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arxiv: 2604.03164 · v1 · submitted 2026-04-03 · 🧮 math.AG · math.AC

Lipschitz saturation of toric singularities in any dimension

Fran\c{c}ois Bernard , Enrique Ch\'avez-Mart\'inez , Arturo E. Giles Flores This is my paper

Pith reviewed 2026-05-13 18:36 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords toric singularitiesLipschitz saturationNewton polyhedranormalizationsemigrouppresaturationcomplex analytic singularities
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The pith

A monomial from the normalization lies in the Lipschitz saturation of a toric singularity precisely when its exponent satisfies Newton polyhedra and lattice conditions.

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

The paper characterizes the semigroup of the Lipschitz saturation for any complex analytic toric singularity. It supplies a necessary and sufficient criterion, expressed through the Newton polyhedron of the singularity and conditions on the ambient lattice, that decides membership for each monomial in the normalization. The criterion immediately produces a finite algorithm that computes the entire saturated semigroup from the defining data. The same description reveals that, once dimension exceeds two, this saturation is strictly smaller than Campillo's presaturation for some singularities.

Core claim

The semigroup of the Lipschitz saturation consists exactly of those monomials in the normalization whose exponents meet the stated Newton-polyhedron and lattice conditions; this holds uniformly in every dimension, and the resulting description separates Lipschitz saturation from presaturation in dimension three and higher.

What carries the argument

The Newton polyhedron of the toric singularity together with the lattice conditions that test membership of each exponent in the Lipschitz saturation semigroup.

If this is right

  • The Lipschitz saturation semigroup becomes computable by a finite, explicit algorithm from the Newton polyhedron alone.
  • In dimension greater than two the Lipschitz saturation is strictly smaller than Campillo's presaturation for some toric singularities.
  • The saturated ring or ideal can be read off directly once the qualifying monomials are identified.
  • The same criterion applies verbatim to the analytic category as well as the algebraic one.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The criterion may serve as a model for describing Lipschitz saturation after resolution of non-toric singularities.
  • It supplies a concrete test that could be used to decide Lipschitz equisingularity for families of toric singularities.
  • The distinction between saturation and presaturation in higher dimensions suggests that similar separations may appear for other notions of saturation.

Load-bearing premise

The Lipschitz saturation of the toric singularity is completely determined by the Newton polyhedron and the lattice conditions on exponents.

What would settle it

A concrete toric singularity in dimension three together with an explicit monomial in its normalization that meets the polyhedral and lattice conditions yet fails to lie in the Lipschitz saturation, or vice versa.

Figures

Figures reproduced from arXiv: 2604.03164 by Arturo E. Giles Flores, Enrique Ch\'avez-Mart\'inez, Fran\c{c}ois Bernard.

Figure 1
Figure 1. Figure 1: Illustration of Proposition 1.4 Remark 1.6. Let A ,→ B be a finite extension of rings with A a Noetherian ring. Then one can show that the process of adding elements bi ∈ B with bi+1 ∈/ A[b1, ..., bi ] ends after a finite number of steps. This means that the process of adding elements of the form presented in Remark 1.3 will end after a finite number of steps. In particular, if Γ is an affine semigroup and… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Example 3.8 Corollary 3.9 Let m, k1, . . . , kr, l1, . . . ls ∈ Γ, such that m − ki ∈ Γ, m − lj ∈ Γ, and X i ki − X j lj ∈ Γ, for all i ∈ {1, . . . , r} and j ∈ {1, . . . , s},then m + P i ki − P j lj ∈ Γ s . Proof : Since m ∈ Γ and P i ki − P j lj ∈ Γ, we obtain m + P i ki − P j lj ∈ Γ. Now, con￾sider {p1, . . . , pn} a set of generators of Γ. Since m, k1, . . . , kr, l1, . . . ls ∈ Γ, the… view at source ↗
read the original abstract

We describe the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. We give a necessary and sufficient condition for a monomial in the normalization to belong to the Lipschitz saturation, in terms of Newton polyhedra and lattice conditions, and deduce a finite algorithm to compute it. We also show that, in dimension greater than two, Campillo's notion of presaturation differs from the Lipschitz saturation, even for complex singularities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript describes the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. It states a necessary and sufficient condition, in terms of Newton polyhedra and lattice conditions, for a monomial in the normalization to lie in the Lipschitz saturation, and derives a finite algorithm from this criterion. It further shows that Campillo's presaturation differs from the Lipschitz saturation in dimensions greater than two, even for complex singularities.

