Computing strong regular characteristic pairs with Groebner bases

TL;DR

This paper introduces a method to decompose polynomial ideals into strong regular characteristic pairs using Groebner bases, enabling effective zero set representations and ideal analysis.

Contribution

It presents a novel algorithm for decomposing polynomial sets into strong regular characteristic pairs via Groebner basis computations, facilitating ideal analysis and solution representations.

Findings

Algorithm successfully decomposes polynomial sets into strong regular characteristic pairs.

Provides two types of zero set representations: via Groebner bases and regular triangular sets.

Experimental results demonstrate the efficiency and effectiveness of the proposed method.

Abstract

The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Groebner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Groebner basis, C is the W-characteristic set of the ideal <G>, the saturated ideal sat(C) of C is equal to <G>, and C is regular. In this paper, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G,C) by computing Groebner bases such that I$\subseteq$sat(C)=<G> and sat(C) divides I, so the ideal I can be split into the saturated ideal sat(C) and the quotient ideal I:sat(C). Based on this strategy of splitting by means of quotient and with Groebner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Groebner 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.

Tables

Table 1. Table 1: Timings for src decomposition (in second)

					Algorithm 3				uvd	Triangularize
Ex	Label	Var	Pol	Dim	Total	GB	SAT	QUO	Total	Total	Regular
1	E1	10	10	1	1.450(2)	0.564	0.221	0.632	0.867(3)	2.041(13)	1.709
2	E5	15	17	4	3.568(1)	1.793	0.461	1.255	3.374(1)	7.684(7)	5.225
3	E11	4	3	1	0.037(1)	0.008	0.008	0.015	0.122(1)	0.079(3)	0.035
4	E34	14	16	0	0.574(0)	0.556	0.	0.	$>1000$	50.650(0)	50.649
5	S1	4	3	2	0.055(2)	0.017	0.007	0.022	0.011(3)	0.054(3)	0.034
6	S6	4	3	2	0.175(3)	0.035	0.018	0.111	0.032(3)	0.080(3)	0.052
7	S7	4	3	1	0.098(1)	0.008	0.052	0.031	0.652(1)	0.127(1)	0.084
8	S8	4	3	2	0.058(2)	0.018	0.004	0.030	0.112(2)	0.099(2)	0.081
9	S14	5	3	2	1.024(6)	0.372	0.299	0.235	3.767(6)	0.184(8)	0.109
10	nueral	4	3	1	1.225(2)	0.445	0.187	0.574	$>1000$	0.222(5)	0.115
11	F663	10	9	2	8.982(1)	1.264	1.684	5.983	$>1000$	4.312(4)	0.935
12	Katsura-4	5	5	0	0.353(1)	0.254	0.	0.086	209.831(1)	6.583(4)	5.343
13	nld-3-4	4	4	0	2.037(8)	0.175	0.624	1.121	1.066(8)	0.754(29)	0.520
14	nld-8-3	3	3	0	2.464(2)	2.254	0.207	0.	3.334(2)	17.360(8)	0.268
15	nql-5-4	5	5	0	2.908(1)	2.088	0.	0.806	25.422(1)	35.508(1)	30.335
16	caprasse-li	4	4	0	104.055(2)	0.066	101.836	2.145	2.883(3)	0.967(7)	0.634
17	Cyclic-5	5	5	0	0.497(4)	0.101	0.058	0.285	$>1000$	1.079(15)	0.798
18	geneig	6	6	0	0.566(1)	0.135	0.003	0.417	$>1000$	–	$>1000$
19	reimer5	5	5	0	32.850(1)	30.983	0.005	1.820	$>1000$	–	$>1000$
20	redcyc6	6	6	0	36.920(6)	0.203	0.556	35.079	$>1000$	–	$>4000$
21	Cyclic-6	6	6	0	368.988(6)	2.455	332.158	33.780	$>1000$	–	$>1000$
22	Bronstein2	4	3	2	0.377(5)	0.188	0.087	0.	150.684(4)	0.430(5)	0.387
23	Morgenstern	9	5	4	0.100(1)	0.028	0.014	0.042	2.812(1)	0.223(1)	0.191
24	Raksanyi	8	4	4	0.057(1)	0.012	0.001	0.028	0.588(1)	0.064(2)	0.047
25	Gerdt 5	13	15	2	3.150(4)	1.163	0.330	1.343	2.830(1)	2.009(3)	1.849
26	Czapor Geddes 3	11	2	9	1098.118(1)	0.597	1097.444	0.095	$>4000$	1.261(1)	0.076

Equations

$$\sqrt{⟨\mathcal{F}⟩}=\sqrt{\mathrm{sat}({\mathcal{T}}_1)}\cap \cdots \cap \sqrt{\mathrm{sat}({\mathcal{T}}_e)}.$$

$$Z(\mathcal{F})=Z({\mathcal{T}}_1/J_1)\cup \cdots \cup Z({\mathcal{T}}_e/J_e),$$

$$\sqrt{⟨\mathcal{F}⟩}=\sqrt{\mathrm{sat}({\mathcal{C}}_1)}\cap \cdots \cap \sqrt{\mathrm{sat}({\mathcal{C}}_e)}=\sqrt{⟨{\mathcal{G}}_1⟩}\cap \cdots \cap \sqrt{⟨{\mathcal{G}}_e⟩}.$$

$$\mathrm{prem}(P,\mathcal{T}):=\mathrm{prem}(\cdots \mathrm{prem}(\mathrm{prem}(P,T_r),T_{r-1}),\dots ,T_1).$$

$$Z(\mathcal{F}/\mathcal{G}):=\{\bar{\mathbf{x}}\in {\bar{\mathbb{K}}}^n:F(\bar{\mathbf{x}})=0,G(\bar{\mathbf{x}})\ne 0,\text{ for all }F\in \mathcal{F},G\in \mathcal{G}\},$$

$$Z(\mathcal{F})=\bigcup _{i=1}^{e}Z({\mathcal{G}}_i)=\bigcup _{i=1}^{e}Z(\mathrm{sat}({\mathcal{C}}_i)).$$

$$Z(\mathcal{F})=\bigcup _{i=1}^{e}Z({\mathcal{C}}_i/\mathrm{ini}({\mathcal{C}}_i))$$

$$\begin{array}[]{cccccccccl}\mathfrak{I}&=&\mathfrak{I}_{1}&=&\langle\mathcal{G}_{1}\rangle&\longrightarrow&\mathcal{C}_{1}&\longrightarrow&\operatorname{sat}(\mathcal{C}_{1})&(\neq\langle 1\rangle)\\ &&\cap\rule{0.5pt}{5.0pt}&&&&&&\cap\rule{0.5pt}{5.0pt}&\\ &=&\mathfrak{I}_{2}&=&\langle\mathcal{G}_{2}\rangle&\longrightarrow&\mathcal{C}_{2}&\longrightarrow&\operatorname{sat}(\mathcal{C}_{2})&(\neq\langle 1\rangle)\\ &&\cap\rule{0.5pt}{5.0pt}&&&&&&\cap\rule{0.5pt}{5.0pt}&\\[-4.0pt] &&\vdots&&&&&&\vdots&\\ &&\cap\rule{0.5pt}{5.0pt}&&&&&&\cap\rule{0.5pt}{5.0pt}&\\ &=&\mathfrak{I}_{m-1}&=&\langle\mathcal{G}_{m-1}\rangle&\longrightarrow&\mathcal{C}_{m-1}&\longrightarrow&\operatorname{sat}(\mathcal{C}_{m-1})\!\!\!&(\neq\langle 1\rangle)\\ &&\cap\rule{0.5pt}{5.0pt}&&&&&&\shortparallel&\\[-11.0pt] &&&&&&&&\shortparallel&\\ &=&\mathfrak{I}_{m}&=&\langle\mathcal{G}_{m}\rangle&\longrightarrow&\mathcal{C}_{m}&\longrightarrow&\operatorname{sat}(\mathcal{C}_{m})&=\,\operatorname{sat}(\mathcal{C}_{m+1})\,=\,\cdots\par\end{array}$$

$$\sqrt{\mathfrak{I}}=\bigcap _{\delta \in \Delta }\sqrt{{\mathfrak{I}}_{\delta }}$$

$$\sqrt{\mathfrak{I}}=\sqrt{\mathrm{sat}(\mathcal{C})}\bigcap \sqrt{\mathfrak{I}:\mathrm{sat}(\mathcal{C})},$$

$${\mathcal{G}}_1=$$

$${\mathfrak{I}}_2={\mathfrak{I}}_1:\mathrm{sat}({\bar{\mathcal{C}}}_1)=⟨{\mathcal{G}}_2⟩$$

$${\mathcal{C}}_2=[C_1,C_2,C_3]=[uv,vP,Q].$$

$$\mathrm{ini}(C_2)\cdot \mathrm{ini}(C_1)=v^{4}u=u{C}_2-v^{3}x{C}_1\in ⟨{\mathcal{C}}_2⟩,$$

$${\mathcal{G}}_4=\{u,P,Q\},{\mathcal{C}}_4=[u,P,Q].$$

$$\Psi =\{({\bar{\mathcal{G}}}_1,{\bar{\mathcal{C}}}_1),({\mathcal{G}}_3,{\mathcal{C}}_3),({\mathcal{G}}_4,{\mathcal{C}}_4)\}$$

$$\mathcal{F}=\{-c{t}^{2}u+t^3-u{v}^2-u{w}^2,-c{t}^{2}v+t^3-u^{2}v-v{w}^2,-c{t}^{2}w+t^3-u^{2}w-v^{2}w\}$$

$$\begin{array}[]{l}\vskip 3.0pt plus 1.0pt minus 1.0pt(\mathcal{G}_{1},\mathcal{C}_{1})=(\{v,u,t\},[v,u,t]),~{}(\mathcal{G}_{2},\mathcal{C}_{2})=(\{w,u,t^{2}\},[w,u,t^{2}]),~{}(\mathcal{G}_{3},\mathcal{C}_{3})=(\{w,v,t^{2}\},[w,v,t^{2}]),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt(\mathcal{G}_{4},\mathcal{C}_{4})=(\{v-w,u-w,wt^{2}c-t^{3}+2w^{3}\},[v-w,u-w,wt^{2}c-t^{3}+2w^{3}]),\\ \vskip 3.0pt plus 1.0pt minus 1.0pt(\mathcal{G}_{5},\mathcal{C}_{5})=(\{v-w,P,Q,t^{2}c-wu+w^{2}\},[v-w,P,Q]),\\ (\mathcal{G}_{6},\mathcal{C}_{6})=(\{G_{1},\ldots,G_{5}\},[G_{1},G_{2},G_{3}]),\end{array}$$

$$P=t^3-w{u}^2-w^{2}u,Q=u^{2}c+wuc-ut-wt.$$

$$Z(\mathcal{F})=\bigcup _{i=1}^{e}Z({\mathcal{G}}_i)=\bigcup _{i=1}^{e}Z(\mathrm{sat}({\mathcal{C}}_i))$$

$$V^{*}=\bigcup_{\scriptsize\begin{array}[]{c}1\leq i\leq s\\ i\neq m\end{array}}V_{i}.$$

$$\sqrt{⟨\mathcal{F}⟩}=\bigcap _{i=1}^e\sqrt{\mathrm{sat}({\mathcal{T}}_i)}=\bigcap _{i=1}^e\sqrt{\mathrm{sat}({\mathcal{C}}_i)}=\bigcap _{i=1}^e\sqrt{⟨{\mathcal{G}}_i⟩}.$$

