Simultaneously Computing a Maximal Independent Set Modulo an Ideal and a Gröbner Basis of the Ideal

Abstract

To solve problems on a positive-dimensional ideal, $I\subset k\left[X\right]$, a maximal independent set $U\subset X$ modulo I, and a Gröbner basis of $I^{\mathrm{e}}$, where $I^{\mathrm{e}}$ is the extension of I to $k\left(U\right)\left[V\right]$ $(V:=$ $X\setminus U)$, are widely used. As far as we know, they are usually computed separately, i.e., U is calculated first and the Gröbner basis is computed after U is obtained. In this paper, we present an efficient algorithm for computing a maximal independent set U modulo I, and a Gröbner basis of $I^{\mathrm{e}}$ simultaneously. Differently from computing them separately, the algorithm takes full advantage of the polynomial information throughout the Gröbner basis computation to obtain U as soon as possible; hence, it significantly improves the computing efficiency.

1. Introduction

As a basic task in computer algebra, polynomial system solving plays an important role in many practical problems that can be described through polynomial systems, which can be divided into zero-dimensional and positive-dimensional ideals. Dealing with problems on a positive-dimensional ideal, $I\subset k\left[X\right]$, such as computing a primary decomposition of I [1] and Noetherian operators of I [2], requires computing a maximal independent set modulo I. Weispfenning presented an algorithm for computing a maximal independent set in 1993 ([3], p. 449), which needs to compute all strongly independent sets from a Gröbner basis of I. Zhang proposed a method for deciding whether an indeterminate set is a maximal independent set by computing a quasi-Gröbner basis of I in 1995 [4]. In 2023, Yang and Tan presented a method to compute maximal independent sets [5], and the complexity is reduced since the method turns a multivariate problem into a single-variable problem.

Gröbner bases of polynomial ideals, first proposed by Buchberger in 1965 [6], are one of the fundamental algorithmic tools for solving polynomial systems and essential in a lot of computational tasks in polynomial algebra and algebraic geometry. Given the positive-dimensional ideal I, with a maximal independent set $U\subset X$ modulo I, known, a Gröbner basis of $I^{\mathrm{e}}$, where $I^{\mathrm{e}}$ is the extension (the ideal generated by the polynomial set $F\in k\left[X\right]$ in $k\left(U\right)\left[V\right]$, where F generates I in k[X]) of I to $k\left(U\right)\left[V\right]$$(V:=X\setminus U)$, is usually necessary for subsequent calculations [7,8]. According to [4], a Gröbner basis of I with respect to a block monomial order, ${\prec }_{U,V}$$(U\ll V)$, is a Gröbner basis of $I^{\mathrm{e}}$. So it is equivalent to computing a Gröbner basis, G, of I with respect to ${\prec }_{U,V}$.

As far as we know, the maximal independent set and the Gröbner basis are generally computed separately. For example, to obtain Rational Representation Sets of the variety of I, Tan [9] computes U using Zhang’s method [4] while Shang [10] and Xiao [11] use Weispfenning’s method. After U is computed, they determine ${\prec }_{U,V}$ without clear rules and compute a Gröbner basis, G, of I with respect to ${\prec }_{U,V}$. In this case, the connection between the calculations of U and G is not considered, which leads to a waste of intermediate calculations. Thus we present an algorithm for computing a maximal independent set U modulo I, and a Gröbner basis, G, of I with respect to some block monomial order, ${\prec }_{U,V}$, simultaneously, i.e., U, ${\prec }_{U,V}$, and G are simultaneously obtained. We compare our algorithm with two widely used methods which compute U and G separately. (1) Compute U, determine ${\prec }_{U,V}$ artificially, and directly compute a Gröbner basis of I with respect to ${\prec }_{U,V}$. This is referred to as the direct method. (2) Compute U from a Gröbner basis of I using Weispfenning’s algorithm, determine ${\prec }_{U,V}$ artificially, and convert the Gröbner basis already computed for obtaining U to a Gröbner basis with respect to ${\prec }_{U,V}$ using the Gröbner Walk [12]. This is referred to as the W-W method. Experiments show advantages of our algorithm compared with the direct and W-W methods.

