A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

Itay Glazer; Yotam I. Hendel

REVIEW 2 major objections 3 minor 36 references

A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

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Pith's one-line read This paper proves that the analytic log-canonical threshold is always positive over local fields of positive characteristic, and gives an optimal effective lower bound of 1/(d·D^m) for regular functions on smooth algebraic varieties.

desk verdict Theorem 1.2 is solid and new, but the effective bound in Theorem 1.4 rests on a gap in Lemma 2.11 that needs a real fix before the paper is complete. read the letter →

arxiv 2511.01270 v2 pith:L4JK2GHO submitted 2025-11-03 math.AG math.LO

Itay Glazer , Yotam I. Hendel This is my paper · ORCID

classification math.AGmath.LO MSC 11S8014B0514G20

keywords log-canonicalthresholdpositivecharacteristiclocalfieldssmallballestimatesWeierstrasspreparationorderofvanishingHilbert–Samuelmultiplicitysublevelsets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the analytic log-canonical threshold — the largest power to which one can raise the reciprocal absolute value of a function and still integrate it locally — is always positive for analytic functions on manifolds over local fields of positive characteristic. In the algebraic setting, where the function and the variety are given by polynomials, the paper derives an explicit, uniform lower bound that depends only on the degrees involved: at least 1/(d·D^m), where d is the degree of the functions, D the degree of the equations cutting out the variety, and m the codimension. The bound is optimal, as a curve defined by iterated D-th powers with a single d-th power function attains equality. These results matter because in positive characteristic resolution of singularities and the usual tools of singularity theory are unavailable, so previously no such uniform integrability statement was known. The proof works through elementary sublevel-measure estimates for monic polynomials over non-Archimedean fields, bypassing the missing resolution machinery.

What carries the argument

The argument rests on two ingredients. First, a small-ball (sublevel) estimate: for a monic polynomial of degree d over a non-Archimedean local field, the Haar measure of the set where |f(x)| ≤ δ is at most d·δ^{1/d}; the same bound holds for a Weierstrass polynomial in several variables once the Weierstrass preparation theorem is used to reduce a general analytic function to a monic polynomial in the last variable. Second, in the algebraic setting, the order of vanishing of a degree-d polynomial along a smooth variety defined by degree-D equations is controlled by the Hilbert–Samuel multiplicity, bounded via Bézout by d·D^m; a conversion lemma (Lemma 2.11) then turns this order bound into t

What would settle it

Check the proof's coordinate-change step on f(x)=x^K(1+x) over any positive-characteristic local field: the coefficients of x^K and x^{K+1} both have absolute value 1, so the function is never distinguished of order K after any allowed shear and rescaling (trivial in one variable). This one-line example settles whether Lemma 2.11's route from order K to lct ≥ 1/K is valid; if the construction fails, the effective lower bound remains unproven, and finding an actual function with lct < 1/K would refute Theorem 1.4.

Watch

Extended reading notes

Core claim

At the heart of the paper is Theorem 1.4: if X is a smooth n-dimensional subvariety of affine space defined by m polynomials of degree at most D, and if φ_1, ..., φ_r are polynomials of degree at most d that do not all vanish on any irreducible component of X, then for every F-point x0 of X the analytic log-canonical threshold lct_F(⟨φ_1,...,φ_r⟩; x0) is at least 1/(d·D^m). The constant is sharp: Example 1.5 shows that on the curve x_{i+1} = x_i^D, the function φ = x_{m+1}^d has threshold exactly 1/(d·D^m). Along the way the paper establishes the more basic positivity statement (Theorem 1.2) for arbitrary analytic functions on analytic manifolds, with no algebraic or degree hypotheses.

Load-bearing premise

The whole argument leans on the claim that an analytic function of order K can, after a shear and a rescaling of some coordinates, be made distinguished—meaning its K-th pure power term has strictly the largest coefficient—so that Weierstrass preparation yields a monic degree-K polynomial; for some functions this step is not guaranteed.

