inequality, Math. Z. 291 (2019), no. 3-4, 1211–1244. [Neu99] J. Neukirch, Algebraic number theory, Grundlehren Math. Wiss...
work page 2000
-
[13]
Stoll, The multiplicity of a holomorphic map , Invent
[Sto66] W. Stoll, The multiplicity of a holomorphic map , Invent. Math. 2 (1966), 15–58. [Sto67] , The continuity of the fiber integral , Math. Z. 95 (1967), 87–138. [TV96] J. Tate and J. F. Voloch, Linear forms in p-adic roots of unity , Internat. Math. Res. Notices (1996), no. 12, 589–601. Departament de Matem`atiques, Universitat Polit `ecnica de Catalu...
work page 1966
-
[1970]
$p$-adic equidistribution and an application to $S$-units
[Sch24] G. Schefer, p-adic equidistribution and an application to S-units, e-print arXiv:2410.24088,
-
[1984]
Bilu, Limit distribution of small points on algebraic tori , Duke Math
[Bil97] Y. Bilu, Limit distribution of small points on algebraic tori , Duke Math. J. 89 (1997), no. 3, 465–476. [Bos69] S. Bosch, Orthonormalbasen in der nichtarchimedischen Funktionent heorie, Manuscripta Math. 1 (1969), 35–57. [Bos14] , Lectures on formal and rigid geometry , Lect. Notes Math., vol. 2105, Springer,
work page 1997
-
[1998]
Limit heights and special values of the Riemann zeta function
[GK17] W. Gubler and K. K¨ unnemann, A tropical approach to nonarchimedean Arakelov geometry , Algebra Number Theory 11 (2017), no. 1, 77–180. [GS23] R. Gualdi and M. Sombra, Limit heights and special values of the Riemann zeta functio n, e-print arXiv:2304.01966,
work page Pith review arXiv 2017
-
[1999]
Ostrowski, Zur arithmetischen Theorie der algebraischen Gr¨ oßen , G¨ ott
[Ost19] A. Ostrowski, Zur arithmetischen Theorie der algebraischen Gr¨ oßen , G¨ ott. Nachr. 1919 (1919), 279–298. [Poi13] J. Poineau, Les espaces de Berkovich sont ang´ eliques, Bull. Soc. Math. France 141 (2013), no. 2, 267–297. 46 GUALDI AND SOMBRA [Roc70] R. T. Rockafellar, Convex analysis , Princeton Math. Ser., vol. 28, Princeton Univ. Press,
work page 1919
-
[2002]
Galois orbits of torsion points over polytopes near atoral sets
[Lin24] C. Lin, Galois orbits of torsion points over polytopes near atoral s ets, e-print arXiv:2412.11156,
Show all 13 references
-
[2011]
Chambert-Loir and A
[CT09] A. Chambert-Loir and A. Thuillier, Mesures de Mahler et ´ equidistribution logarithmique , Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 977–1014. [DH24] V. Dimitrov and P. Habegger, Galois orbits of torsion points near atoral sets , Algebra Number Theory 18 (2024), n...
2009
-
[2012]
Chambert-Loir, Mesures et ´ equidistribution sur les espaces de Berkovich, J
[Cha06] A. Chambert-Loir, Mesures et ´ equidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235. [CLS11] D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties , Grad. Stud. Math., vol. 124, Amer. Math. Soc.,
2006
-
[2014]
Chambert-Loir and A
[CD12] A. Chambert-Loir and A. Ducros, Formes diff´ erentielles r´ eelles et courants sur les espaces de Berkovich , e-print arXiv:1204.6277,
-
[2018]
10, 2403–2443
[Gua18b] , Heights of hypersurfaces in toric varieties , Algebra Number Theory 12 (2018), no. 10, 2403–2443. [Gub98] W. Gubler, Local heights of subvarieties over non-Archimedean fields , J. Reine Angew. Math. 498 (1998), 61–113. [Gub07] , Tropical varieties for non-Archimedean...
2018
-
[2024]
D’Andrea and M
[DS15] C. D’Andrea and M. Sombra, A Poisson formula for the sparse resultant , Proc. Lond. Math. Soc. (3) 110 (2015), no. 4, 932–964. [Ful93] W. Fulton, Introduction to toric varieties , Ann. of Math. Stud., vol. 131, Princeton Univ. Press,
2015
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.