This paper is arranged as follows. In Section 2, we introduce relative notations, definitions, and conclusions. In Section 3, we describe the algorithms in detail. Experimental results are listed in Section 4.

2. Preliminaries

In this paper, k denotes a field of the characteristic 0, $X:=\{x_1,x_2,\cdots ,x_n\}$ is an indeterminate set, $k\left[X\right]$ is the polynomial ring over k in X, and $T\left(X\right)$ denotes the set of all monomials in X. $plex(x_1,x_2,\cdots ,x_n)$ and $grlex(x_1,x_2,\cdots ,x_n)$ denote the pure lexicographic and the graded reverse lexicographic monomial orders on $T\left(X\right)$ with $x_1\succ x_2\succ \cdots \succ x_n$, respectively. Let $I\subset k\left[X\right]$ be an ideal, $F\subset k\left[X\right]$ be a polynomial set, and ≺ be a monomial order on $T\left(X\right)$. Then $L{M}_{\prec }\left(I\right)$ and $L{M}_{\prec }\left(F\right)$ denote the sets of leading monomials of all polynomials in I and F with respect to ≺, respectively. For $U\subset X$, $k\left[U\right]$ is the polynomial ring over k in U, $k\left(U\right)$ denotes the rational function field over k in U, and $T\left(U\right)$ denotes the set of all monomials in U. Suppose that $V:=X\setminus U$. Then $k\left(U\right)\left[V\right]$ denotes the polynomial ring over $k\left(U\right)$ in V, and $I^{\mathrm{e}}$ denotes the extension of I to $k\left(U\right)\left[V\right]$. Given $F\subset k\left[X\right]$ and $f\in k\left[X\right]$, $F_V$ and $f_V$ denote F and f as a polynomial set of $k\left(U\right)\left[V\right]$ and a polynomial in $k\left(U\right)\left[V\right]$, respectively.

Moreover, the cardinality of a set S is written as $♯S$, and the i-th element of a list or an ordered set, L, is denoted by $L_i$. $\mathbb{N}$ is the non-negative integer set, ${\mathbb{N}}_{+}$ is the positive integer set, and ${\mathbb{R}}_{+}$ is the non-negative real number set. We denote $e_i$, the vector of which the i-th element is 1 and the others are 0.

Definition 1 

([3]). Let $I\subset k\left[X\right]$ be a proper ideal. $U\subset X$ is called independent modulo I, if

$$I\cap k\left[U\right]=\left\{0\right\}.$$

Otherwise, U is called  dependent  modulo I. Moreover, U is called  maximal  whenever it is maximal with respect to the inclusion of sets.

Definition 2 

([3]). Let $I\subset k\left[X\right]$ be a proper ideal and ≺ be a monomial order on $T\left(X\right)$. $U\subset X$ is called strongly independent modulo I, with respect to ≺ if

$$L{M}_{\prec }\left(I\right)\cap T\left(U\right)=\varnothing .$$

Moreover, U is called  maximal  whenever it is maximal with respect to the inclusion of sets.

Proposition 1 

([3]). Let $I\subset k\left[X\right]$ be a proper ideal, ≺ be a monomial order on $T\left(X\right)$, and $G\subset k\left[X\right]$ be a Gröbner basis of I with respect to ≺. Then $U\subset X$ is strongly independent modulo I, with respect to ≺ if

$$L{M}_{\prec }\left(G\right)\cap T\left(U\right)=\varnothing .$$

Definition 3 

([3]). Let $U\subset X$ and $V:=X\setminus U$. Let ${\prec }_V$ and ${\prec }_U$ be monomial orders on $T\left(V\right)$ and $T\left(U\right)$, respectively. Any monomial $t\in T\left(X\right)$ can be written uniquely as $t=t_1{t}_2$, where $t_1\in$ $T\left(V\right)$, $t_2\in T\left(U\right)$. Then a block monomial order on $T\left(X\right)$ with $U\ll V$ (for arbitrary $u\in U$ and $v\in V$, $u\prec v$), denoted by ${\prec }_{U,V}$, is defined by $r\prec s$ if

$$\begin{array}{cc}& r_1{\prec }_V{s}_1,\mathit{or}\hfill \\ & r_1=s_1\mathit{and}{r}_2{\prec }_U{s}_2.\hfill \end{array}$$

