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arxiv: 2506.18891 · v2 · submitted 2025-06-23 · 🧮 math.AC

On lower bounds for the F-pure threshold of equigenerated ideals

Benjamin Baily This is my paper

Pith reviewed 2026-05-19 07:34 UTC · model grok-4.3

classification 🧮 math.AC MSC 13A35
keywords F-pure thresholdtest idealsequigenerated idealshomogeneous idealsTakagi-Watanabe inequalitypositive characteristiccommutative algebra
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The pith

Equality holds in the Takagi-Watanabe F-pure threshold bound exactly for a classified family of equigenerated homogeneous ideals, with a new lower bound given via the height of the associated test ideal.

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

The paper classifies which equigenerated homogeneous ideals achieve equality in the known lower bound fpt(I) >= height(I)/d. It also establishes a new lower bound for fpt(I) expressed in terms of the height of the test ideal of I raised to the power fpt(I). These results apply to ideals in polynomial rings over fields of positive characteristic. A sympathetic reader cares because the F-pure threshold measures singularities in this setting, and precise classification plus tighter bounds improve explicit computations and understanding of when minimal thresholds occur.

Core claim

Let k be a field of positive characteristic and R = k[x_0, ..., x_n]. We consider ideals I subset R generated by homogeneous polynomials of degree d. Takagi and Watanabe proved that fpt(I) >= height(I)/d; we classify ideals I for which equality is attained. Inspired by a result of de Fernex, Ein, and Mustata, we give a lower bound on fpt(I) in terms of the height of tau(I^{fpt(I)}).

What carries the argument

The F-pure threshold fpt(I) together with the test ideal tau(I raised to the power fpt(I)) and the height of that test ideal.

If this is right

  • Equality in fpt(I) = height(I)/d holds precisely for the classified equigenerated ideals.
  • Every equigenerated ideal satisfies the new lower bound relating fpt(I) to height of tau(I^{fpt(I)}).
  • The results give tighter control over possible values of the F-pure threshold for given height and generator degree.

Where Pith is reading between the lines

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

  • The classification may help compute exact F-pure thresholds in low-dimensional examples by reducing to the equality cases.
  • The test-ideal height approach could connect to similar bounds for other invariants such as log canonical thresholds in mixed characteristic.
  • If the new bound is sharp for some non-equality cases, it suggests a way to iterate the construction for even tighter estimates.

Load-bearing premise

The Takagi-Watanabe inequality holds for equigenerated homogeneous ideals and test ideals behave as expected when raised to the F-pure threshold power before taking height.

What would settle it

A specific equigenerated homogeneous ideal in a small polynomial ring over a positive-characteristic field whose explicitly computed F-pure threshold violates either the classified equality cases or the new lower bound derived from the test-ideal height.

read the original abstract

Let $k$ be a field of positive characteristic and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we classify ideals $I$ for which equality is attained. Inspired by a result of de Fernex, Ein, and Musta\c{t}\u{a}, we give a lower bound on $\mathrm{fpt}(I)$ in terms of the height of $\tau(I^{\mathrm{fpt}(I)})$.

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

0 major / 3 minor

Summary. The manuscript classifies equigenerated homogeneous ideals I in the polynomial ring R = k[x_0, …, x_n] over a field k of positive characteristic for which equality holds in the Takagi-Watanabe inequality fpt(I) ≥ height(I)/d. It also establishes a new lower bound on fpt(I) expressed in terms of the height of the test ideal τ(I^{fpt(I)}), inspired by results of de Fernex, Ein, and Mustaţă.

Significance. If the results hold, the classification of equality cases resolves when the Takagi-Watanabe bound is sharp for equigenerated ideals, providing a concrete tool for identifying minimal F-pure thresholds. The new lower bound refines existing estimates by incorporating the test ideal and may improve computations in positive-characteristic singularity theory. The work rests on standard properties of test ideals without introducing free parameters or circular definitions.

minor comments (3)
  1. The introduction would benefit from a brief outline of the proof strategy for the classification, even if details appear later, to help readers assess the scope immediately.
  2. Notation for the test ideal τ and the powering I^{fpt(I)} should be defined or recalled at the first use in the main body rather than relying solely on the abstract.
  3. Consider adding one or two explicit low-dimensional examples (e.g., monomial ideals) that attain equality, with direct verification of both the original bound and the new one, to illustrate the classification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately describes our classification of equigenerated homogeneous ideals attaining equality in the Takagi-Watanabe bound and the new lower bound on fpt(I) in terms of the height of the test ideal. We appreciate the recognition of the work's potential utility in positive-characteristic singularity theory.

Circularity Check

0 steps flagged

No significant circularity; builds on external prior results

full rationale

The paper classifies equality cases in the established Takagi-Watanabe inequality fpt(I) >= height(I)/d for equigenerated homogeneous ideals of degree d and derives a new lower bound on fpt(I) via height of tau(I^{fpt(I)}), drawing inspiration from de Fernex-Ein-Mustaţă. These steps rest on the prior external inequality (by different authors) and standard properties of test ideals under powering and height operations in positive characteristic. No derivation reduces by construction to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain; the central claims retain independent content from the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the Takagi-Watanabe inequality and standard properties of test ideals in positive characteristic; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption Takagi-Watanabe inequality fpt(I) >= height(I)/d holds for equigenerated homogeneous ideals
    Abstract invokes this as the starting point for both the classification and the new bound.

pith-pipeline@v0.9.0 · 5622 in / 1223 out tokens · 31336 ms · 2026-05-19T07:34:14.406743+00:00 · methodology

discussion (0)

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Reference graph

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