Editorial extensions

If this is right

For any smooth algebraic F-variety and any regular function φ that does not vanish identically on a component, |φ|_F^{-s} is locally integrable for every s < 1/(d·D^m), uniformly at all F-points.
The lower bound is optimal in general: the curve x_{i+1}=x_i^D with φ=x_{m+1}^d realizes lct = 1/(d·D^m).
The positivity theorem covers arbitrary analytic functions on arbitrary F-analytic manifolds, with no bound depending on degrees.
For an ideal J generated by several functions, lct_F(J; x0) is bounded below by 1/(d·D^m) whenever the generating polynomials satisfy the stated degree and vanishing conditions.
These bounds supply a positive exponent of integrability for pushforwards of smooth compactly supported measures by analytic maps in positive characteristic, generalizing characteristic-zero results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's key step, Lemma 2.11, assumes that a generic shear and rescaling makes an order-K analytic function distinguished in the last variable; for a function like x^K(1+x) this step visibly fails, so the effective bound may rest on a repairable but currently unproven claim—or the bound may hold by a different route.
The small-ball estimation technique may be flexible enough to yield sublevel bounds for functions that are not Weierstrass polynomials, such as sums of monomials or products, potentially giving analogous thresholds without the algebraic assumptions.
The optimality example suggests a general heuristic: the smallest threshold on a variety tends to be realized along a branch where the function has maximal order, governed by the product of degrees; this could guide conjectures for the related F-pure threshold, which the paper notes is a different invariant.
The uniform-in-point nature of the algebraic bound may help in establishing global integrability theorems for distributions on reductive groups over positive-characteristic local fields, where previously no such uniform arithmetical control existed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the positivity theorem is the real content and it is convincing. The effective bounds are plausible but the proof of Lemma 2.11, which is the bridge from order to lct, does not actually establish that the function is distinguished. The paper deserves review, but the author will need to repair that lemma.

What's new: over a fixed local field of positive characteristic there is no Hironaka resolution and no D-module machinery, so even positivity of the analytic log-canonical threshold was apparently not known. The paper proves it via a Remez-type small-ball estimate for Weierstrass polynomials. Lemmas 2.6 and 2.7 are clean and correct; the proof of Theorem 1.2 is short and self-contained. That alone is a nice contribution.

The effective lower bound in Theorem 1.4, lct ≥ 1/(d·D^m), is a useful quantitative statement, and Example 1.5 shows it is optimal. But the proof depends on Lemma 2.11, and there the argument goes wrong. The shear T(x_i + c_i x_n) followed by rescaling the first n−1 coordinates does not make the function distinguished of order K in x_n. Distinguishedness requires the coefficient of x_n^K — a power series in the other variables — to have Gauss norm strictly larger than all coefficients of higher x_n powers. The proof only shows that a scalar (the constant term of that coefficient) is nonzero, and then claims maximality. That is not strict inequality, and the rescaling leaves x_n alone, so higher pure x_n powers are not controlled. The stress-test example f = x^K − y^K + t y^{K+1} over F_q((t)) with K = q−1 is a legitimate counterexample to the construction: for every unit c, after the shear the y^K and y^{K+1} coefficients both end up with norm 1 when the valuation of c^K−1 is 1. So the proof as written does not work.

That said, the lemma is likely true. The paper itself quotes Proposition 2.2 (BGR), which gives an analytic diffeomorphism of the form (x_i + x_n^{d_i}) making any nonzero Tate algebra element distinguished of the correct order. Applying that instead of the shear would fix the proof. But the current text doesn't do it, so the effective bound rests on an unproved step. Lemma 2.10's Bezout inequality also deserves a precise reference, though I don't doubt the bound.