Full text

1 Introduction

Triangular sets [28, 34, 17, 31] and Gröbner bases [6, 7, 5, 10] are special kinds of well-structured sets of multivariate polynomials that can be used to represent and to study zeros of arbitrary polynomial sets and ideals. A large variety of problems in commutative algebra and algebraic geometry [10, 11] may readily be solved by transforming the involved sets of polynomials into triangular sets or Gröbner bases. It is widely known that the theories and methods of triangular sets are different from those of Gröbner bases conceptually and operationally. The questions that have motivated our work here and in [32, 33] are what inherent relationship there may exist between triangular sets and Gröbner bases and how to connect or combine the two algorithmic approaches to amplify their applicability and power. These questions have been touched in [18, 23, 12, 20] for bivariate, zero-dimensional, and other kinds of special polynomial ideals. For general polynomial ideals of arbitrary dimension, it is shown in [2, 32] that intrinsical connections between Ritt characteristic sets and lexicographical Gröbner bases exist and the concept of W-characteristic sets plays an essential role in exploiting such connections. Apart from the lack of investigation on the connection aspect, the literature on triangular sets and Gröbner bases is extremely rich (see [2, 5, 7, 4, 8, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33] and references therein).

Multivariate polynomials in a triangular set may be ordered strictly according to their leading variables, with respect to a fixed variable ordering, so the number of the polynomials cannot be bigger than that of the variables in any triangular set. On the other hand, Gröbner bases are defined with respect to a fixed term order determined by the variable ordering and the number of elements in a Gröbner basis can be arbitrarily large. For any polynomial ideal with given set $\mathcal{F}$ of generators, one can compute, by using any of the available algorithms, a Gröbner basis that generates the same ideal as $\mathcal{F}$. To represent the zeros of $\mathcal{F}$ using triangular sets, in general one needs more than one triangular set, so decomposition takes place. When a triangular set $\mathcal{T}$ is of concern, the leading coefficients of the polynomials in $\mathcal{T}$ with respect to their leading variables, called the initials of the polynomials in $\mathcal{T}$, play a fundamental role. The saturated ideal of $\mathcal{T}$ by the product of the initials of the polynomials in $\mathcal{T}$ is simply called the saturated ideal of $\mathcal{T}$ and denoted as $\operatorname{sat}(\mathcal{T})$. Its radical is the largest ideal whose zero set contains the set of those zeros of $\mathcal{T}$ which are not zeros of any of the initials.

Triangular sets may be ordered according to the ranks (leading variables and degrees) and then the leading terms of their polynomials. Let the minimal triangular set contained in the reduced lexicographical (lex) Gröbner basis of a polynomial ideal (or trivially $[1]$ if the ideal is unit) be called the W-characteristic set of the ideal. By strong regular Gröbner basis, we mean a reduced lex Gröbner basis $\mathcal{G}$ such that the W-characteristic set $\mathcal{C}$ of the ideal $\langle\mathcal{G}\rangle$ is regular and $\operatorname{sat}(\mathcal{C})=\langle\mathcal{G}\rangle$; we call $(\mathcal{G},\mathcal{C})$ a strong regular characteristic pair, or an src pair for short. The pair $(\mathcal{G},\mathcal{C})$ is an interesting object of study because the strong regular Gröbner basis $\mathcal{G}$ and the regular triangular set $\mathcal{C}$ therein not only have remarkable properties but also provide two different yet correlated representations for the zeros of the ideal $\langle\mathcal{G}\rangle$.

What interests us most is algorithmic decomposition of arbitrary polynomial sets into strong regular Gröbner bases, or equivalently into src pairs, for which we have the following general approach. From any polynomial set $\mathcal{F}$, one can compute finitely many regular sets $\mathcal{T}_{1},\ldots,\mathcal{T}_{e}$ (also called regular chains; [16]) such that

[TABLE]

There are two families of algorithms for such regular triangular decomposition. One family of algorithms was proposed initially by Kalkbrener [17], and developed further by Moreno Maza and coauthors [25, 1, 8, 9] with algorithmic techniques from the method of Lazard [19]. These algorithms were designed mainly for computing regular triangular representations of the form (1). The other family of algorithms was proposed by the second author [31, 30] to compute regular zero decompositions of the form

[TABLE]

where each $J_{i}$ is the product of the initials of the polynomials in the regular set $\mathcal{T}_{i}$ and $\operatorname{\mathsf{Z}}(\mathcal{T}_{i}/J_{i})$ denotes the set of all common zeros of the polynomials in $\mathcal{T}_{i}$ which do not make $J_{i}$ vanish. It is easy to see that decomposition (2) implies representation (1). However, representation (1) does not necessarily lead to decomposition (2) and the generators of the saturated ideals $\operatorname{sat}(\mathcal{T}_{i})$ in (1) are not explicitly provided. Nevertheless, for each $\operatorname{sat}(\mathcal{T}_{i})$ a Gröbner basis can be computed straightforwardly from $\mathcal{T}_{i}$.

Another alternative approach proposed recently by Mou and the authors [33] permits one to decompose an arbitrary polynomial set into normal or regular characteristic pairs directly. This approach is based on a structure theorem about irregular W-characteristic sets [32] for the splitting of ideals and relies strongly upon Gröbner basis computation; it is independent of pseudo-division-based triangular decomposition.

In this paper, we show that from any polynomial ideal $\mathfrak{I}$ with given generators one can either detect that $\mathfrak{I}$ is unit, or construct an src pair $(\mathcal{G},\mathcal{C})$ such that $\mathfrak{I}\subseteq\operatorname{sat}(\mathcal{C})$ and $\operatorname{sat}(\mathcal{C})$ divides $\mathfrak{I}$ by computing Gröbner bases. After $(\mathcal{G},\mathcal{C})$ (called an src divisor of $\mathfrak{I}$) is constructed, one can divide the saturated ideal $\operatorname{sat}(\mathcal{C})$ out of $\mathfrak{I}$ to obtain an ideal $\mathfrak{J}$ by taking ideal quotient: $\mathfrak{J}=\mathfrak{I}:\operatorname{sat}(\mathcal{C})$. Then the radical ideal of $\mathfrak{I}$ is decomposed as the intersection of the radical ideals of $\operatorname{sat}(\mathcal{C})$ and $\mathfrak{J}$. Two key techniques are used in search for an src divisor of $\mathfrak{I}$: one is to produce an src pair by means of computing $\operatorname{sat}(\operatorname{wcs}(\cdots\operatorname{sat}(\operatorname{wcs}(\mathfrak{I}))))$, where $\operatorname{wcs}(\mathfrak{I})$ denotes the W-characteristic set of $\mathfrak{I}$. The other is to use the relation $\operatorname{sat}(\mathcal{C}^{*})\neq\mathfrak{I}$, when the produced src pair is not an src divisor of $\mathfrak{I}$, to determine a polynomial $F\in\operatorname{sat}(\mathcal{C}^{*})\setminus\mathfrak{I}$ with $\mathcal{C}^{*}=\operatorname{wcs}(\mathfrak{I})$. In this case, there exists a positive integer $q$ such that $H=F[\prod_{C\in\mathcal{C}^{*}}\operatorname{ini}(C)]^{q}\in\langle\mathcal{C}^{*}\rangle\subseteq\mathfrak{I}$ and an src divisor of $\mathfrak{I}$ may be found among the src pairs computed from $\langle\mathfrak{I}\cup\{H_{i}\}\rangle$ for $H_{i}\not\in\mathfrak{I}$ and $H_{i}\mid H$. Once $\mathfrak{J}$ is obtained, one can iterate the find-and-divide process with $\mathfrak{J}$ instead of $\mathfrak{I}$. 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 src pairs $(\mathcal{G}_{1},\mathcal{C}_{1}),\ldots,(\mathcal{G}_{e},\mathcal{C}_{e})$ such that

[TABLE]

In the above decomposition computed by the ideal-division-based algorithm, each $\mathcal{G}_{i}$ is actually the reduced lex Gröbner basis of a saturated ideal $\operatorname{sat}(\mathcal{T}_{i})$, where $\mathcal{T}_{i}$ is the W-characteristic set of a certain ideal that contains $\langle\mathcal{F}\rangle$. The regular set $\mathcal{C}_{i}$ is the W-characteristic set of $\operatorname{sat}(\mathcal{T}_{i})$, obtained from $\mathcal{G}_{i}$ as a by-product for free. Moreover, regular sets computed by our algorithm are normal in most cases and they are often simpler than the corresponding regular sets computed by pseudo-division222Most of the available methods for triangular decomposition use pseudo-division to eliminate variables. In doing pseudo-division, the dividend has to be multiplied by some power of the initial of the dividing polynomial. Repeated multiplication of the powers of initials necessarily creates extraneous factors, making the sizes of intermediate polynomials increase rapidly. It happens often that decomposition cannot continue after a few successive pseudo-divisions because polynomials in the pseudo-reminder sequence become larger and larger. or subresultant-based algorithms, because the former are minimal triangular sets taken from lex Gröbner bases. In general, the generating sets of the saturated ideals of normal triangular sets are much easier to compute than those of abnormal ones. More importantly, our algorithm generates few redundant components, so the decomposition process usually terminates in a few iterations. The effectiveness of the decomposition algorithm has been demonstrated by our experimental results.

The rest of the paper is organized as follows. After a brief introduction to Gröbner bases, triangular decomposition, and characteristic decomposition in Section 2, we show in Section 3 how to construct from an arbitrary ideal $\mathfrak{I}$ an src pair $(\mathcal{G},\mathcal{C})$ such that $\mathfrak{I}\subseteq\operatorname{sat}(\mathcal{C})=\langle\mathcal{G}\rangle$. In Section 4, it is explained how an src divisor of $\mathfrak{I}$ can be produced from any ideal $\mathfrak{I}\subsetneq\mathbb{K}[\bm{x}]$ and the ideal-division-based algorithm for strong regular characteristic decomposition is described. The algorithm and its performance are illustrated by examples and experimental results with a preliminary implementation of the algorithm in Section 5. The paper contains a summary of contributions in Section 6 and some remarks on special cases in Appendix A.