${\prec }_V$ and ${\prec }_U$ are called  the restrictions  of ${\prec }_{U,V}$ to $T\left(V\right)$ and $T\left(U\right)$, respectively. Particularly, we denote ${\prec }_{U,V}^{grl}$ the block monomial order of which ${\prec }_V$ $=grlex\left(V\right)$ and ${\prec }_U=grlex\left(U\right)$.

Given a positive-dimensional ideal I, according to [13], a strongly independent set modulo I, is independent modulo I. However, a maximal strongly independent set modulo I, is not always maximally independent modulo I. For example, let $I:=⟨zx-x^2,zy-xy⟩\subset k[x,y,z]$ and $\prec :=plex(z,y,x)$. Then $G:=\{zx-x^2,zy-xy\}$ is a Gröbner basis of I with respect to ≺. Since $L{M}_{\prec }\left(G\right)=\{zx,zy\}$, $\left\{z\right\}$ is a maximal strongly independent set modulo I, with respect to ≺. By a simple calculation, we have that $\{y,z\}$ is independent modulo I; thus $\left\{z\right\}$ is not maximally independent modulo I. Fortunately, the situation differs for block monomial orders.

Lemma 1. 

Let $I\subset k\left[X\right]$ be a positive-dimensional ideal, $U\subset X$, and $V:=X\setminus U$ and ${\prec }_{U,V}$ be a block monomial order on $T\left(X\right)$. If U is a maximal strongly independent set modulo I, with respect to ${\prec }_{U,V}$, then U is maximally independent modulo I.

Proof. 

Suppose $U:=\{u_1,u_2,\cdots ,u_r\}$, $V:=\{v_1,v_2,\cdots ,v_{n-r}\}$, where $1\le r<n$, and G is a Gröbner basis of I with respect to ${\prec }_{U,V}$. Since U is maximally strongly independent modulo I, with respect to ${\prec }_{U,V}$, by Definition 2, U is strongly independent modulo I, with respect to ${\prec }_{U,V}$, and $U\cup \left\{v_i\right\}$ is not strongly independent modulo I, with respect to ${\prec }_{U,V}$, $i=1,2,\cdots ,n-r$. So according to Proposition 1, we have $L{M}_{{\prec }_{U,V}}\left(G\right)\cap T\left(U\right)=\varnothing$ and $L{M}_{{\prec }_{U,V}}\left(G\right)\cap T(U,v_i)\ne \varnothing$ for $i=1,2,\cdots ,$ $n-r$. These two facts imply that G contains polynomials of the form

$$\begin{array}{cc}\hfill f_1& :=c_1·v_1^{a_1}·u_1^{b_{1,1}}{u}_2^{b_{1,2}}\cdots u_r^{b_{1,r}}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{\prec }_{U,V},\hfill \\ \hfill f_2& :=c_2·v_2^{a_2}·u_1^{b_{2,1}}{u}_2^{b_{2,2}}\cdots u_r^{b_{2,r}}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{\prec }_{U,V},\hfill \\ \hfill & ⋮\hfill \\ \hfill f_{n-r}& :=c_{n-r}·v_{n-r}^{a_{n-r}}·u_1^{b_{n-r,1}}{u}_2^{b_{n-r,2}}\cdots u_r^{b_{n-r,r}}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{\prec }_{U,V},\hfill \end{array}$$

