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A Syzygial Method for Equidimensional Decomposition

Rafael Mohr

TL;DR

This work addresses the problem of equidimensional decomposition of algebraic sets by introducing a non-incremental, syzygy-based algorithm grounded in Vasconcelos' local regular sequence theorem. It uses an affine-cell data structure and GetSyz-driven Gröbner-basis computations to avoid elimination and full free resolutions, producing an irredundant Kalkbrener partition of $\mathbf{V}(F)$. The main contributions are the KalkPart algorithm with Hull/Remove that terminates and is correct, and a practical implementation showing competitive performance across benchmarks. The approach offers scalable, Syzygy-driven decomposition that enhances applicability to real algebraic geometry tasks and related computational problems.

Abstract

Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gröbner bases. This algorithm avoids the use of elimination, homological algebra and processing the input equations one-by-one present in previous algorithms. We experimentally demonstrate the practical interest of our algorithm compared to the state of the art.

A Syzygial Method for Equidimensional Decomposition

TL;DR

This work addresses the problem of equidimensional decomposition of algebraic sets by introducing a non-incremental, syzygy-based algorithm grounded in Vasconcelos' local regular sequence theorem. It uses an affine-cell data structure and GetSyz-driven Gröbner-basis computations to avoid elimination and full free resolutions, producing an irredundant Kalkbrener partition of . The main contributions are the KalkPart algorithm with Hull/Remove that terminates and is correct, and a practical implementation showing competitive performance across benchmarks. The approach offers scalable, Syzygy-driven decomposition that enhances applicability to real algebraic geometry tasks and related computational problems.

Abstract

Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gröbner bases. This algorithm avoids the use of elimination, homological algebra and processing the input equations one-by-one present in previous algorithms. We experimentally demonstrate the practical interest of our algorithm compared to the state of the art.
Paper Structure (5 sections, 7 theorems, 24 equations, 1 table, 3 algorithms)

This paper contains 5 sections, 7 theorems, 24 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Algebraic Description & Preliminaries
  3. The Data Structure
  4. The Algorithms
  5. Benchmarks

Key Result

Proposition 2.3

For two ideals $I,J\subset R$ we have In particular, the irreducible components of $\mathbf{V}(\left(I:J^{\infty}\right))$ are those irreducible components of $\mathbf{V}(I)$ not contained in $\mathbf{V}(J)$.

Theorems & Definitions (23)

  • Definition 2.1: Locally Closed Set
  • Definition 2.2: Saturation
  • Proposition 2.3: see e.g. Chapter 4 in cox2015a
  • Definition 2.4: Equidimensionality
  • Definition 2.5: Local Regular Sequence
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • Corollary 2.8
  • proof
  • ...and 13 more