By double ideal quotient, we mean $(I:(I:J))$ where ideals $I$ and $J$. In our previous work [11], double ideal quotient and its variants are shown to be very useful for checking prime divisor and generating primary component. Combining those properties, we can compute "direct localization" effectively, comparing with full primary decomposition. In this paper, we apply modular techniques effectively to computation of such double ideal quotient and its variants, where first we compute them modulo several prime numbers and then lift them up over rational numbers by Chinese Remainder Theorem and rational reconstruction. As a new modular technique for double ideal quotient and its variants, we devise criteria for output from modular computations. Also, we apply modular techniques to intermediate primary decomposition. We examine the effectiveness of our modular techniques for several examples by preliminary computational experiences on Singular.
翻译:以双重理想商数计算, 我们指的是 $( I: (I: J: ) $( 美元) 和$( J) 美元 。 在我们先前的工作 [11] 中, 双理想商数及其变异数被证明非常有助于检查原始差数和生成原始元件。 结合这些属性, 我们可以有效地计算“ 直接本地化 ”, 与全部初级分解相比较。 在本文中, 我们有效地应用模块化技术来计算这种双重理想商数及其变异数, 首先我们用中文保存理论和合理重建来计算数的模数, 然后把它们提升到比合理数字高。 作为双重理想商数及其变异的新的模块化技术, 我们设计了模块化计算结果的标准。 此外, 我们运用模块化技术来进行中间初级分解。 我们通过在 Singula 上的初步计算经验来检查模块化技术的效能。