2 Preliminaries

In this section we recall some basic notions which will be used in the following sections. For those notions which are not formally introduced in the paper, the reader may consult the references [2, 31, 5, 10].

2.1 Triangular sets, triangular decompositions, and Gröbner bases

Let $\mathbb{K}$ be a field of characteristic 0 and $\mathbb{K}[x_{1},\ldots,x_{n}]$ be the ring of polynomials in $n$ ordered variables $x_{1}<\cdots<x_{n}$ with coefficients in $\mathbb{K}$. Throughout the paper, we write $\bm{x}$ for $(x_{1},\ldots,x_{n})$. Let $F$ be a polynomial in $\mathbb{K}[\bm{x}]\setminus\mathbb{K}$. With respect to the variable ordering, the greatest variable appearing in $F$ is called the leading variable of $F$ and denoted as $\operatorname{lv}(F)$. Assume that $\operatorname{lv}(F)=x_{i}$; then $F$ can be written as $F=Ix_{i}^{k}+R$, where $I\in\mathbb{K}[x_{1},\ldots,x_{i-1}]$, $R\in\mathbb{K}[x_{1},\ldots,x_{i}]$, and $\deg(R,x_{i})<k=\deg(F,x_{i})$ (the degree of $F$ in $x_{i}$). The polynomial $I$ is called the initial of $F$, denoted as $\operatorname{ini}(F)$. For any polynomial set $\mathcal{F}\subseteq\mathbb{K}[\bm{x}]$, $\operatorname{ini}(\mathcal{F})$ stands for $\{\operatorname{ini}(F):F\in\mathcal{F}\}$.

Definition 1**.**

A finite, nonempty, ordered set $[T_{1},\ldots,T_{r}]$ of polynomials in $\mathbb{K}[\bm{x}]\setminus\mathbb{K}$ is called a triangular set if $\operatorname{lv}(T_{1})<\cdots<\operatorname{lv}(T_{r})$.

We denote by $\operatorname{prem}(P,Q)$ the pseudo-reminder of $P\in\mathbb{K}[\bm{x}]$ with respect to $Q\in\mathbb{K}[\bm{x}]\setminus\mathbb{K}$ in $\operatorname{lv}(Q)$. Let $\mathcal{T}=[T_{1},\ldots,T_{r}]\subseteq\mathbb{K}[\bm{x}]$ be any triangular set; the pseudo-reminder of $P$ with respect to $\mathcal{T}$ is defined as

[TABLE]

The variables in $\{x_{1},\ldots,x_{n}\}\setminus\{\operatorname{lv}(T_{1}),\ldots,\operatorname{lv}(T_{r})\}$ are called the parameters of $\mathcal{T}$. For any two polynomial sets $\mathcal{F},\mathcal{G}\subset\mathbb{K}[\bm{x}]$, define

[TABLE]

where $\bar{\mathbb{K}}$ is the algebraic closure of $\mathbb{K}$. Sometimes we write $\operatorname{\mathsf{Z}}(\mathcal{F}/\prod_{G\in\mathcal{G}}G)$ for $\operatorname{\mathsf{Z}}(\mathcal{F}/\mathcal{G})$ and write $\operatorname{\mathsf{Z}}(\mathcal{F})$ for $\operatorname{\mathsf{Z}}(\mathcal{F}/\emptyset)$.

Let $\mathcal{F}\subseteq\mathbb{K}[\bm{x}]$ be any polynomial set and denote by $\langle\mathcal{F}\rangle$ the ideal generated by $\mathcal{F}$ in $\mathbb{K}[\bm{x}]$ and by $\sqrt{\langle\mathcal{F}\rangle}$ the radical of $\langle\mathcal{F}\rangle$. For any $\mathcal{P}\subseteq\mathbb{K}[\bm{x}]$, $\langle\mathcal{F}\rangle:\langle\mathcal{P}\rangle$ denotes the ideal quotient of $\langle\mathcal{F}\rangle$ by $\langle\mathcal{P}\rangle$. The saturated ideal of a triangular set $\mathcal{T}=[T_{1},\ldots,T_{r}]$ is defined as $\operatorname{sat}(\mathcal{T}):=\{P\in\mathbb{K}[\bm{x}]:\,\exists~{}i\text{ such that }PJ^{i}\in\langle\mathcal{T}\rangle\}$, where $J=\operatorname{ini}(T_{1})\cdots\operatorname{ini}(T_{r})$. For any $c\in\mathbb{K}\setminus\{0\}$, we consider $[c]$ also as a triangular set, which is trivial, and define $\operatorname{sat}([c])=\langle 1\rangle$.

Definition 2**.**

Let $\mathcal{T}=[T_{1},\ldots,T_{r}]$ be any nontrivial triangular set in $\mathbb{K}[\bm{x}]$. $\mathcal{T}$ is said to be regular, or called a regular set or a regular chain, if $\operatorname{ini}(T_{i})$ is neither zero nor a zero-divisor in $\mathbb{K}[\bm{x}]/\operatorname{sat}(\mathcal{T}_{i-1})$ for all $i=1,\ldots,r$.

Regular sets or chains [30, 2] are special triangular sets with nice properties which have been extensively studied. In particular, it is proved in [2, 31] that a triangular set $\mathcal{T}$ is regular if and only if $\operatorname{sat}(\mathcal{T})=\{P\in\mathbb{K}[\bm{x}]:\,\operatorname{prem}(P,\mathcal{T})=0\}$. The triangular set $\mathcal{T}$ is called a normal set (or said to be normal) if $\operatorname{ini}(\mathcal{T})$ does not involve any of the leading variables of the polynomials in $\mathcal{T}$. Obviously, any normal set is regular, while a regular set is not necessarily normal.

For a given term order $<$, the greatest term in a polynomial $F\in\mathbb{K}[\bm{x}]$ with respect to $<$ is called the leading term of $F$ and denoted as $\operatorname{lt}(F)$. In this paper, we are concerned only with $<_{\rm lex}$, the lex term order.

Definition 3**.**

Let $\mathfrak{I}\subseteq\mathbb{K}[\bm{x}]$ be an ideal, $<$ be a term order, and $\langle\operatorname{lt}(\mathfrak{I})\rangle$ stand for the ideal generated by the leading terms of all the polynomials in $\mathfrak{I}$. A finite set $\{G_{1},\ldots,G_{s}\}\subseteq\mathfrak{I}$ is called a Gröbner basis of $\mathfrak{I}$ with respect to $<$ if $\langle\operatorname{lt}(G_{1}),\ldots,\operatorname{lt}(G_{s})\rangle=\langle\operatorname{lt}(\mathfrak{I})\rangle$.

Let $\mathcal{G}=\{G_{1},\ldots,G_{s}\}$ be a Gröbner basis of an ideal $\mathfrak{I}\subseteq\mathbb{K}[\bm{x}]$ with respect to a fixed term order $<$. For any polynomial $F\in\mathbb{K}[\bm{x}]$, there exists a unique polynomial $R\in\mathbb{K}[\bm{x}]$, called the normal form of $F$ with respect to $\mathcal{G}$ and denoted as $\operatorname{nform}(F,\mathcal{G})$, such that $F-R\in\mathfrak{I}$ and no term of $R$ is divisible by any of $\operatorname{lt}(G_{1}),\ldots,\operatorname{lt}(G_{s})$. If $F=R$, then $F$ is said to be B-reduced with respect to $\mathcal{G}$.

Definition 4**.**

A Gröbner basis $\{G_{1},\ldots,G_{s}\}$ is said to be reduced if every $G_{i}$ is monic and no term of $G_{i}$ is divisible by any $\operatorname{lt}(G_{j})$ for all $j\neq i$ and $i,j=1,\ldots,s$.

In the rest of this paper, the variable ordering will be fixed and all Gröbner bases mentioned are meant reduced lex Gröbner bases.

2.2 W-characteristic sets, characteristic pairs, and characteristic decompositions

For any polynomial ideal, one can compute its unique reduced lex Gröbner basis and from the Gröbner basis, one can extract a minimal triangular set. This special triangular set, defined formally as the W-characteristic set of the ideal, possesses remarkable properties and plays a key role in our work on src pairs.

Definition 5** ([32, Def. 3.1]).**

Let $\mathcal{F}$ be a polynomial set in $\mathbb{K}[\bm{x}]$, $\mathcal{G}$ be the Gröbner basis of $\langle\mathcal{F}\rangle$, $\Theta_{0}=\mathcal{G}\cap\mathbb{K}$, $\mathcal{G}^{\langle i\rangle}=\{G\in\mathcal{G}:\,\operatorname{lv}(G)=x_{i}\}$, $\Theta_{i}$ be the set consisting of the smallest polynomial in $\mathcal{G}^{\langle i\rangle}$ if $\mathcal{G}^{\langle i\rangle}\neq\emptyset$, or $\emptyset$ otherwise ($1\leq i\leq n$). The set $\Theta_{0}\cup\Theta_{1}\cup\cdots\cup\Theta_{n}$ of polynomials, ordered with increasing leading variables when $\mathcal{G}\neq\{1\}$, is called the W-characteristic set of $\langle\mathcal{F}\rangle$.

Proposition 6**.**

Let $\mathfrak{I}$ be any ideal in $\mathbb{K}[\bm{x}]$ with $\mathfrak{I}^{(0)}=\mathfrak{I}\cap\mathbb{K}=\emptyset$, $\mathfrak{I}^{(i)}=\mathfrak{I}\cap\mathbb{K}[x_{1},\ldots,x_{i}]$, $\mathfrak{I}^{\langle i\rangle}=\{F\in\mathfrak{I}^{(i)}\setminus\mathfrak{I}^{(i-1)}:F\text{ is monic and B-reduced with respect to all }P\in\mathfrak{I}^{(i-1)}\}$, and $\Omega_{i}$ be the set consisting of the smallest polynomial in $\mathfrak{I}^{\langle i\rangle}$ with respect to the lex term order if $\mathfrak{I}^{\langle i\rangle}\neq\emptyset$, or $\emptyset$ otherwise $(1\leq i\leq n)$. Then the set $\Omega_{1}\cup\cdots\cup\Omega_{n}$ of polynomials, ordered with increasing leading variables, is the W-characteristic set of $\mathfrak{I}$.

Proof.

Let $\mathcal{G}$ be the reduced lexicographical Gröbner basis of $\mathfrak{I}$. We need to show that $\Omega_{1}\cup\cdots\cup\Omega_{n}$ defined in the proposition is the minimal triangular set contained in $\mathcal{G}$. Note that $\mathcal{G}^{\langle i\rangle}=\mathcal{G}^{(i)}\setminus\mathcal{G}^{(i-1)}\subseteq\mathfrak{I}^{\langle i\rangle}$ for $1\leq i\leq n$, where $\mathcal{G}^{(0)}=\emptyset$ and $\mathcal{G}^{(i)}=\mathcal{G}\cap\mathbb{K}[x_{1},\ldots,x_{i}]$. For each $i$, if $\Omega_{i}=\emptyset$, then $\mathfrak{I}^{\langle i\rangle}=\emptyset$, so $\mathcal{G}^{\langle i\rangle}=\emptyset$.