(1)

where $a_i\in {\mathbb{N}}_{+}$, $b_{i,j}\in \mathbb{N}$, and $c_i\ne 0$ is the leading coefficient of $f_i$ with respect to ${\prec }_{U,V}$, $i=1,2,\cdots ,$ $n-r$, and $j=1,2,\cdots ,r$. Let ${\prec }_V$ be the restriction of ${\prec }_{U,V}$ to $T\left(V\right)$. According to  Lemma 8.93 from [3], $G\subset k\left(U\right)\left[V\right]$ is a Gröbner basis of $I^{\mathrm{e}}$ with respect to ${\prec }_V$. Denote $t_i$ as a non-leading monomial of $f_i$ with respect to ${\prec }_{U,V}$, written as $t_i=t_{i,V}{t}_{i,U}$, where $t_{i,V}\in T\left(V\right)$, $t_{i,U}\in T\left(U\right)$ for $i=1,2,\cdots ,n-r$. According to Definition 3, $t_{i,V}{⪯}_V{v}_i^{a_i}$. So $L{M}_{{\prec }_V}\left({\left(f_i\right)}_V\right)=v_i^{a_i}$. Let $F:=\{f_1,f_2,\cdots ,f_{n-r}\}$. Then $L{M}_{{\prec }_V}\left(F_V\right)=\{v_1^{a_1},v_2^{a_2},\cdots ,v_{n-r}^{a_{n-r}}\}$. By the Finiteness Theorem ([14], p. 39), $I^{\mathrm{e}}$ is zero-dimensional. According to Proposition 1.2.8 from [4], U is maximally independent modulo I.    □

According to Lemma 1, if U is a maximal strongly independent set modulo I, with respect to a block monomial order, ${\prec }_{U,V}$, then U is a maximal independent set modulo I. Moreover, by Proposition 1, it is sufficient to compute a Gröbner basis of I with respect to ${\prec }_{U,V}$. However, after the Gröbner basis is computed, the fact is that U is usually not a maximal strongly independent set modulo I, with respect to ${\prec }_{U,V}$. To resolve this issue, i.e., to ensure that $L{M}_{{\prec }_{U,V}}\left(G\right)\cap T\left(U\right)=\varnothing$ and $L{M}_{{\prec }_{U,V}}\left(G\right)\cap T(U,v)\ne \varnothing$ for each $v\in V$, where G is the Gröbner basis computed, it must be held that $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$ and $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T(U,v)\ne \varnothing$ for each $v\in V$, where F is a polynomial set (not a Gröbner basis) during the Gröbner basis computation. Therefore, we need the concept of a maximal strongly quasi-independent set modulo a polynomial set.

Definition 4. 

Let $F\subset k\left[X\right]$ be a polynomial set and ≺ be a monomial order on $T\left(X\right)$. $U\subset X$ is called  strongly quasi-independent  modulo F, with respect to ≺ if

$$L{M}_{\prec }\left(F\right)\cap T\left(U\right)=\varnothing .$$

Moreover, U is called  maximal  whenever it is maximal with respect to the inclusion of sets. For simplicity, a (maximal) strongly quasi-independent set is written as an (M)SQIS.

Particularly, (M)SQISs, U, with respect to ≺ have certain properties if ≺ is a block monomial order “with $U\ll V$”. Firstly, let us discuss such SQISs through Propositions 2 and 3.

Proposition 2. 

Let $F\subset k\left[X\right]$ be a polynomial set, $U\subset X$, $V:=X\setminus U$, and ${\prec }_{U,V}$ be a block monomial order on $T\left(X\right)$. Then U is an SQIS modulo F, with respect to ${\prec }_{U,V}$, i.e., $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$, if $F\cap k\left[U\right]=\varnothing$.

Proof. Necessity: 

Since $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$, for each $f\in F$, we have $L{M}_{{\prec }_{U,V}}\left(f\right)\notin T\left(U\right)$. So $f\notin k\left[U\right]$, and $F\cap k\left[U\right]=\varnothing$.

Sufficiency: Aiming for a contradiction, suppose that $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)\ne \varnothing$. Then there exists $f\in F$, such that $L{M}_{{\prec }_{U,V}}\left(f\right)\in T\left(U\right)$. Thus, by Definition 3, for each non-leading monomial, t, of f with respect to ${\prec }_{U,V}$, written as $t=t_1{t}_2$, where $t_1\in T\left(V\right)$, $t_2\in T\left(U\right)$, we have $t_1{⪯}_{V}1$, i.e., $t\in T\left(U\right)$. So $f\in k\left[U\right]$, and $F\cap k\left[U\right]\ne \varnothing$. This contradiction shows that $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$.    □

Proposition 3. 

