Computing the decomposition group of a zero-dimensional ideal by elimination method

TL;DR

This paper introduces a new elimination-based method to compute the decomposition group of zero-dimensional radical ideals using Gröbner bases, enabling better understanding and decomposition of polynomial ideals in computer algebra.

Contribution

The paper presents a novel approach to represent the decomposition group as a direct sum of symmetric groups, enhancing theoretical understanding without complexity analysis.

Findings

Decomposition group expressed as a direct sum of symmetric groups.

Method facilitates computation of triangular sets in polynomial decomposition.

Provides a new algebraic tool for polynomial ideal analysis.

Abstract

In this note, we show that the decomposition group $Dec(I)$ of a zero-dimensional radical ideal $I$ in ${\bf K}[x_1,\ldots,x_n]$ can be represented as the direct sum of several symmetric groups of polynomials based upon using Gr\"{o}bner bases. The new method makes a theoretical contribution to discuss the decomposition group of $I$ by using Computer Algebra without considering the complexity. As one application, we also present an approach to yield new triangular sets in computing triangular decomposition of polynomial sets ${\mathbb P}$ if $Dec(<{\mathbb P}>)$ is known.