Now suppose that $\Omega_{i}\neq\emptyset$ for some $i$ and let $\Omega_{i}=\{F_{i}\}$. Then $F_{i}$ is B-reduced with respect to all the polynomials in $\mathfrak{I}^{(i-1)}\supseteq\mathcal{G}^{(i-1)}$. If $\mathcal{G}^{\langle i\rangle}=\emptyset$, then $F_{i}\neq 0$ is B-reduced with respect to $\mathcal{G}^{(i)}$. This leads to a contradiction because $F_{i}\in\mathfrak{I}^{(i)}$. Hence $\mathcal{G}^{\langle i\rangle}\neq\emptyset$. Let $G_{i}$ be the smallest polynomial in $\mathcal{G}^{\langle i\rangle}$. Then $\operatorname{lt}(F_{i})\leq\operatorname{lt}(G_{i})$ with respect to the lexicographical term order. If $\operatorname{lt}(F_{i})<\operatorname{lt}(G_{i})$, then $F_{i}$ is B-reduced with respect to $\mathcal{G}^{(i)}$. This contradicts with the fact that $F_{i}\in\mathfrak{I}^{(i)}$. Therefore $\operatorname{lt}(F_{i})=\operatorname{lt}(G_{i})>\operatorname{lt}(F_{i}-G_{i})$, for $F_{i}$ and $G_{i}$ are monic. Since $F_{i}$ is the smallest polynomial in $\mathfrak{I}^{\langle i\rangle}$, $F_{i}-G_{i}\in\mathfrak{I}^{(i-1)}$. As $F_{i}$ and $G_{i}$ are both B-reduced with respect to $\mathcal{G}^{(i-1)}$, so is $F_{i}-G_{i}$. It follows that $F_{i}-G_{i}\equiv 0$, and thus $F_{i}=G_{i}$. ∎

The ordered set $\Omega_{1}\cup\cdots\cup\Omega_{n}$ in the above proposition equals the W-characteristic set of $\mathfrak{I}$, so it can be taken as an equivalent definition for the W-characteristic set of any nontrivial polynomial ideal. This alternative definition, which is independent of Gröbner basis and is worked out in response to a comment of Mingsheng Wang, does not indicate how to construct the W-characteristic set of the ideal.

Proposition 7** ([32, Prop. 3.1]).**

Let $\mathcal{F}$ be a polynomial set in $\mathbb{K}[\bm{x}]$ and $\mathcal{C}$ be the W-characteristic set of $\langle\mathcal{F}\rangle\subseteq\mathbb{K}[\bm{x}]$. Then:

1. $(a)$

for any $F\in\langle\mathcal{F}\rangle$, $\operatorname{prem}(F,\mathcal{C})=0$; 2. $(b)$

$\langle\mathcal{C}\rangle\subseteq\langle\mathcal{F}\rangle\subseteq\operatorname{sat}(\mathcal{C})$; 3. $(c)$

$\operatorname{\mathsf{Z}}(\mathcal{C}/\operatorname{ini}(\mathcal{C}))\subseteq\operatorname{\mathsf{Z}}(\mathcal{F})\subseteq\operatorname{\mathsf{Z}}(\mathcal{C})$.

We say that the variable ordering condition is satisfied for a triangular set $\mathcal{T}$ if all the parameters of $\mathcal{T}$ are ordered smaller than the leading variables of the polynomials in $\mathcal{T}$.

Theorem 8** ([32, Thm. 3.9]).**

Let $\mathcal{C}=[C_{1},\ldots,C_{r}]$ be the W-characteristic set of $\langle\mathcal{F}\rangle\subseteq\mathbb{K}[\bm{x}]$. If the variable ordering condition is satisfied for $\mathcal{C}$ and $\mathcal{C}$ is not normal, then there exists an integer $k~{}(1\leq k<r)$ such that $[C_{1},\ldots,C_{k}]$ is normal and $[C_{1},\ldots,C_{k+1}]$ is not regular.

Some structural properties about pseudo-divisibility among polynomials in the Gröbner bases can be found in [32]. Based on those properties, an effective algorithm for normal triangular decomposition of polynomial sets has been proposed in [33].

Definition 9**.**

A pair $(\mathcal{G},\mathcal{C})$ of polynomial sets in $\mathbb{K}[\bm{x}]$ is called a characteristic pair if $\mathcal{G}$ is a Gröbner basis and $\mathcal{C}$ is the W-characteristic set of $\langle\mathcal{G}\rangle$. We say that the characteristic pair $(\mathcal{G},\mathcal{C})$ is strong if $\operatorname{sat}(\mathcal{C})=\langle\mathcal{G}\rangle$.

A characteristic pair $(\mathcal{G},\mathcal{C})$ is said to be regular or normal if $\mathcal{C}$ is regular or normal, respectively. Let $(\{1\},[1])$ be regarded as a trivial regular/normal characteristic pair. If a characteristic pair is strong and regular, then the Gröbner basis $\mathcal{G}$ in the pair is said to be strong regular. By regular or normal characteristic decomposition of a polynomial set $\mathcal{F}\subseteq\mathbb{K}[\bm{x}]$, we mean a finite set of regular or normal characteristic pairs $(\mathcal{G}_{1},\mathcal{C}_{1}),\ldots,(\mathcal{G}_{e},\mathcal{C}_{e})$ satisfying the ideal relations in (3).

The expression (3) can be rewritten in terms of the zero sets or varieties as

[TABLE]

When the regular or normal characteristic pairs $(\mathcal{G}_{i},\mathcal{C}_{i})$ are computed by the algorithms described in [30, 33], the zero relation

[TABLE]

also holds.

To compute a desired characteristic decomposition of a polynomial set $\mathcal{F}$, we can first proceed to decompose $\mathcal{F}$ into finitely many Gröbner bases $\mathcal{G}_{i}$ of certain kinds and then form the characteristic pairs $(\mathcal{G}_{i},\mathcal{C}_{i})$ by simply extracting the W-characteristic sets $\mathcal{C}_{i}$ from $\mathcal{G}_{i}$.

3 Computing strong regular characteristic pairs

The main objective of this section is to show how to construct from an arbitrary ideal $\mathfrak{I}$ an src pair $(\mathcal{G},\mathcal{C})$ such that $\mathfrak{I}\subseteq\operatorname{sat}(\mathcal{C})$ by computing Gröbner bases. The construction enables us to devise a novel algorithm for decomposing any polynomial set into finitely many src pairs.

Definition 10**.**

A Gröbner basis $\mathcal{G}$ is said to be characterizable if $\langle\mathcal{G}\rangle=\operatorname{sat}(\mathcal{C})$, where $\mathcal{C}$ is the W-characteristic set of $\langle\mathcal{G}\rangle$.

Obviously, the W-characteristic set $\mathcal{C}$ extracted from the Gröbner basis $\mathcal{G}$ is unique. The following proposition shows that if $\operatorname{sat}(\mathcal{C})=\langle\mathcal{G}\rangle$, then $\mathcal{C}$ must be regular and thus $(\mathcal{G},\mathcal{C})$ is an src pair. In other words, the Gröbner basis $\mathcal{G}$ in any src pair $(\mathcal{G},\mathcal{C})$ is characterizable. Otherwise, $\operatorname{sat}(\mathcal{C})\neq\langle\mathcal{G}\rangle$; in this case, $\mathcal{G}$ may not necessarily be determined by $\mathcal{C}$ (i.e., there may be two Gröbner bases $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ such that $\langle\mathcal{G}_{1}\rangle$ and $\langle\mathcal{G}_{2}\rangle$ have the same W-characteristic set $\mathcal{C}$).

Proposition 11**.**

The W-characteristic set of the ideal generated by any characterizable Gröbner basis is regular.

Proof.

Let $\mathcal{G}$ be any characterizable Gröbner basis and $\mathcal{C}=[C_{1},\ldots,C_{r}]$ be the W-characteristic set of $\langle\mathcal{G}\rangle$. For any $P\in\operatorname{sat}(\mathcal{C})$, we have $P\in\langle\mathcal{G}\rangle$ since $\langle\mathcal{G}\rangle=\operatorname{sat}(\mathcal{C})$. From Proposition 7(a) one can see that $\operatorname{prem}(P,\mathcal{C})=0$, which means that $\operatorname{sat}(\mathcal{C})\subseteq\{P\in\mathbb{K}[\bm{x}]:\operatorname{prem}(P,\mathcal{C})=0\}\subseteq\operatorname{sat}(\mathcal{C})$. Therefore, by [2, Thm. 6.1] $\mathcal{C}$ is regular. ∎

Theorem 12** (Strong Regularization).**

Let $\mathcal{C}_{1}$ be the W-characteristic set of an arbitrary ideal $\mathfrak{I}\subset\mathbb{K}[\bm{x}]$, $\mathcal{G}_{i}$ be the Gröbner basis of $\operatorname{sat}(\mathcal{C}_{i-1})$, and $\mathcal{C}_{i}$ be the W-characteristic set of $\langle\mathcal{G}_{i}\rangle$ for $i\geq 2$. Then there exists an integer $m\geq 2$ such that $\mathfrak{I}\subseteq\operatorname{sat}(\mathcal{C}_{1})\subseteq\operatorname{sat}(\mathcal{C}_{m})$ and either $\mathcal{C}_{m}$ is a regular set, or $\operatorname{sat}(\mathcal{C}_{m})=\langle 1\rangle$.

Proof.

From Proposition 7(a) and $\operatorname{sat}(\mathcal{C}_{i-1})=\langle\mathcal{G}_{i}\rangle\subseteq\operatorname{sat}(\mathcal{C}_{i})$ we know that $\operatorname{sat}(\mathcal{C}_{1})\subseteq\operatorname{sat}(\mathcal{C}_{2})\subseteq\cdots\subseteq\operatorname{sat}(\mathcal{C}_{i})\subseteq\cdots$ for $i\geq 2$. Then by the Ascending Chain Condition there exists an $m\geq 2$ such that $\operatorname{sat}(\mathcal{C}_{m-1})=\operatorname{sat}(\mathcal{C}_{m})=\cdots$. It follows that the Gröbner basis $\mathcal{G}_{m}$ is characterizable unless $\operatorname{sat}(\mathcal{C}_{m})=\langle 1\rangle$. Therefore, one sees from Proposition 11 that $\mathcal{C}_{m}$ is regular or $\operatorname{sat}(\mathcal{C}_{m})=\langle 1\rangle$. ∎

The process of constructing the chain of W-characteristic sets $\mathcal{C}_{i}$ of $\mathfrak{I}_{i}=\langle\mathcal{G}_{i}\rangle$ for $i=1,\ldots,m$ such that $\mathcal{C}_{m}$ is regular and $\operatorname{sat}(\mathcal{C}_{m})=\operatorname{sat}(\mathcal{C}_{m-1})$ (called strong regularization) shown in Theorem 12 is depicted by the diagram in Figure 1.