Significance. If the claimed equivalence holds, the work supplies a combinatorial, algorithmically computable description of an analytic invariant for toric singularities. This is valuable for explicit calculations in higher dimensions, where direct analytic verification of Lipschitz extensions is difficult, and the separation from presaturation clarifies distinctions among saturation notions.

major comments (2)
  1. [§4, Theorem 4.3] The sufficiency direction of the main result (that the stated Newton-polyhedron and lattice conditions imply membership in the analytic Lipschitz saturation) is load-bearing. The argument must explicitly construct or guarantee a Lipschitz map on the ambient space whose restriction induces the monomial; in dimension >=3 the normalization map can introduce local analytic obstructions not obviously controlled by polyhedral data alone.
  2. [§5, Algorithm 5.1] The finite algorithm deduced from the criterion is asserted to terminate, but the proof that the lattice conditions can be checked in finitely many steps (via boundedness of relevant polyhedral faces or support functions) is not detailed enough to confirm it works uniformly across all dimensions.
minor comments (2)
  1. [§1] The introduction would benefit from a short table comparing the various saturation notions (Lipschitz, Campillo presaturation, etc.) and their defining properties.
  2. [§2] Notation for the semigroup of the normalization and the associated polyhedra is introduced gradually; a consolidated list of symbols at the end of §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The two major points raised concern the sufficiency proof in Theorem 4.3 and the termination argument for Algorithm 5.1. We address each below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] The sufficiency direction of the main result (that the stated Newton-polyhedron and lattice conditions imply membership in the analytic Lipschitz saturation) is load-bearing. The argument must explicitly construct or guarantee a Lipschitz map on the ambient space whose restriction induces the monomial; in dimension >=3 the normalization map can introduce local analytic obstructions not obviously controlled by polyhedral data alone.

    Authors: We agree that an explicit link between the combinatorial conditions and the existence of a Lipschitz extension is essential. In the toric case the normalization is monomial, so the analytic Lipschitz saturation reduces to controlling orders of vanishing along the toric divisors. The Newton-polyhedron condition together with the lattice inequalities on the support functions precisely ensure that the difference of valuations admits a Lipschitz extension to the ambient space; this follows from the fact that toric singularities admit monomial resolutions and that Lipschitz maps can be constructed by patching along the fan using the boundedness of the relevant support functions. We will insert a short auxiliary lemma (or expanded paragraph) immediately after the statement of Theorem 4.3 that spells out this reduction, citing the relevant toric geometry facts. This will make the sufficiency direction fully explicit without altering the main argument. revision: partial

  2. Referee: [§5, Algorithm 5.1] The finite algorithm deduced from the criterion is asserted to terminate, but the proof that the lattice conditions can be checked in finitely many steps (via boundedness of relevant polyhedral faces or support functions) is not detailed enough to confirm it works uniformly across all dimensions.

    Authors: We accept that the termination argument requires more detail to be uniform in dimension. The relevant faces of the Newton polyhedron are compact in the directions transverse to the support, and the linear inequalities defining the lattice conditions therefore admit only finitely many solutions inside any bounded region; the support-function bounds coming from the polyhedron itself give an explicit (dimension-independent) radius beyond which no further lattice points can satisfy the inequalities. We will expand the proof of Algorithm 5.1 with a dedicated paragraph that records this boundedness estimate and verifies that the enumeration is finite for any fixed toric singularity, independent of ambient dimension. revision: yes

Circularity Check

0 steps flagged

No circularity: combinatorial criterion derived from standard toric and Lipschitz definitions

full rationale

The paper states a necessary and sufficient condition for monomials in the normalization to lie in the Lipschitz saturation, expressed via Newton polyhedra and lattice conditions, together with a finite algorithm and a separation from Campillo presaturation in dimension >2. These statements rest on the standard definitions of toric singularities, their normalization, Newton polyhedra, and the Lipschitz saturation concept taken from prior literature. No step in the abstract or described claims reduces the target membership condition to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation whose content is itself unverified. The sufficiency direction is presented as a proved equivalence rather than an identity by construction, and the work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on prior definitions of toric singularities, normalization, Newton polyhedra, and Lipschitz saturation from the literature in algebraic geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard definitions of complex analytic toric singularities, their normalization, and the Lipschitz saturation concept hold as in prior literature.
    Invoked to state the condition and algorithm for the semigroup.

pith-pipeline@v0.9.0 · 5367 in / 1202 out tokens · 52480 ms · 2026-05-13T18:36:57.599760+00:00 · methodology

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