Who this is for: anyone working on integrability of Fourier transforms, jet schemes, or singularities over F_ℓ. Theorem 1.2 is the headline. The paper is worth refereeing; a serious referee would ask for the Lemma 2.11 repair (or a citation to Proposition 2.2), and then the effective results would be in good shape. I'd send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper studies the analytic log-canonical threshold lct_F(f;x0) over local fields F of positive characteristic. Theorem 1.2 asserts positivity for any nonzero analytic ideal on an analytic manifold, and Theorems 1.3 and 1.4 assert uniform and effective lower bounds for regular functions on smooth algebraic varieties, with the explicit bound 1/(d·D^m) in the affine setting of Theorem 1.4. The strategy is to reduce to a Weierstrass polynomial and use a Remez-type small-ball estimate (Lemmas 2.6 and 2.7). In the algebraic case, Lemma 2.10 bounds the order of vanishing via Hilbert–Samuel multiplicity and Bézout, and Lemma 2.11 is then used to convert an order bound K into the lower bound lct ≥ 1/K.

Significance. If the proof is completed, the paper supplies useful foundational results in positive characteristic, where Hironaka resolution and model-theoretic transfer are unavailable. The small-ball estimates in Lemmas 2.6 and 2.7 are elementary and correct, and Example 1.5 shows that the effective bound in Theorem 1.4 is optimal. However, the proof of Lemma 2.11, which is load-bearing for the effective theorem, has a serious gap. The overall approach is promising and likely salvageable, but the current version does not establish Theorem 1.4.

major comments (2)

[§2.1, Lemma 2.11] The step claiming that the rescaled function \tilde g is distinguished of order K is unjustified. Definition 2.1(4)(b) requires ||a_K|| > ||a_s|| for all s>K, but the proof only makes a_K a unit (norm 1). The final rescaling leaves x_n unscaled: pure x_n^s coefficients for s>K are multiplied only by the common factor \varpi^{-k0}. They were known only to have norm <1 before this rescaling, so when k0≥1 they can have norm ≥1 afterwards. Concretely, take F=F_q((t)), K=q-1, and g=x^K-y^K+t y^{K+1}. For every unit c, after T(x,y)=(x+cy,y) the y^K coefficient is c^K-1 ∈ tO_F, so k0≥1, while the y^{K+1} coefficient is t. After the proof's rescaling with k0=1, both coefficients have norm 1, so strict dominance fails. Thus Lemma 2.11 is not proved as written.
[§2.1, Theorem 1.4] Lemma 2.11 is the only step converting the order bound ord_{X,x0}(φ) ≤ d·D^m from Lemma 2.10 into the lower bound lct_F ≥ 1/(d·D^m). Since the proof of Lemma 2.11 is invalid, Theorem 1.4 is unsupported. The statement of Lemma 2.11 may still be true, but the manuscript does not provide a correct argument; this is a load-bearing gap, not a cosmetic issue.

minor comments (3)

[§1, first paragraph] 'In this Appendix we study...' appears to be a leftover from the appendix format; it should read 'In this paper we study...'.
[Lemma 2.11, proof] The notation q_F^{k0} appears to mean q_F^{-k0}; as written, the valuations are inverted (e.g., '|Σ c^I b_I|_F = q_F^{k0}' and 'maximal absolute value q_F^{k0}').
[Theorems 1.3 and 1.4] The phrase 'do not simultaneously vanish on some irreducible component of X' is ambiguous; it should be 'do not simultaneously vanish on any irreducible component of X' to match the intended condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; the only self-citations are contextual and non-load-bearing.

full rationale

The claimed derivation chain goes Theorem 1.4 -> Lemma 2.10 (order bound via Hilbert-Samuel multiplicity and Bezout, cited to external [HS06]) plus Lemma 2.11 (order-to-lct conversion via Weierstrass preparation [BGR84], Gauss-norm normalization, and the small-ball estimate Lemma 2.7). Each load-bearing step is obtained from external theorems or direct estimates, not from the target lct bound. No parameter is fitted to the result and then renamed a prediction; Example 1.5 only illustrates optimality, and no equation in the proof is equivalent by construction to the theorem it is used to prove. The author self-citations ([GH19] for the notion of complexity, [GH21,GHS] for characteristic-0 analogues) are contextual and do not carry the main argument. The reader's/skeptic's objection about the strict-dominance condition in Lemma 2.11 is a possible proof gap or correctness issue, not a circular reduction of the theorem to its own input, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper uses no fitted parameters and introduces no new entities. It relies on standard rigid-analytic preparation theorems, a standard multiplicity/order equivalence, a plausible but under-referenced Bezout-type multiplicity inequality, and standard Haar-measure facts. The only non-standard load-bearing assumption is the Bezout inequality in Lemma 2.10.

