Comprehensive Systems for Primary Decompositions of Parametric Ideals
<|reference_start|>Comprehensive Systems for Primary Decompositions of Parametric Ideals: We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner 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 $\varphi_\alpha(\mathcal{Q})$ is a primary decomposition for any $\alpha\in C\cap \mathcal{O}$ via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.<|reference_end|>
@article{ishihara2024comprehensive,
    title={Comprehensive Systems for Primary Decompositions of Parametric Ideals},
    author={Yuki Ishihara, Kazuhiro Yokoyama},
    journal={arXiv preprint arXiv:2408.15917},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.15917},
    primaryClass={cs.SC math.AC}
}