Let $F\subset k\left[X\right]$ be a polynomial set, $U\subset X$, $V:=X\setminus U$, and ${\prec }_{U,V}$ be a block monomial order on $T\left(X\right)$. If U is an SQIS modulo F, with respect to ${\prec }_{U,V}$, then for each block monomial order, ${\prec }_{U,V}^{\prime }\ne {\prec }_{U,V}$, with $U\ll V$, U is also an SQIS modulo F, with respect to ${\prec }_{U,V}^{\prime }$.

Proof. 

Since U is an SQIS modulo F, with respect to ${\prec }_{U,V}$, by Definition 4, we have $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$. According to Proposition 2, $F\cap k\left[U\right]=\varnothing$, and moreover, $L{M}_{{\prec }_{U,V}^{\prime }}\left(F\right)\cap T\left(U\right)=\varnothing$. So U is an SQIS modulo F, with respect to ${\prec }_{U,V}^{\prime }$.    □

Secondly, we discuss such MSQISs in the rest of this section. The following Theorem 1 reveals the relation between an MSQIS U modulo F, with respect to a block monomial order, ${\prec }_{U,V}$, and the dependent set modulo the ideal I, where $F\subset I$.

Theorem 1. 

Let $I\subset k\left[X\right]$ be a positive-dimensional ideal and $F\subset I$ be a polynomial set. Let $U\subset X$, $V:=X\setminus U$ and ${\prec }_{U,V}$ be a block monomial order on $T\left(X\right)$. If U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$, then for any $\overline{U}⊋U$, $\overline{U}$ is a dependent modulo I.

Proof. 

On one hand, if U is a dependent modulo I, by Definition 1, $I\cap k\left[U\right]\ne \left\{0\right\}$. Since $\overline{U}⊋U$, $I\cap k\left[\overline{U}\right]\ne \left\{0\right\}$. So $\overline{U}$ is a dependent modulo I.

On the other hand, if U is independent modulo I, $I\cap k\left[U\right]=\left\{0\right\}$, i.e., for each $f\in I$, $f\notin$ $k\left[U\right]$. According to Proposition 2, $L{M}_{{\prec }_{U,V}}\left(f\right)\notin T\left(U\right)$. Thus $L{M}_{{\prec }_{U,V}}\left(I\right)\cap T\left(U\right)=\varnothing$, and U is strongly independent modulo I, with respect to ${\prec }_{U,V}$. Since U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$, we have $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(\overline{U}\right)\ne \varnothing$. Then $L{M}_{{\prec }_{U,V}}\left(I\right)\cap T\left(\overline{U}\right)\ne \varnothing$ since $F\subset I$, and this means U is maximally strongly independent modulo I, with respect to ${\prec }_{U,V}$. By Lemma 1, U is maximally independent modulo I, and $\overline{U}$ is a dependent modulo I, since $\overline{U}⊋U$.    □

Definition 5 and Proposition 4 provide a way to compute an MSQIS.

Definition 5. 

Let $F\subset k\left[X\right]$ be a polynomial set. The label  of $x\in X$ on F is defined as

$$lb{l}_{x,F}:=\{f\in F|f\notin k[X\setminus x]\}.$$

Moreover, the label list $[lb{l}_{x_1,F},lb{l}_{x_2,F},\cdots ,lb{l}_{x_n,F}]$ is written as $Lb{l}_{X,F}$.

Remark 1. 

We propose a new kind of indeterminate order on $Lb{l}_{X,F}$: $x_i\succ x_j$ if $♯lb{l}_{x_i,F}<♯lb{l}_{x_j,F}$ for $1\le i,j\le n$. Throughout this paper, as long as F is updated, $Lb{l}_{X,F}$ will be modified, and the indeterminates of X will be reordered in descending order. Experiments show that such indeterminate order can improve the efficiency of the main algorithm (Algorithm 1).

