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arxiv: 2605.17350 · v1 · pith:3GU7XWACnew · submitted 2026-05-17 · 🧮 math.AG

Generic mathcal{A}-finite determinacy and singularities of homogeneous polynomial mappings

N. G. Grulha Jr. , J. V. Pissolato , M. A. S. Ruas This is my paper

Pith reviewed 2026-05-19 23:02 UTC · model grok-4.3

classification 🧮 math.AG
keywords A-finite determinacyhomogeneous polynomial mappingsgeneric propertiessingularitiesalgebraic geometrydeterminacy criteriacomplex mappings
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The pith

Homogeneous polynomial mappings are generically A-finitely determined in dimensions 2 to 4.

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

This paper extends results on generic properties of homogeneous polynomial mappings between complex spaces of the same dimension. Starting from a 2019 proof for maps from three-space to three-space, the authors apply the same strategy and geometric criterion to establish that A-finite determinacy holds generically for n at least 2. Several other generic properties of the mappings extend only as far as dimension 4. A reader would care because these statements describe the typical singularity behavior of polynomial maps, which controls their local classification and deformation in algebraic geometry.

Core claim

Using the geometric criterion for A-finite determinacy, the authors prove that this property is generic for homogeneous polynomial mappings from C^n to C^n for n greater than or equal to 2, while additional generic properties of the mappings can be shown only for n up to 4.

What carries the argument

The geometric criterion for A-finite determinacy, which reduces genericity questions to algebraic conditions on the homogeneous mappings.

If this is right

  • A-finite determinacy holds generically for homogeneous maps in every dimension at least 2.
  • Several other generic singularity properties extend to dimensions greater than or equal to 2.
  • The same strategy that worked in dimension 3 produces the higher-dimensional statements directly.
  • Most homogeneous polynomial mappings therefore possess only isolated finite singularities in these dimensions.

Where Pith is reading between the lines

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

  • The same criterion might continue to work for mildly non-homogeneous maps after small perturbations.
  • Explicit computational checks in dimension 5 could locate the precise cutoff for each property.
  • Results of this type could guide classification of real polynomial maps or maps between spaces of unequal dimension.

Load-bearing premise

The geometric criterion for A-finite determinacy applies without modification to the higher-dimensional homogeneous cases.

What would settle it

A homogeneous polynomial mapping in dimension 5 that satisfies the generic conditions yet fails to be A-finitely determined would disprove the claimed extension.

read the original abstract

We make a detailed investigation of the generic properties that polynomial mappings possess. An important starting point is the work by Farnik, Jelonek and Ruas in 2019, where they prove some of those properties in the context of homogeneous polynomial mappings of $\mathbb{C}^3$ to $\mathbb{C}^3$, and conclude the genericity of $\mathcal{A}$-finite determinacy by applying the geometric criterion. Using their strategy, we further extend and generalize some of their key findings to dimensions greater than or equal to $2$, though some of those properties can only be extended up to dimension $4$.

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 paper extends results from Farnik, Jelonek and Ruas (2019) on generic properties of homogeneous polynomial mappings f: C^3 → C^3, including genericity of A-finite determinacy via a geometric criterion, to homogeneous maps in dimensions n ≥ 2 (with some properties holding only up to n = 4) by reusing the same strategy and criterion.

Significance. If the extension of the geometric criterion is valid without modification, the work would generalize key genericity statements for A-equivalence and singularities of homogeneous polynomials to higher dimensions, strengthening tools in algebraic geometry for studying jet-space stratifications and orbit dimensions under the contact group action.

major comments (2)
  1. [generalization section (statements for n ≥ 2 and n ≤ 4)] The central claim that the geometric criterion for A-finite determinacy from Farnik-Jelonek-Ruas (2019) applies unchanged to homogeneous maps in dimensions n ≥ 4 is load-bearing but unsupported by explicit verification. The manuscript states it follows the same strategy yet provides no check that the dimension of the tangent space to the A-orbit or the codimensions in the jet-space stratification remain unaltered when source and target dimensions increase (see the section on generalization to dimensions ≥ 2 and the statements limited to dimension 4).
  2. [main results on A-finite determinacy] The acknowledged cutoff at dimension 4 for some properties is consistent with possible changes in stratification components or orbit dimensions, but the paper does not exhibit the key lemmas equating geometric conditions to finite determinacy for n > 3, undermining the extension claim.
minor comments (2)
  1. [Abstract] The abstract refers to 'some of those properties' being limited to dimension 4 without specifying which ones, reducing clarity for readers.
  2. [preliminaries] Notation for the contact group action and A-orbit tangent space should be recalled explicitly when reusing the 2019 criterion to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help us improve the clarity and rigor of the generalization from the results in Farnik, Jelonek and Ruas (2019). We address the major comments below.

