Core Concepts

An effective method for primary decomposition of symmetric ideals by utilizing the symmetric structure of the ideals.

Abstract

The paper proposes an effective algorithm for primary decomposition of symmetric ideals. The key insights are:
Primary decomposition of a symmetric ideal has a symmetric structure - if {Q1, ..., Qk} is a minimal primary decomposition of a symmetric ideal I, then {σ(Q1), ..., σ(Qk)} is also a minimal primary decomposition of I for any permutation σ in the symmetric group Sn.
To handle the non-uniqueness of primary decompositions, the authors introduce the concept of "the quotient set of primary components" of I, which is uniquely determined from I. They show that the symmetric group Sn acts on this quotient set.
The authors devise an effective algorithm for computing the primary decomposition of a symmetric ideal by leveraging the symmetric structure. The algorithm only needs to compute a partial set of primary components and can derive the full primary decomposition from them using the group action.
The authors also provide practical improvements to the algorithm by specializing the Shimoyama-Yokoyama algorithm for symmetric ideals. This includes computing a symmetric system of separators and a symmetric saturated separating ideal.
Experiments show that the specialized symmetric algorithm outperforms the general primary decomposition algorithm, especially for ideals with many primary components and high symmetry.

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Key Insights Distilled From

by Yuki Ishihar... at **arxiv.org** 04-17-2024

Deeper Inquiries

The proposed algorithm can be extended to handle more general group actions beyond the symmetric group, such as the general linear group GL(n, K), by modifying the group action in the algorithm. In the case of GL(n, K), where A(f(x1, ..., xn)) = f(A(x1, ..., xn)) for a matrix A ∈ GL(n, K) and a polynomial f ∈ K[X], the algorithm can be adapted to incorporate this group action. By considering the action of GL(n, K) on the polynomial ring K[X], the algorithm can be adjusted to compute primary decompositions efficiently for ideals that exhibit this more general group symmetry.

The ideas presented in this paper can indeed be applied to improve the primary decomposition of other algebraic structures with symmetry, such as modules or tensor ideals. By leveraging the concept of symmetric ideals and utilizing specialized algorithms for primary decomposition tailored to symmetric structures, similar improvements can be made in the primary decomposition of these algebraic structures. The key lies in identifying the symmetry present in the structure and adapting the algorithm to exploit this symmetry for more efficient and effective decomposition.

The efficient primary decomposition of symmetric ideals has various potential applications in different areas of mathematics and computer science. In invariant theory, where symmetry plays a crucial role, the algorithm can be used to analyze and decompose symmetric ideals efficiently, leading to advancements in the study of invariant objects and properties. In Galois theory, the algorithm can aid in understanding the structure of ideals generated by symmetric polynomials, contributing to the exploration of Galois groups and their properties. Furthermore, in computer science, the algorithm can be utilized in computational algebra systems to handle symmetric structures efficiently, enabling faster computations and analysis of symmetric ideals in various applications. Overall, the algorithm's applications extend to diverse fields where symmetry and algebraic structures are prevalent, offering a valuable tool for researchers and practitioners.

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