Algorithm 1 (ComputeUandG)

Input:  $I:=⟨F_0⟩\subset k\left[X\right]$ a positive-dimensional ideal Output:  U a maximal independent set modulo I                 G the reduced Gröbner basis of I w.r.t. ${\prec }_{U,V}^{grl}$$(V:=X\setminus U)$ Line 1:  $F_{all}:=F_0$, $U:=MSQIS\_Alg1\left(F_{all}\right)$ and $V:=X\setminus U$. Line 2: $(G,P):=Update(\varnothing ,\varnothing ,F_0,{\prec }_{U,V}^{grl})$, $P_p:=SelCritPairs(P,{\prec }_{U,V}^{grl})$ and $P_r:=P\setminus P_p$. Line 3: while  $P_p\ne \varnothing$  do Line 4:             $H:=F_{4}Reduce(P_p,G,{\prec }_{U,V}^{grl})$, and $F_{all}:=F_{all}\cup H$. Line 5:             if  $H\cap k\left[U\right]\ne \varnothing$  then Line 6:                  $U:=MSQIS\_Alg2(F_{all},U)$. Line 7:                  if  $U=\varnothing$  then Line 8:                      $U:=MSQIS\_Alg1\left(F_{all}\right)$, $V:=X\setminus U$, and turn back to Line 2. Line 9:             else Line 10:                   $(G,P):=Update(G,P_r,H,{\prec }_{U,V}^{grl})$, $P_p:=SelCritPairs(P,{\prec }_{U,V}^{grl})$ and                                  $P_r:=P\setminus P_p$. Line 11: return  U, G.

Proposition 4. 

Let $F\subset k\left[X\right]$ be a polynomial set, $U\subset X$, and $V:=X\setminus U$. If ${\displaystyle \bigcup _{v\in V}}lb{l}_{v,F}=F$ and ${\displaystyle \bigcup _{\tilde{v}\in \tilde{V}}}lb{l}_{\tilde{v},F}\ne F$ for any $\tilde{V}⊊V$, then for each block monomial order ${\prec }_{U,V}$ on $T\left(X\right)$ with $U\ll V$, U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$.

Proof. 

Since ${\displaystyle \bigcup _{v\in V}}lb{l}_{v,F}=F$, for each $f\in F$, there exists $v\in V$, such that $f\in lb{l}_{v,F}$. So $f\notin k[X\setminus v]$ according to Definition 5, i.e., $F\cap k\left[U\right]=\varnothing$. According to Proposition 2, $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$. Thus U is an SQIS modulo F, with respect to ${\prec }_{U,V}$. Moreover, let $\tilde{V}:=X\setminus \tilde{U}$ for each $\tilde{U}⊋U$. Then $\tilde{V}⊊V$. Since ${\displaystyle \bigcup _{\tilde{v}\in \tilde{V}}}lb{l}_{\tilde{v},F}\ne F$, there exists $f\in F$, such that $f\notin lb{l}_{\tilde{v},F}$ for each $\tilde{v}\in \tilde{V}$. So $f\in k\left[\tilde{U}\right]$ and $F\cap k\left[\tilde{U}\right]\ne \varnothing$. Thus, by Proposition 2, $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(\tilde{U}\right)\ne \varnothing$, and U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$.    □

Proposition 5 ensures the construction of the main algorithm (Algorithm 1).

Proposition 5. 

Let $F_1,F_2\subset k\left[X\right]$ be polynomial sets with $F_2⊋F_1$ and $H:=F_2\setminus F_1$. Let $U\subset X$, $V:=X\setminus U$ and ${\prec }_{U,V}$ be a block monomial order on $T\left(X\right)$. Suppose U is an SQIS modulo $F_1$, with respect to ${\prec }_{U,V}$. Then

(1) U is not an SQIS modulo $F_2$, with respect to ${\prec }_{U,V}$ iff $H\cap k\left[U\right]\ne \varnothing$, which is called the altering condition.