Remark 13**.**

From [26, Thm. 4.4] we know that for any normal set $\mathcal{T}$, if its parameters are ordered smaller than the other variables, then the W-characteristic set $\mathcal{C}$ of $\operatorname{sat}(\mathcal{T})$ is also normal. This corresponds in some way to the case of Theorem 12 when the variable ordering condition is satisfied and $\mathcal{C}_{1}$ is assumed to be regular; in this case $\mathcal{C}_{2}$ is always regular and thus $m=2$. However, under the hypothesis of Theorem 12, $\mathcal{C}_{2}$ is not necessarily regular (i.e., $m$ may be greater than $2$), as shown by Example 15 (where $m=3$). Example 16 shows that Theorem 4.4 in [26] does not necessarily hold for regular set $\mathcal{T}$ without the variable ordering condition.

For any triangular set $\mathcal{T}$, $\operatorname{sat}(\mathcal{T})$ is said to be equiprojectable if there exists a regular set $\bar{\mathcal{T}}$ such that $\operatorname{sat}(\mathcal{T})=\operatorname{sat}(\bar{\mathcal{T}})$ [3]. The following corollary follows directly from [26, Thm. 4.4].

Corollary 14**.**

For any triangular set $\mathcal{T}$ satisfying the variable ordering condition, $\operatorname{sat}(\mathcal{T})$ is equiprojectable if and only if the W-characteristic set $\mathcal{C}$ of $\operatorname{sat}(\mathcal{T})$ is regular and $\operatorname{sat}(\mathcal{T})=\operatorname{sat}(\mathcal{C})$.

Corollary 14 points out explicitly how to construct $\bar{\mathcal{T}}$ from $\mathcal{T}$ and thus how to check whether $\operatorname{sat}(\mathcal{T})$ is equiprojectable.

Example 15**.**

Let $\mathcal{C}_{1}=[x^{2}-x,(y^{2}-x)(y-1),(y-1)z]\subseteq\mathbb{K}[x,y,z]$ with $x<y<z$. It is easy to verify that $\mathcal{C}_{1}$ is a Gröbner basis, it is a triangular set, and it is also the W-characteristic set of $\mathfrak{I}=\langle\mathcal{C}_{1}\rangle$, but $\mathcal{C}_{1}$ is not regular. The Gröbner basis of the saturated ideal of $\mathcal{C}_{1}$ may be easily computed as $\mathcal{G}_{2}=\{z,x(x-1),y^{2}-x,x(y+1)\}$. The W-characteristic set of $\langle\mathcal{G}_{2}\rangle=\operatorname{sat}(\mathcal{C}_{1})$ is $\mathcal{C}_{2}=[x(x-1),x(y+1),z]$. Obviously $\mathcal{C}_{2}$ is not regular and thus $\operatorname{sat}(\mathcal{C}_{1})$ is not equiprojectable, while the W-characteristic set $\mathcal{C}_{3}=[x-1,y+1,z]$ of $\operatorname{sat}(\mathcal{C}_{2})$ is regular. Therefore, for this example the integer $m$ in Theorem 12 is equal to $3$.

Example 16**.**

Consider the regular set $\mathcal{T}=[y^{2},x^{2}z+xy]$ in $\mathbb{K}[x,y,z]$ with $y<x<z$ (cf. [33, Ex. 3.20]). The reduced lex Gröbner basis $\mathcal{G}$ of the saturated ideal $\operatorname{sat}(\mathcal{T})$ of $\mathcal{T}$ is $\{y^{2},yz,xz+y,z^{2}\}$, so the W-characteristic set $\mathcal{C}$ of $\langle\mathcal{G}\rangle$ is $[y^{2},yz]$. One can easily see that $\mathcal{C}$ is not regular.

Algorithm 1 can be used to compute an src pair from any polynomial ideal. Its termination and correctness follow directly from Theorem 12.

4 Strong regular characteristic decomposition

In this section, we present an algorithm to compute strong regular characteristic decompositions of polynomial sets using Gröbner basis and ideal computations. The main ingredients of the algorithm are two subalgorithms: one for computing src pairs with ideal saturation and the other for computing src divisors with ideal quotient.

For any two ideals $\mathfrak{I}$ and $\mathfrak{J}$, we say that $\mathfrak{J}$ divides $\mathfrak{I}$ if $\mathfrak{I}:\mathfrak{J}\neq\mathfrak{I}$.

Definition 17**.**

Let $\mathfrak{I}$ be an ideal in $\mathbb{K}[\bm{x}]$ and $\mathcal{C}$ be the W-characteristic set of $\mathfrak{I}$. $\mathcal{C}$ is said to be strong if $\operatorname{sat}(\mathcal{C})=\mathfrak{I}$, or morbid if $\operatorname{sat}(\mathcal{C})$ does not divide $\mathfrak{I}$.

Let $\mathcal{C}$ be the W-characteristic set of an ideal $\mathfrak{I}\subseteq\mathbb{K}[\bm{x}]$. If $\mathcal{C}$ is strong, then $\mathcal{C}$ carries all the information of $\mathfrak{I}$. If $\mathcal{C}$ is morbid, then the associated primes of $\mathfrak{I}$ are all properly contained in those of $\operatorname{sat}(\mathcal{C})$; in this case the structure of $\mathfrak{I}$ is so complicated that $\mathcal{C}$ carries almost no information of $\mathfrak{I}$.

Proposition 18**.**

Let $\mathfrak{I}$ and $\mathfrak{J}$ be two ideals in $\mathbb{K}[\bm{x}]$ such that $\mathfrak{I}\subseteq\mathfrak{J}$. Then $\sqrt{\mathfrak{I}}=\sqrt{\mathfrak{J}}\cap\sqrt{\mathfrak{I}:\mathfrak{J}}$.

Proof.

On one hand, for any polynomial $P\in\mathfrak{I}$, we have $P\in\mathfrak{J}$ and $P\in\mathfrak{I}:\mathfrak{J}$ since $\mathfrak{I}\subseteq\mathfrak{J}$ and $\mathfrak{I}\subseteq\mathfrak{I}:\mathfrak{J}$. It follows that $P\in\mathfrak{J}\cap(\mathfrak{I}:\mathfrak{J})$, which implies that $\sqrt{\mathfrak{I}}\subseteq\sqrt{\mathfrak{J}}\cap\sqrt{\mathfrak{I}:\mathfrak{J}}$. On the other hand, for any polynomial $P\in\sqrt{\mathfrak{J}}\cap\sqrt{\mathfrak{I}:\mathfrak{J}}$, we have $P\in\sqrt{\mathfrak{J}}$ and $P\in\sqrt{\mathfrak{I}:\mathfrak{J}}$. As $P\in\sqrt{\mathfrak{J}}$, there exists an $m$ such that $P^{m}\in\mathfrak{J}$; as $P$ also belongs to $\sqrt{\mathfrak{I}:\mathfrak{J}}$, one can find an $l$ such that $P^{l}H\in\mathfrak{I}$ for any polynomial $H\in\mathfrak{J}$. Therefore, $P^{l}P^{m}\in\mathfrak{I}$ and thus $P\in\sqrt{\mathfrak{I}}$. Hence $\sqrt{\mathfrak{J}}\cap\sqrt{\mathfrak{I}:\mathfrak{J}}\subseteq\sqrt{\mathfrak{I}}$. ∎

For two ideals $\mathfrak{I}$ and $\mathfrak{J}$ in $\mathbb{K}[\bm{x}]$ with $\mathfrak{I}\subseteq\mathfrak{J}$, by Proposition 18, $\sqrt{\mathfrak{I}}=\sqrt{\mathfrak{J}}\cap\sqrt{\mathfrak{I}:\mathfrak{J}}$. This decomposition is trivial if $\mathfrak{I}:\mathfrak{J}=\mathfrak{I}$. For any given ideal $\mathfrak{I}\subsetneq\mathbb{K}[\bm{x}]$, we will show in the following subsection how to construct an ideal $\mathfrak{J}\subseteq\mathbb{K}[\bm{x}]$ such that $\mathfrak{I}\subseteq\mathfrak{J}$ and $\mathfrak{I}:\mathfrak{J}\neq\mathfrak{I}$. With the ideal $\mathfrak{J}$ specially constructed from $\mathfrak{I}$, the splitting of $\mathfrak{I}$ to $\mathfrak{J}$ and $\mathfrak{I}:\mathfrak{J}$ is nontrivial and $\sqrt{\mathfrak{J}}$ and $\sqrt{\mathfrak{I}:\mathfrak{J}}$ can be further decomposed. The technique of splitting ideals by taking quotient according to Proposition 18 originates from the idea of dividing a known subvariety out of any given variety explained in [31, pp. 196–197].

4.1 Finding saturated ideals of regular sets that divide a given ideal

Definition 19**.**

An src pair $(\mathcal{G},\mathcal{C})$ of polynomial sets is called an src divisor of a polynomial ideal $\mathfrak{I}$ in $\mathbb{K}[\bm{x}]$ if $\langle\mathcal{G}\rangle=\operatorname{sat}(\mathcal{C})$ divides $\mathfrak{I}$.

Given an ideal $\mathfrak{I}\subsetneq\mathbb{K}[\bm{x}]$, we want to find an src divisor $(\mathcal{G},\mathcal{C})$ of $\mathfrak{I}$. Suppose that $\mathfrak{I}\neq\langle 1\rangle$ and compute an src pair $(\bar{\mathcal{G}},\bar{\mathcal{C}}):=\operatorname{\sf srcPair}(\mathfrak{I})$ (using Algorithm 1) such that $\mathfrak{I}\subseteq\operatorname{sat}(\bar{\mathcal{C}})$. If $\mathfrak{I}:\operatorname{sat}(\bar{\mathcal{C}})\neq\mathfrak{I}$, then $(\mathcal{G},\mathcal{C})=(\bar{\mathcal{G}},\bar{\mathcal{C}})\neq(\{1\},[1])$ is an src divisor of $\mathfrak{I}$.

