The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Gröbner basis of the ideal. A pair $(\mathcal{G},\mathcal{C})$ of polynomial sets is a strong regular characteristic pair if $\mathcal{G}$ is a reduced lexicographical Gröbner basis, $\mathcal{C}$ is the W-characteristic set of the ideal $〈\mathcal{G}〉$, the saturated ideal $\mathrm{sat}(\mathcal{C})$ of $\mathcal{C}$ is equal to $〈\mathcal{G}〉$, and $\mathcal{C}$ is regular. In this paper, we show that for any polynomial ideal $\mathfrak{I}$ with given generators one can either detect that $\mathfrak{I}$ is unit, or construct a strong regular characteristic pair $(\mathcal{G},\mathcal{C})$ by computing Gröbner bases such that $\mathfrak{I}\subseteq \mathrm{sat}(\mathcal{C})=〈\mathcal{G}〉$ and $\mathrm{sat}(\mathcal{C})$ divides $\mathfrak{I}$, so the ideal $\mathfrak{I}$ can be split into the saturated ideal $\mathrm{sat}(\mathcal{C})$ and the quotient ideal $\mathfrak{I}:\mathrm{sat}(\mathcal{C})$. Based on this strategy of splitting by means of quotient and with Gröbner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set $\mathcal{F}$ into finitely many strong regular characteristic pairs, from which two representations for the zeros of $\mathcal{F}$ are obtained: one in terms of strong regular Gröbner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.

多项式理想的W特征集是理想的简化词典法Gröbner基础中包含的最小三角形集。一双$（\mathcal{G}，\mathcal{C}）$ 多项式集是一个强大的正则特征对，如果 $\mathcal{G}$ 是简化的字典式格罗布纳基础， $\mathcal{C}$ 是理想的W特征集 $〈\mathcal{G}〉$，饱和的理想 $\mathrm{坐着}（\mathcal{C}）$ 的 $\mathcal{C}$ 等于 $〈\mathcal{G}〉$和 $\mathcal{C}$是正常的。在本文中，我们表明对于任何多项式理想$\mathfrak{一世}$ 使用给定的发电机，可以检测到 $\mathfrak{一世}$ 是单元，或构造一个强大的规则特征对 $（\mathcal{G}，\mathcal{C}）$ 通过计算Gröbner基 $\mathfrak{一世}\subseteq \mathrm{坐着}（\mathcal{C}）=〈\mathcal{G}〉$ 和 $\mathrm{坐着}（\mathcal{C}）$ 分界 $\mathfrak{一世}$，所以理想 $\mathfrak{一世}$ 可以分为饱和理想 $\mathrm{坐着}（\mathcal{C}）$ 和商理想 $\mathfrak{一世}：\mathrm{坐着}（\mathcal{C}）$。基于这种商数分解策略，并利用Gröbner基础和理想计算，我们设计了一种简单的算法来分解任意多项式集$\mathcal{F}$ 分成有限的许多强正则特征对，从中有两个表示形式的零 $\mathcal{F}$可以得到：一个是根据强规则的Gröbner基，另一个是根据规则的三角形集。我们给出了关于强规则特征对和特征分解的一些性质，并通过实例和实验结果说明了该算法及其性能。