assumptions (5)

standard math Weierstrass preparation theorem and distinguishedness for Tate algebras (BGR84, §5.2.2 Thm 1 and Prop 2.2)
Reduces an arbitrary analytic function to a Weierstrass polynomial in Cor 2.4 and Lemma 2.11; classical rigid-analytic theorem.
domain assumption Hilbert–Samuel multiplicity of a hypersurface equals the order of its defining function at a smooth point, and satisfies the Bezout-type bound e_HS(X∩{φ=0},x0) ≤ deg(φ)·∏deg(g_j)
Lemma 2.10; the first equality is cited to [HS06, Ex. 11.2.8], but the Bezout inequality is invoked without a specific proof or reference and is load-bearing for the 1/(d·D^m) bound.
standard math Haar measure on a local field has ball measures μ(B(x,r)) ≤ r under the normalization μ(O_F)=1
Used in small-ball Lemmas 2.6 and 2.7.
domain assumption Local integrability of |f|^{-s} is invariant under F-analytic diffeomorphisms, multiplication by nonzero scalars, and anisotropic rescaling of coordinates
Used in Cor 2.4 and Lemma 2.11 to change coordinates and rescale; standard for locally constant Haar measure.
standard math Finite field extensions allow splitting a monic polynomial into linear factors, and the norm extends
Used in Lemma 2.6.

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Pith. "Pith review of A lower bound on the analytic log-canonical threshold over local fields of positive characteristic." pith.science (2026). https://pith.science/paper/L4JK2GHO

@misc{pith2026251101270,
  author       = {Pith},
  title        = {Pith review of: A lower bound on the analytic log-canonical threshold over local fields of positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4JK2GHO}},
  note         = {Machine review of arXiv:2511.01270}
}

abstract

Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\rightarrow F$, the $F$-analytic log-canonical threshold $\mathrm{lct}_{F}(f;x_{0})$ is the supremum over the values $s\geq0$ such that $\left|f\right|_{F}^{-s}$ is integrable near $x_{0}\in X$. We show that $\mathrm{lct}_{F}(f;x_{0})>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound $\mathrm{lct}_{F}(f;x_{0})>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Aizenbud, D

A. Aizenbud, D. Gourevitch, D. Kazhdan, and E. Sayag. Effective local differential topology of algebraic varieties over local fields of positive characteristic. in preparation

[2] [2]

Aizenbud, D

A. Aizenbud, D. Gourevitch, D. Kazhdan, and E. Sayag. The jet schemes of the nilpotent cone of gl _ n over F _ and analytic properties of the C hevalley map. in preparation

[3] [3]

V. I. Arnold, S. M. Guse n-Zade, and A. N. Varchenko. Singularities of differentiable maps. V ol. II , volume 83 of Monographs in Mathematics . Birkh\" a user Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi

1988

[4] [4]

I. N. Bern s te n. Analytic continuation of generalized functions with respect to a parameter. Funkcional. Anal. i Prilo z en. , 6(4):26--40, 1972

1972

[5] [5]

Bosch, U

S. Bosch, U. G\"untzer, and R. Remmert. Non- A rchimedean analysis , volume 261 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry

1984

[6] [6]

Bhatt, D

B. Bhatt, D. J. Hern\'andez, L. E. Miller, and M. Musta t a . Log canonical thresholds, F -pure thresholds, and nonstandard extensions. Algebra Number Theory , 6(7):1459--1482, 2012

2012

[7] [7]

Blickle, M

M. Blickle, M. Musta t a , and K. E. Smith. F -thresholds of hypersurfaces. Trans. Amer. Math. Soc. , 361(12):6549--6565, 2009

2009

[8] [8]

N. Budur. Singularity invariants related to M ilnor fibers: survey. In Zeta functions in algebra and geometry , volume 566 of Contemp. Math. , pages 161--187. Amer. Math. Soc., Providence, RI, 2012

2012