Otherwise, $\mathfrak{I}:\operatorname{sat}(\bar{\mathcal{C}})=\mathfrak{I}$. Let $\mathcal{G}^{*}$ be the Gröbner basis of $\mathfrak{I}$ and extract the W-characteristic set $\mathcal{C}^{*}$ of $\mathfrak{I}$ from $\mathcal{G}^{*}$. Recall that $\mathfrak{I}\subseteq\operatorname{sat}(\mathcal{C}^{*})$ and $\mathfrak{I}\neq\langle 1\rangle$. If $\operatorname{sat}(\mathcal{C}^{*})=\mathfrak{I}$, then $\operatorname{sat}(\bar{\mathcal{C}})=\operatorname{sat}(\mathcal{C}^{*})$ according to Theorem 12 (or Fig. 1), so $\mathfrak{I}:\operatorname{sat}(\bar{\mathcal{C}})=\operatorname{sat}(\mathcal{C}^{*}):\operatorname{sat}(\mathcal{C}^{*})=\langle 1\rangle\neq\mathfrak{I}$, which contradicts the assumption that $\mathfrak{I}:\operatorname{sat}(\bar{\mathcal{C}})=\mathfrak{I}$. Therefore, $\operatorname{sat}(\mathcal{C}^{*})\neq\mathfrak{I}$ and one can find a polynomial $F\in\operatorname{sat}(\mathcal{C}^{*})$, $F\notin\mathfrak{I}$. Let $J=\prod_{C\in\mathcal{C}^{*}}\operatorname{ini}(C)$. Then there exists a positive integer $q$ such that $FJ^{q}\in\langle\mathcal{C}^{*}\rangle\subseteq\mathfrak{I}$. Now choose polynomials $H_{1},\ldots,H_{t}\in\mathbb{K}[\bm{x}]\setminus\mathfrak{I}$ such that333Natural choices for $\{H_{1},\ldots,H_{t}\}$ in line 2 of Algorithm 2 are $\{F,\,\operatorname{ini}(C):C\in\mathcal{C}^{*},\,\operatorname{ini}(C)\not\in\mathbb{K}\}$ and $\{H\in\mathbb{K}[\bm{x}]:H\mid F\prod_{C\in\mathcal{C}^{*}}\operatorname{ini}(C),H~{}\mbox{is irreducible (or squarefree)}\}$. Our experiments are done with the latter choice for squarefree $H$. $H_{1}\cdots H_{t}=0\iff FJ^{q}=0$.

1. (1)

Let $(\tilde{\mathcal{G}},\tilde{\mathcal{C}})\neq(\{1\},[1])$ be an src pair computed by Algorithm $\operatorname{\sf srcPair}$ from $\langle\mathfrak{I}\cup\{H_{i}\}\rangle$ for some $1\leq i\leq t$. If $\mathfrak{I}:\operatorname{sat}(\tilde{\mathcal{C}})\neq\mathfrak{I}$, then $(\mathcal{G},\mathcal{C})=(\tilde{\mathcal{G}},\tilde{\mathcal{C}})$ is an src divisor of $\mathfrak{I}$. 2. (2)

Otherwise, $\operatorname{sat}(\tilde{\mathcal{C}})$ does not divide $\mathfrak{I}$, where $(\tilde{\mathcal{G}},\tilde{\mathcal{C}})$ is computed from $\langle\mathfrak{I}\cup\{H_{i}\}\rangle$ as in (1), for all $1\leq i\leq t$. In this case, proceed further to find an src divisor of $\mathfrak{I}$ from $\langle\mathfrak{I}\cup\{H_{i},H_{ij}\}\rangle$ similarly, where $H_{ij}\in\mathbb{K}[\bm{x}]\setminus\langle\mathfrak{I}\cup\{H_{i}\}\rangle$ such that $H_{i1}\cdots H_{it_{i}}=0\iff F_{i}\prod_{C\in\mathcal{C}^{*}_{i}}\operatorname{ini}(C)=0$, $F_{i}\in\operatorname{sat}(\mathcal{C}^{*}_{i})\setminus\langle\mathfrak{I}\cup\{H_{i}\}\rangle$, and $\mathcal{C}^{*}_{i}$ is the W-characteristic set of $\langle\mathfrak{I}\cup\{H_{i}\}\rangle$.

The above process can continue recursively until an src divisor of $\mathfrak{I}$ is found. We formulate the process as Algorithm 2.

Proof of Termination and Correctness of Algorithm 2.

The process of searching for an src divisor of the ideal $\mathfrak{I}$ in Algorithm 2 can be viewed as of building up a multi-branch tree $\Gamma$ with $\mathfrak{I}$ associated to its root, $\mathfrak{I}_{j_{1}}$ associated to its nodes of the first level, $\mathfrak{I}_{j_{1}j_{2}}$ associated to its nodes of the second level, etc. In general, the nodes $j_{1}\cdots j_{l}$ of the $l$th level of $\Gamma$ are associated with $\mathfrak{I}_{j_{1}\cdots j_{l}}$. For each index $j_{1}\cdots j_{l}$ there is an src pair $(\mathcal{G}_{j_{1}\cdots j_{l}},\mathcal{C}_{j_{1}\cdots j_{l}})$ such that either (1) $\mathfrak{I}_{j_{1}\ldots j_{l}}=\langle 1\rangle$, or (2) $\mathfrak{I}:\operatorname{sat}(\mathcal{C}_{j_{1}\cdots j_{l}})\neq\mathfrak{I}$, or (3) $\mathfrak{I}:\operatorname{sat}(\mathcal{C}_{j_{1}\cdots j_{l}})=\mathfrak{I}$. In case (1) or (2), the tree $\Gamma$ terminates to grow and the node $j_{1}\cdots j_{l}$ becomes a leaf. When (2) happens, an src divisor of $\mathfrak{I}$ is found. In case (3), new nodes $j_{1}\cdots j_{l}i$ of the $(l+1)$th level are generated by some factors $H_{j_{1}\cdots j_{l}i}\not\in\mathfrak{I}_{j_{1}\cdots j_{l}}$ of $F[\prod_{C\in\mathcal{C}^{*}_{j_{1}\cdots j_{l}}}\operatorname{ini}(C)]^{q}\in\langle\mathcal{C}^{*}_{j_{1}\cdots j_{l}}\rangle\subseteq\mathfrak{I}_{j_{1}\cdots j_{l}}$, where $F\in\operatorname{sat}(\mathcal{C}^{*}_{j_{1}\cdots j_{l}})\setminus\mathfrak{I}_{j_{1}\cdots j_{l}}$, $q$ is a positive integer, and $\mathcal{C}^{*}_{j_{1}\cdots j_{l}}$ is the W-characteristic set of $\mathfrak{I}_{j_{1}\cdots j_{l}}$. The new nodes are associated with $\langle\mathfrak{I}_{j_{1}\cdots j_{l}}\cup\{H_{j_{1}\cdots j_{l}i}\}\rangle$ ($s_{j_{1}\cdots j_{l}}\leq i\leq t_{j_{1}\cdots j_{l}}$).

(Termination) As $H_{j_{1}\cdots j_{l}i}\not\in\mathfrak{I}_{j_{1}\cdots j_{l}}$ and $\mathfrak{I}_{j_{1}\cdots j_{l}i}=\langle\mathfrak{I}_{j_{1}\cdots j_{l}}\cup\{H_{j_{1}\cdots j_{l}i}\}\rangle$, the containment $\mathfrak{I}\subset\mathfrak{I}_{j_{1}}\subset\cdots\subset\mathfrak{I}_{j_{1}\cdots j_{l}}\subset\mathfrak{I}_{j_{1}\cdots j_{l}i}$ is strict according to the Ascending Chain Condition for polynomial ideals. Hence the length of every branch of the tree $\Gamma$ from the root to a leaf is finite, so the algorithm must terminate.

(Correctness) Recall Proposition 18 and observe that $\prod\nolimits_{i=s_{j_{1}\cdots j_{l}}}^{t_{j_{1}\cdots j_{l}}}H_{j_{1}\cdots j_{l}i}\in\sqrt{\mathfrak{I}_{j_{1}\cdots j_{l}}}$. Let $\Delta$ be the set of indices of all the leaves of $\Gamma$. Then, after the growth of the tree $\Gamma$ from a node is completed,

[TABLE]

always holds. Therefore, at the termination of the algorithm, either an src divisor $(\mathcal{G},\mathcal{C})$ is found, or $\langle 1\rangle$ is associated to all the leaves of $\Gamma$; the latter contradicts $\mathfrak{I}\neq\langle 1\rangle$. The src divisor produced in line 2, 2, or 2 is what we want to find. ∎

The following theorem follows directly from Algorithm 2.

Theorem 20**.**

From any ideal $\mathfrak{I}\neq\langle 1\rangle$ in $\mathbb{K}[\bm{x}]$, one can construct an src divisor of $\mathfrak{I}$.

4.2 An ideal-division-based algorithm for src decomposition

In this subsection, we describe an algorithm to decompose any polynomial set, or the ideal $\mathfrak{I}$ it generates, into finitely many src pairs. Based on the strategy of ideal splitting with quotient, the algorithm works by finding an src divisor $(\mathcal{G},\mathcal{C})$ of $\mathfrak{I}$ and then dividing the saturated ideal of $\mathcal{C}$ out of $\mathfrak{I}$ iteratively.

Consider an ideal $\mathfrak{I}$, initially with input $\mathcal{F}\subseteq\mathbb{K}[\bm{x}]$ as its generating set of polynomials. Let $\Psi$ be the set of src pairs already computed; it is $\emptyset$ initially. Suppose that $\mathfrak{I}\neq\langle 1\rangle$ and compute $(\mathcal{G},\mathcal{C}):=\operatorname{\sf srcDivisor}(\mathfrak{I})$; in this case, $\mathfrak{I}$ is split into $\operatorname{sat}(\mathcal{C})$ and $\mathfrak{I}:\operatorname{sat}(\mathcal{C})$ and an src pair $(\mathcal{G},\mathcal{C})\neq(\{1\},[1])$ is obtained and adjoined to $\Psi$. From Proposition 18 we know that

[TABLE]

so the procedure can continue to decompose $\mathfrak{I}:\operatorname{sat}(\mathcal{C})$ instead of $\mathfrak{I}$.

The above process of decomposition will terminate in finitely many iterations. We formulate the process as Algorithm 3.

Proof of Termination and Correctness of Algorithm 3.

(Termination) Decomposition in Algorithm 3 is an iterative process: every time after the ideal $\mathfrak{I}$ is decomposed, a new ideal $\bar{\mathfrak{I}}\supsetneq\mathfrak{I}$ is obtained and then further decomposed in the same way. Since every new ideal generated in line 3 is strictly enlarged, by the Ascending Chain Condition the algorithm terminates.

(Correctness) When $\langle\mathcal{F}\rangle=\langle 1\rangle$ in line 3, $\Psi=\emptyset$ is returned. Hence we only need to show that when $\langle\mathcal{F}\rangle\neq\langle 1\rangle$, $\Psi$ is an src decomposition of $\mathcal{F}$, namely all the pairs in $\Psi$ are src pairs and the relation (3) holds. It is clear that only in line 3 is adjoined to $\Psi$ a new pair $(\mathcal{G},\mathcal{C})$ which is computed by Algorithm 2. Therefore, all the pairs in $\Psi$ are src pairs.

