2024/08/28 by Yuki Ishihara, Ishihara, Yuki, Kazuhiro Yokoyama +1
Computer Science · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.2408.15917
openalex publication_date 2024/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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, Q), where for each parameter value in C, the ideal has the stable decomposition Q. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set O such that φα(Q) is a primary decomposition for any α∈ C∩ O via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.