read point-by-point responses

  1. Referee: [generalization section (statements for n ≥ 2 and n ≤ 4)] The central claim that the geometric criterion for A-finite determinacy from Farnik-Jelonek-Ruas (2019) applies unchanged to homogeneous maps in dimensions n ≥ 4 is load-bearing but unsupported by explicit verification. The manuscript states it follows the same strategy yet provides no check that the dimension of the tangent space to the A-orbit or the codimensions in the jet-space stratification remain unaltered when source and target dimensions increase (see the section on generalization to dimensions ≥ 2 and the statements limited to dimension 4).

    Authors: We agree that the manuscript would be strengthened by an explicit verification that the tangent space dimensions to the A-orbit and the relevant codimensions in the jet-space stratification are unaltered for the dimensions under consideration. Although the geometric criterion itself is formulated in a dimension-independent manner and the strategy follows the 2019 paper directly, we will add a dedicated paragraph in the generalization section that recalls the formulas for these dimensions from the reference and confirms they yield the same conclusions for n up to 4 (with the acknowledged limitations beyond that range arising from changes in stratification components). revision: yes

  2. Referee: [main results on A-finite determinacy] The acknowledged cutoff at dimension 4 for some properties is consistent with possible changes in stratification components or orbit dimensions, but the paper does not exhibit the key lemmas equating geometric conditions to finite determinacy for n > 3, undermining the extension claim.

    Authors: We acknowledge that explicitly displaying the adapted lemmas would make the extension more transparent. The cutoff at dimension 4 is indeed due to the point at which orbit dimensions and stratification components change, as noted in the manuscript. In the revised version we will insert explicit statements of the key lemmas (adapted from the 2019 work) that equate the geometric conditions to A-finite determinacy, indicating the precise range of n for which each holds. revision: yes

Circularity Check

1 steps flagged

Extension of generic A-finite determinacy relies on geometric criterion from 2019 overlapping-author paper without explicit re-verification for n>3

specific steps
  1. self citation load bearing [Abstract]
    "An important starting point is the work by Farnik, Jelonek and Ruas in 2019, where they prove some of those properties in the context of homogeneous polynomial mappings of C^3 to C^3, and conclude the genericity of A-finite determinacy by applying the geometric criterion. Using their strategy, we further extend and generalize some of their key findings to dimensions greater than or equal to 2, though some of those properties can only be extended up to dimension 4."

    The genericity of A-finite determinacy and related generic properties for homogeneous polynomial mappings in dimensions n>=2 are obtained by directly invoking and extending the geometric criterion derived in the 2019 paper by overlapping authors. No independent re-derivation or verification is quoted showing that the contact-group orbit dimensions, jet-space stratification, or tangent-space conditions remain equivalent once source/target dimensions exceed 3; the acknowledged cutoff at dimension 4 is consistent with the criterion potentially failing to transfer unchanged.

full rationale

The paper's central strategy is to reuse the geometric criterion for A-finite determinacy established in Farnik-Jelonek-Ruas 2019 (overlapping author Ruas) and apply the same approach to homogeneous maps in dimensions >=2 (with some results limited to n=4). The abstract explicitly positions the 2019 work as the starting point and states that the new results follow from using their strategy. This constitutes self-citation load-bearing on the key sufficiency/necessity link between geometric conditions and finite determinacy, but the manuscript still performs concrete extensions and acknowledges dimension cutoffs, so the central claim retains some independent content rather than reducing purely by definition or fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work assumes standard results from algebraic geometry and singularity theory; no new free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption The geometric criterion for A-finite determinacy holds in the generalized setting.
    Invoked to conclude genericity in higher dimensions based on 2019 work.

pith-pipeline@v0.9.0 · 5640 in / 1139 out tokens · 27574 ms · 2026-05-19T23:02:21.061281+00:00 · methodology

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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