Now we prove that $\sqrt{\langle\mathcal{F}\rangle}=\bigcap_{\mathcal{C}\in\Psi}\sqrt{\operatorname{sat}(\mathcal{C})}$. For those src pairs generated in line 3, one can easily see that $\mathfrak{I}\subseteq\operatorname{sat}(\mathcal{C})$; thus from Proposition 18 we know that the relation (6) holds.

The relation (6) implies that every polynomial $F\in\sqrt{\mathfrak{I}}$ is in the intersection of $\sqrt{\operatorname{sat}(\mathcal{C})}$ and $\sqrt{\bar{\mathfrak{I}}}$, where $\bar{\mathfrak{I}}$ is a new ideal obtained which remains for further processing. This proves that the relation $\sqrt{\langle\mathcal{F}\rangle}=\bigcap_{\mathcal{C}\in\Psi}\sqrt{\operatorname{sat}(\mathcal{C})}$ holds when the algorithm terminates with $\bar{\mathfrak{I}}=\langle 1\rangle$. Since $\langle\mathcal{G}\rangle=\operatorname{sat}(\mathcal{C})$ for every src pair $(\mathcal{G},\mathcal{C})\in\Psi$, $\sqrt{\langle\mathcal{F}\rangle}=\bigcap_{\mathcal{G}\in\Psi}\sqrt{\langle\mathcal{G}\rangle}$. This completes the proof of the relation (3). ∎

5 Examples and experiments

5.1 Examples for src decomposition

Example 21**.**

Let $\mathcal{F}=\{uxy,vy^{2}+y,vx^{2}+y^{2}\}\subseteq\mathbb{K}[u,v,x,y]$ with $u<v<x<y$. The Gröbner basis of $\mathfrak{I}_{1}=\langle\mathcal{F}\rangle$ can be easily computed as

[TABLE]

and the W-characteristic set of $\langle\mathcal{G}_{1}\rangle$ is $\mathcal{C}_{1}=[uvx^{2},y-v^{2}x^{2}]$. The saturated ideal of $\mathcal{C}_{1}$ is $\operatorname{sat}(\mathcal{C}_{1})=\langle x^{2},y\rangle$. One sees that $(\bar{\mathcal{G}}_{1},\bar{\mathcal{C}}_{1})$ is an src pair, where $\bar{\mathcal{G}}_{1}=\{x^{2},y\}$ is the Gröbner basis and $\bar{\mathcal{C}}_{1}=[x^{2},y]$ is the W-characteristic set of $\operatorname{sat}(\mathcal{C}_{1})$. It is easy to verify that

[TABLE]

is strictly larger than $\mathfrak{I}_{1}$, where $\mathcal{G}_{2}=\{uv,vP,Q\}$ is the Gröbner basis of $\mathfrak{I}_{2}$ and $P=v^{3}x^{2}+1$, $Q=y-v^{2}x^{2}$. Hence $(\bar{\mathcal{G}}_{1},\bar{\mathcal{C}}_{1})$ is adjoined to $\Psi$ and the procedure continues to decompose $\mathfrak{I}_{2}$ instead of $\mathfrak{I}_{1}$. The W-characteristic set $\mathcal{C}_{2}$ of $\mathfrak{I}_{2}$ consists of the same polynomials as $\mathcal{G}_{2}$:

[TABLE]

Since $\operatorname{sat}(\mathcal{C}_{2})=\langle 1\rangle$, the pair $(\{1\},[1])$ is obtained and $\mathfrak{I}_{2}:\langle 1\rangle=\mathfrak{I}_{2}$. As $1\in\operatorname{sat}(\mathcal{C}_{2})$,

[TABLE]

so we can take, e.g., $H_{1}=v$ and $H_{2}=u$. Compute the Gröbner basis $\mathcal{G}_{3}=\{v,y\}$ of $\langle\mathcal{G}_{2}\cup\{H_{1}\}\rangle$ and extract the W-characteristic set $\mathcal{C}_{3}=[v,y]$ of $\langle\mathcal{G}_{3}\rangle$ from $\mathcal{G}_{3}$; one finds that $\mathcal{C}_{3}$ is regular and $\operatorname{sat}(\mathcal{C}_{3})=\langle v,y\rangle=\langle\mathcal{G}_{3}\rangle$ and $\mathfrak{I}_{2}:\operatorname{sat}(\mathcal{C}_{3})=\langle u,P,Q\rangle$ is strictly larger than $\mathfrak{I}_{2}$. Then the src pair $(\mathcal{G}_{3},\mathcal{C}_{3})$ is obtained and adjoined to $\Psi$. The procedure continues with $\mathfrak{I}_{3}=\langle u,P,Q\rangle$.

Simple computation shows that the Gröbner basis and the W-characteristic set of $\mathfrak{I}_{3}$ contain the same polynomials:

[TABLE]

One can check that $(\mathcal{G}_{4},\mathcal{C}_{4})$ is an src pair and $\mathfrak{I}_{3}:\operatorname{sat}(\mathcal{C}_{4})=\langle 1\rangle$. Therefore, $(\mathcal{G}_{4},\mathcal{C}_{4})$ is added to $\Psi$ and the procedure terminates. Finally, an src decomposition

[TABLE]

of $\mathcal{F}$ is obtained.

Example 22**.**

Consider the polynomial set

[TABLE]

(which is Ex 9 in Table 1) with variable ordering $w<v<u<t<c$. The ideal generated by $\mathcal{F}$, which is of dimension 2 and not radical, consists of $8$ primary components (none of them is embedded). The polynomial set $\mathcal{F}$ can be decomposed by Algorithm 3 into 6 src pairs

[TABLE]

where the polynomials $G_{1},\ldots,G_{5}$ consist of 6, 3, 4, 10, 3 terms respectively and

[TABLE]

It may be observed that (1) $\mathcal{C}_{i}$ is normal for all $i$, (2) $\sqrt{\langle\mathcal{G}_{i}\rangle}$ is prime for $i=1,\ldots,5$, and (3) $\sqrt{\langle\mathcal{G}_{6}\rangle}$ is composed of 3 prime ideals.

The cyclic-$n$ systems are well-known examples for which triangular decompositions based on pseudo-division are more difficult to compute than Gröbner bases. Our algorithm can compute src pairs for cyclic-6 in a few minutes, while other triangular decomposition algorithms cannot (see Ex 21 in Table 1).

5.2 Implementation and experimental results

We have implemented Algorithm 3 as a Maple function srcDec and carried out experiments with the implementation on an Intel(R) Core(TM) i5-4210U CPU at 1.70 GHz$\times 4$ with 7.7 GB RAM under Ubuntu 16.04 LTS. The implementation is based on the functions for Gröbner basis computation available in the FGb library and Maple’s built-in packages. Selected results of experiments on some test examples are presented in Table 1: Ex 1–4 are taken from the Epsilon library, Ex 5–9 from [27], Ex 10 from [8], Ex 11 from the FGb library, and Ex 12–16 can be found at http://www.lifl.fr/~lemaire/BCLM09/BCLM09-systems.txt, Ex 17–21 at http://homepages.math.uic.edu/~jan/Demo/TITLES.html, and Ex 22–27 at http://www-sop.inria.fr/saga/POL/BASE/3.posso. The function srcDec is implemented for direct decomposition of polynomial sets into src pairs. To observe the performance of our algorithm, we made comparative experiments on srcDec in Maple 18 with two other relevant functions for unmixed decomposition of polynomial sets into Gröbner bases of saturated ideals of triangular sets: one is the Epsilon function uvd which is implemented for decomposing an arbitrary algebraic variety into unmixed subvarieties. The other function computes first the decomposition of a polynomial set into regular sets using the RegularChains function Triangularize and then the Gröbner bases of the saturated ideals of the computed regular sets.

In Table 1, Label indicates the label used in the above-cited references and Var, Pol, and Dim denote the number of variables, the number of polynomials in the example, and the dimension of the ideal generated by the polynomials, respectively. Total, GB, SAT, and QUO under Algorithm 3 record respectively the total time (followed by the number of src pairs in parentheses) for src decomposition using Algorithm 3, the time for computing all the Gröbner bases, saturated ideals, and ideal quotients; Total under uvd records the total time for unmixed decomposition. Total and Regular under Triangularize record the total time for unmixed decomposition (followed by the number of components in parentheses) and the time for regular decomposition respectively.

The time-consuming steps in algorithm $\operatorname{\sf srcDec}$ are for the computation of the Gröbner bases of the input ideals, the Gröbner bases of ideal quotients, and the Gröbner bases of saturated ideals. The regular sets computed by the algorithm are normal in most cases and they are often simpler than the corresponding regular sets computed by pseudo-division or subresultant-based algorithms, so that the Gröbner bases of the saturated ideals of the regular sets produced by our algorithm srcDec tend to be easier to compute than those produced by uvd or Triangularize for about two thirds of the test examples. More importantly, the number of src pairs in an src decomposition computed by srcDec is usually smaller than the number of components in the corresponding unmixed decomposition computed by uvd or regular decomposition by Triangularize, as shown by the experimental data in Table 1. This is because our ideal-division-based algorithm generates few redundant components. For example, the computation of the src decomposition using srcDec takes much less time than that of the regular decomposition using Triangularize for Ex 14, as the src decomposition contains only two src pairs.

6 Conclusion

In this paper, it is shown that a strong regular characteristic (src) pair, and therefore an src divisor $(\mathcal{G},\mathcal{C})$ of $\mathfrak{I}$, can be constructed from any polynomial ideal $\mathfrak{I}$ with given generating set $\mathcal{F}$. The constructed src divisor may be used to split the ideal $\mathfrak{I}$ into the saturated ideal $\operatorname{sat}(\mathcal{C})$ and the quotient ideal $\mathfrak{I}:\operatorname{sat}(\mathcal{C})$. The process of construction and splitting can be repeated for $\mathfrak{I}:\operatorname{sat}(\mathcal{C})$ instead of $\mathfrak{I}$ and recursively, yielding an algorithm capable of decomposing the polynomial set $\mathcal{F}$ into finitely many src pairs $(\mathcal{G}_{1},\mathcal{C}_{1}),\ldots,(\mathcal{G}_{e},\mathcal{C}_{e})$ such that

[TABLE]

or equivalently (3) holds. The relation (7) provides two representations for the zero set of $\mathcal{F}$: one in terms of the Gröbner bases $\mathcal{G}_{1},\ldots,\mathcal{G}_{e}$ and the other in terms of the regular sets $\mathcal{C}_{1},\ldots,\mathcal{C}_{e}$. Several nice properties about strong regular characteristic pairs and characteristic decompositions have been presented, and the implementation and performance of our proposed algorithm have been illustrated by examples and experimental results.

