参数理想的 Comprehensive Systems 与准素分解
Comprehensive Systems for Primary Decompositions of Parametric Ideals
AI总结：
针对有理系数多项式环中参数理想准素分解的计算难题，提出基于 comprehensive Gröbner systems 的可行性概念与算法，通过计算 comprehensive system 实现稳定的参数准素分解，并经实例验证了其有效性。
摘要：
我们提出了一种基于 comprehensive Gröbner systems 计算参数准素分解的有效方法。通常，在有理系数多项式环 $\mathbb{Q}[A,X]$ 中计算给定理想的参数准素分解非常困难，其中 $A$ 为参数集，$X$ 为普通变量集。困难原因之一与专门化多项式的不可约性有关。为此，我们引入了关于理想结构在其准素分解意义上稳定性的“可行性”新概念，并给出了一种新算法来计算由对 $(C, \mathcal{Q})$ 组成的所谓 comprehensive system，其中对于 $C$ 中的每个参数值，理想具有稳定的分解 $\mathcal{Q}$。我们可以将该 comprehensive system 称为理想的参数准素分解。此外，还能计算出一个稠密集 $\mathcal{O}$，使得对于任意 $\alpha\in C\cap \mathcal{O}$，$\phi_\alpha(\mathcal{Q})$ 均为通过不可约多项式得到的准素分解。另外，我们给出了几个计算实例来验证新分解方法的有效性。
英文摘要：
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 $φ_α(\mathcal{Q})$ 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.
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