A computational approach to higher Jacobian ideals

Ramirez, Tabitha; Shen, Hao; Ye, Fei

doi:10.2140/involve.2025.18.873

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A computational approach to higher Jacobian ideals

Tabitha Ramirez, Hao Shen and Fei Ye

Vol. 18 (2025), No. 5, 873–883

DOI: 10.2140/involve.2025.18.873

Abstract

Employing a technique involving Kronecker tensor and matrix vectorization, we compute the second order Jacobian ideal of a hypersurface $V (F) \subset ℂ^{3}$ . Our calculations reveal that the Jacobian ideal, when considered modulo the ideal $(F)$ , undergoes a decomposition where a power of the Jacobian ideal becomes a factor. As an application, we confirm a conjecture of Hussain, Ma, Yau and Zuo (J. Algebra 618 (2023), 165–194, Conjecture 1.5) for a hypersurface with an isolated singularity in $ℂ^{3}$ .

Keywords

Nash blow-up, higher Jacobian ideal, hypersurface singularity

Mathematical Subject Classification

Primary: 14B05, 32S05

Milestones

Received: 12 January 2024

Revised: 13 May 2024

Accepted: 15 May 2024

Published: 13 November 2025

Communicated by Ravi Vakil

Authors

Tabitha Ramirez
The City College of New York New York, NY United States

Hao Shen
Hamilton College Clinton, NY United States

Fei Ye
Department of Mathematics and Computer Science Queensborough Community College of CUNY New York, NY United States

Vol. 18, No. 5, 2025