The contributions of this paper include: (1) two main theorems (Theorems 12 and 20) showing how to construct an src pair and an src divisor of an arbitrary polynomial ideal; (2) an algorithm for decomposing any polynomial set $\mathcal{F}$ into src pairs $(\mathcal{G}_{i},\mathcal{C}_{i})$ such that (3) and (7) hold; (3) some experiments with a preliminary implementation of the decomposition algorithm.

The triangular sets in an src decomposition are normal in most cases (cf. [33, 32]). It turns out that comprehensive triangular decompositions [8] and/or Gröbner systems [24] can be reproduced rather easily from src pairs computed by our algorithm, and we are working on the details. The W-characteristic set of an ideal may be morbid and it is not yet clear when morbidity happens. How to establish equivalent conditions for a W-characteristic set to be morbid and how to retrieve information of an ideal from its morbid W-characteristic set are some of the questions that remain for further investigation.

The authors wish to thank the referees for their insightful comments which helped bring the paper to the present form. This work also benefited from frequent discussions the authors had with Chenqi Mou.

Appendix A

A.1. Strong regular characteristic decomposition of radical ideals

Lemma 23**.**

Let $\mathfrak{I}$ and $\mathfrak{J}$ be two radical ideals in $\mathbb{K}[\bm{x}]$. Then the algebraic varieties $\operatorname{\mathsf{Z}}(\mathfrak{I}:\mathfrak{J})$ and $\operatorname{\mathsf{Z}}(\mathfrak{J})$ do not have any common irredundant irreducible component.

Proof.

Let $\operatorname{\mathsf{Z}}(\mathfrak{I}:\mathfrak{J})=V_{1}\cup\cdots\cup V_{s}$ be an irredundant irreducible decomposition of the algebraic variety $\operatorname{\mathsf{Z}}(\mathfrak{I}:\mathfrak{J})$. Suppose that $\operatorname{\mathsf{Z}}(\mathfrak{I}:\mathfrak{J})$ and $\operatorname{\mathsf{Z}}(\mathfrak{J})$ have a common irredundant irreducible component, say $V_{m}$ for some positive integer $m$ ($\leq s$), and let

[TABLE]

Then $\operatorname{\mathsf{Z}}(\mathfrak{I})\setminus\operatorname{\mathsf{Z}}(\mathfrak{J})\subseteq V^{*}\subsetneq V_{1}\cup\cdots\cup V_{s}=\operatorname{\mathsf{Z}}(\mathfrak{I}:\mathfrak{J})$; in this case, $\operatorname{\mathsf{Z}}(\mathfrak{I})\setminus\operatorname{\mathsf{Z}}(\mathfrak{J})\subset V^{*}$ implies that $\overline{\operatorname{\mathsf{Z}}(\mathfrak{I})\setminus\operatorname{\mathsf{Z}}(\mathfrak{J})}\subseteq\overline{V^{*}}=V^{*}$, which contradicts with the fact that $\operatorname{\mathsf{Z}}(\mathfrak{I}:\mathfrak{J})=\overline{\operatorname{\mathsf{Z}}(\mathfrak{I})\setminus\operatorname{\mathsf{Z}}(\mathfrak{J})}$. ∎

Proposition 24**.**

Let $\mathfrak{I}$ and $\mathfrak{J}$ be two radical ideals in $\mathbb{K}[\bm{x}]$. Then:

1. $(a)$

the saturated ideal of the W-characteristic of $\mathfrak{I}$ is radical; 2. $(b)$

the ideal quotient $\mathfrak{I}:\mathfrak{J}$ is also radical.

Proof.

$(a)$ Let $\mathcal{T}$ be the W-characteristic set of $\mathfrak{I}$. To show that $\operatorname{sat}(\mathcal{T})$ is radical, it suffers to show that $\operatorname{sat}(\mathcal{T})=\sqrt{\operatorname{sat}(\mathcal{T})}$. The inclusion $\operatorname{sat}(\mathcal{T})\subseteq\sqrt{\operatorname{sat}(\mathcal{T})}$ is obvious. For any $P\in\sqrt{\operatorname{sat}(\mathcal{T})}$, we have $P^{l}I^{m}\in\langle\mathcal{T}\rangle\subseteq\mathfrak{I}$ for some nonnegative integers $m$ and $l$, where $I=\prod_{T\in\mathcal{T}}\operatorname{ini}(T)$. Thus ${(IP)}^{\max(m,l)}\in\mathfrak{I}$; it follows that $IP\in\sqrt{\mathfrak{I}}=\mathfrak{I}$, so that $\operatorname{prem}(IP,\mathcal{T})=0$, which means that $P\in\operatorname{sat}(\mathcal{T})$. Therefore, $\operatorname{sat}(\mathcal{T})=\sqrt{\operatorname{sat}(\mathcal{T})}$.

$(b)$ To prove that $\mathfrak{I}:\mathfrak{J}$ is radical, we only need to prove that $\sqrt{\mathfrak{I}:\mathfrak{J}}\subseteq\mathfrak{I}:\mathfrak{J}$. For any $F\in\sqrt{\mathfrak{I}:\mathfrak{J}}$ and $J\in\mathfrak{J}$, there exists a nonnegative integer $l$ such that $F^{l}J\in\mathfrak{I}$; it follows that ${(FJ)}^{l}\in\mathfrak{I}$, so that $FJ\in\sqrt{\mathfrak{I}}=\mathfrak{I}$. Hence $F\in\mathfrak{I}:\mathfrak{J}$. ∎

By Proposition 24, for any radical ideal $\mathfrak{I}$ the saturated ideal $\operatorname{sat}(\mathcal{C})$ of every regular set $\mathcal{C}$ appearing in an src decomposition $\Psi$ of $\mathfrak{I}$ is radical. According to Lemma 23, for any two src pairs $(\mathcal{G}_{i},\mathcal{C}_{i})$ and $(\mathcal{G}_{j},\mathcal{C}_{j})$ ($i\neq j$) in $\Psi$ the two varieties $\operatorname{\mathsf{Z}}(\operatorname{sat}(\mathcal{C}_{i}))$ and $\operatorname{\mathsf{Z}}(\operatorname{sat}(\mathcal{C}_{j}))$ do not have any common irredundant irreducible component. In other words, the ideals $\operatorname{sat}(\mathcal{C}_{i})$ and $\operatorname{sat}(\mathcal{C}_{j})$ have no common minimal associated prime. Therefore, the number of src pairs in $\Psi$ must be smaller than that of irredundant irreducible components of the variety $\operatorname{\mathsf{Z}}(\mathcal{F})$. We are unable to establish such a tight bound for nonradical ideals because when $\mathfrak{I}$ and $\mathfrak{J}$ in Lemma 23 are not radical, $\mathfrak{I}:\mathfrak{J}$ and $\mathfrak{J}$ may have a common minimal associated prime. Taking $\mathfrak{I}=\langle x^{3}y\rangle$ and $\mathfrak{J}=\langle x\rangle$ as an example, we have $\mathfrak{I}:\mathfrak{J}=\langle x^{2}y\rangle$, which has a minimal associated prime $\langle x\rangle$ as $\mathfrak{J}$.

A.2. Triangular decomposition and characteristic decomposition

Let $\mathfrak{I}\subset\mathbb{K}[\bm{x}]$ be a polynomial ideal with generating set $\mathcal{F}$ and $\Psi=\{(\mathcal{G}_{1},\mathcal{C}_{1}),\ldots,(\mathcal{G}_{e},\mathcal{C}_{e})\}$ be an src decomposition of $\mathcal{F}$ computed by Algorithm 3. Then $\{\mathcal{C}_{1},\ldots,\mathcal{C}_{e}\}$ is a regular triangular decomposition of $\mathcal{F}$ obtained from $\Psi$ with no further computation. Therefore, the decomposition $\Psi$ provides two representations for $\operatorname{\mathsf{Z}}(\mathcal{F})$: one in terms of the strong regular Gröbner bases $\mathcal{G}_{1},\ldots,\mathcal{G}_{e}$, and the other in terms of the regular triangular sets $\mathcal{C}_{1},\ldots,\mathcal{C}_{e}$.

Theorem 25**.**

Let $\{\mathcal{T}_{1},\ldots,\mathcal{T}_{e}\}$ be a regular triangular decomposition of $\mathcal{F}\subseteq\mathbb{K}[\bm{x}]$ and each $(\mathcal{G}_{i},\mathcal{C}_{i})$ be constructed from $\mathcal{T}_{i}$ according to Theorem 4.4 in [26]. Assume that the variable ordering condition is satisfied for all $\mathcal{T}_{i}$. Then $\{(\mathcal{G}_{1},\mathcal{C}_{1}),\ldots,(\mathcal{G}_{e},\mathcal{C}_{e})\}$ is a strong normal characteristic decomposition of $\mathcal{F}$.

Proof.

As $\{\mathcal{T}_{1},\ldots,\mathcal{T}_{e}\}$ is a triangular decomposition of $\mathcal{F}$, we have $\sqrt{\langle\mathcal{F}\rangle}=\bigcap_{i=1}^{e}\sqrt{\operatorname{sat}(\mathcal{T}_{i})}$. By Theorem 4.4 in [26] the W-characteristic set $\mathcal{C}_{i}$ of $\operatorname{sat}(\mathcal{T}_{i})$ is normal and $\operatorname{sat}(\mathcal{C}_{i})=\langle\mathcal{G}_{i}\rangle$, so each $(\mathcal{G}_{i},\mathcal{C}_{i})$ is a strong normal characteristic pair for $i=1,\ldots,e$. Using the equalities $\operatorname{sat}(\mathcal{T}_{i})=\operatorname{sat}(\mathcal{C}_{i})$ and $\langle\mathcal{G}_{i}\rangle=\operatorname{sat}(\mathcal{T}_{i})$, one can easily prove that

[TABLE]

Therefore, $\{(\mathcal{G}_{1},\mathcal{C}_{1}),\ldots,(\mathcal{G}_{e},\mathcal{C}_{e})\}$ is a strong normal characteristic decomposition of $\mathcal{F}$. ∎

More generally, for any decomposition of an ideal $\mathfrak{I}=\langle\mathcal{F}\rangle\subseteq\mathbb{K}[\bm{x}]$ into $e$ ideals $\mathfrak{I}_{1},\ldots,\mathfrak{I}_{e}$ such that $\sqrt{\mathfrak{I}}=\bigcap_{i=1}^{e}\sqrt{\mathfrak{I}_{i}}$, one can apply Algorithm 3 to each generating set of $\mathfrak{I}_{i}$ for $i=1,\ldots,e$ and then combine the results for all $i$ to yield a strong regular characteristic decomposition of $\mathcal{F}$, from which a regular triangular decomposition of $\mathcal{F}$ is obtained as by-product.