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Comprehensive Systems for Primary Decompositions of Parametric Ideals

Yuki Ishihara, Kazuhiro Yokoyama

Abstract

We present an effective method for computing parametric primary decomposition via comprehensive Gröbner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients $\mathbb{Q}[A,X]$ where $A$ is the set of parameters and $X$ is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs $(C, \mathcal{Q})$, where for each parameter value in $C$, the ideal has the stable decomposition $\mathcal{Q}$. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set $\mathcal{O}$ such that $\varphi_α(\mathcal{Q})$ is a primary decomposition for any $α\in C\cap \mathcal{O}$ via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.

Comprehensive Systems for Primary Decompositions of Parametric Ideals

Abstract

We present an effective method for computing parametric primary decomposition via comprehensive Gröbner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients where is the set of parameters and is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs , where for each parameter value in , the ideal has the stable decomposition . We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set such that is a primary decomposition for any via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.
Paper Structure (16 sections, 26 theorems, 38 equations, 1 table, 3 algorithms)

This paper contains 16 sections, 26 theorems, 38 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Basis
  3. Primary Decomposition
  4. Comprehensive Gröbner System
  5. Stability of Ideal Operations
  6. Comprehensive Gröbner system for Ideal Operations
  7. Main Algorithms
  8. Algorithms for Feasible CPDS
  9. Algorithm for a minimal feasible CPDS
  10. Primarity of Parametric Primary Ideal
  11. Primarity of Parametric Ideal
  12. Hilbert's Irreducibility Theorem
  13. Computer Experiments
  14. Timings of Several Examples
  15. Simple Example for Hibert Subset of Prime Parametric Ideal
  16. ...and 1 more sections

Key Result

Proposition 2.13

Let $G$ be a Gröbner basis of an ideal $\langle F\rangle$ in $K[A,X]$ with respect to a block ordering $X\succ\succ A$. If $\varphi_\alpha({\mathop{\mathrm{lc}}} _\succ (g)) \neq 0$ for each $g \in G\setminus K[A]$, then $\varphi_\alpha (G)$ is a Gröbner basis of $\langle \varphi_\alpha(F)\rangle$ i

Theorems & Definitions (87)

  • Definition 2.1: Primary Decomposition Definition 4.1.1, greuel2002singular
  • Example 2.2
  • Remark 2.3
  • Definition 2.4: Maximal Independent Set greuel2002singular, Definition 3.5.3
  • Example 2.5
  • Definition 2.6: Localy Closed Set Brunat, Section 2
  • Remark 2.7
  • Remark 2.8
  • Example 2.9
  • Remark 2.10
  • ...and 77 more