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arxiv: 1907.02378 · v1 · pith:UV6RGSCPnew · submitted 2019-07-04 · 🧮 math.AG

The Bruce-Roberts number of a function on a hypersurface with isolated singularity

Juan J. Nu\~no-Ballesteros , Bruna Or\'efice-Okamoto , B\'arbara K. L. Pereira , Jo\~ao N. Tomazella This is my paper

Pith reviewed 2026-05-25 09:09 UTC · model grok-4.3

classification 🧮 math.AG
keywords bruce-roberts numbermilnor numbertjurina numberhypersurface singularitylogarithmic characteristic varietycohen-macaulay
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The pith

For functions on isolated hypersurface singularities with finite Bruce-Roberts number, that number equals the sum of three Milnor numbers minus the Tjurina number.

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

The paper proves an explicit algebraic formula relating the Bruce-Roberts number of a function f on a hypersurface X to the Milnor numbers of f, of the pair (φ,f), and of X itself, minus the Tjurina number of X. It further establishes that the logarithmic characteristic variety of X is Cohen-Macaulay. These identities hold whenever the Bruce-Roberts number is finite and remove the weighted-homogeneity restriction present in earlier statements of the same results.

Core claim

If (X,0) is an isolated hypersurface singularity defined by φ and f has finite Bruce-Roberts number μ_BR(f,X), then μ_BR(f,X) equals μ(f) + μ(φ,f) + μ(X,0) − τ(X,0). In addition, the logarithmic characteristic variety LC(X,0) is Cohen-Macaulay.

What carries the argument

The Bruce-Roberts number μ_BR(f,X), which counts the dimension of a certain quotient module attached to the pair (f,X) and serves as the central invariant whose value is expressed in terms of Milnor and Tjurina numbers.

Load-bearing premise

The Bruce-Roberts number of f on X is finite and X is an isolated hypersurface singularity.

What would settle it

An explicit pair (f,X) with finite μ_BR(f,X) for which the numerical equality μ_BR(f,X) = μ(f) + μ(φ,f) + μ(X,0) − τ(X,0) fails to hold.

read the original abstract

Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $\mu_{BR}(f,X)$ is finite. We first prove that $\mu_{BR}(f,X)=\mu(f)+\mu(\phi,f)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

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 / 1 minor

Summary. The paper proves two results for an isolated hypersurface singularity (X,0) defined by φ : (C^n,0) → (C,0) and a function f with finite Bruce-Roberts number μ_BR(f,X): the equality μ_BR(f,X) = μ(f) + μ(φ,f) + μ(X,0) − τ(X,0), where μ and τ denote the indicated Milnor and Tjurina numbers, and the assertion that the logarithmic characteristic variety LC(X,0) is Cohen-Macaulay. Both statements remove the weighted-homogeneous hypothesis required in earlier work by the authors.

Significance. If correct, the explicit formula supplies a practical relation among standard singularity invariants that accounts for the difference μ(X,0) − τ(X,0) in the non-weighted-homogeneous setting, while the Cohen-Macaulay property of LC(X,0) supplies a structural fact useful for further computations in the theory of logarithmic derivations and characteristic varieties.

minor comments (1)
  1. The abstract states that μ and τ apply to “a function or an isolated complete intersection singularity,” but does not specify the precise definition of the mixed term μ(φ,f); a brief parenthetical or reference in the abstract would clarify the notation for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No circularity; derivation self-contained via independent invariants

full rationale

The central result expresses μ_BR(f,X) explicitly as μ(f) + μ(φ,f) + μ(X,0) − τ(X,0) using the standard, independently defined Milnor and Tjurina numbers; these are not defined in terms of μ_BR nor fitted from it. The Cohen-Macaulay claim on LC(X,0) is likewise a direct structural statement. The generalization from the authors' prior weighted-homogeneous case is achieved precisely by inserting the difference μ(X,0)−τ(X,0), which is an external correction term rather than a self-referential loop. No equation reduces to its own inputs by construction, no parameter is renamed as a prediction, and the cited prior work supplies only the special-case baseline, not the load-bearing argument for the general case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on the standard definitions and basic properties of Milnor and Tjurina numbers for isolated singularities in complex analytic geometry; no free parameters or new entities introduced in the abstract.

axioms (1)
  • standard math Milnor and Tjurina numbers are well-defined for isolated hypersurface and complete-intersection singularities
    Invoked directly in the stated equality

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