(2) If the altering condition is true, then for any block monomial order, ${\prec }_{U,V}^{\prime }\ne {\prec }_{U,V}$, U is not an MSQIS modulo $F_2$, with respect to ${\prec }_{U,V}^{\prime }$.

Proof. (1) Necessity: 

Since U is an SQIS modulo $F_1$, with respect to ${\prec }_{U,V}$ but not an SQIS modulo $F_2$, with respect to ${\prec }_{U,V}$, by Definition 4, $L{M}_{{\prec }_{U,V}}\left(F_1\right)\cap T\left(U\right)=\varnothing$ and $L{M}_{{\prec }_{U,V}}\left(F_2\right)\cap T\left(U\right)\ne \varnothing$. Since $H:=F_2\setminus F_1$, we have $L{M}_{{\prec }_{U,V}}\left(H\right)\cap T\left(U\right)\ne \varnothing$. According to Proposition 2, $H\cap k\left[U\right]\ne \varnothing$.

Sufficiency: Since $H\cap k\left[U\right]\ne \varnothing$, we have $L{M}_{{\prec }_{U,V}}\left(H\right)\cap T\left(U\right)\ne \varnothing$ by Proposition 2, and $L{M}_{{\prec }_{U,V}}\left(F_2\right)\cap T\left(U\right)\ne \varnothing$ since $F_2\supset H$. Thus U is not an SQIS modulo $F_2$, with respect to ${\prec }_{U,V}$.

(2) Since the altering condition holds, U is not an SQIS modulo $F_2$, with respect to ${\prec }_{U,V}$ from (1). By Proposition 3, U is not an SQIS modulo $F_2$, with respect to ${\prec }_{U,V}^{\prime }$. So according to Definition 4, U is certainly not an MSQIS modulo $F_2$, with respect to ${\prec }_{U,V}^{\prime }$.    □

3. The Idea and Algorithms

In this section, we present an algorithm for computing a maximal independent set U modulo a positive-dimensional ideal, $I:=⟨F⟩\subset k\left[X\right]$, and a Gröbner basis of I with respect to a block monomial order, ${\prec }_{U,V}$, simultaneously.

3.1. The Idea

Let us describe the procedure of the algorithm briefly.

Step 1: Compute $U\subset X$, such that there exists a block monomial order, ${\prec }_{U,V}$, with respect to which U is an MSQIS modulo F. Then,

(a)

By Definition 4 and Proposition 3, U is an SQIS modulo F, with respect to ${\prec }_{U,V}^{grl}$;

(b)

By Theorem 1, any $\overline{U}⊋U$ is a dependent modulo I;

Step 2: Execute the Buchberger operations on F with respect to ${\prec }_{U,V}^{grl}$. Let $S:=\left\{\right(f,g\left)\right|f,g\in F,f\ne g\}$ be the critical pair (the critical pair of two polynomials, f and g, is an element $(f,g)$ of $k\left[X\right]\times k\left[X\right]$) set of F. Then select a prior subset, $\tilde{S}$, of S with a certain strategy and compute the normal form set H of the S-polynomials of all $s\in \tilde{S}$ modulo F, with respect to ${\prec }_{U,V}^{grl}$.

Step 3: Let $F^{\prime }:=F\cup H$ and check whether the altering condition $H\cap k\left[U\right]\ne \varnothing$ is true. If it is not true, then go to Step 2 and continue to execute the Buchberger operations on $F^{\prime }$ with respect to ${\prec }_{U,V}^{grl}$. Otherwise, compute $U^{\prime }\subset X$, such that there exists a block monomial order, ${\prec }_{U^{\prime },V^{\prime }}$, with respect to which $U^{\prime }$ is an MSQIS modulo $F^{\prime }$. In this case, (a) and (b) hold for $U^{\prime }$, $F^{\prime }$, and ${\prec }_{U^{\prime },V^{\prime }}^{grl}$. Then go to Step 2 and switch to execute the Buchberger operations on F with respect to ${\prec }_{U^{\prime },V^{\prime }}^{grl}$. U and $U^{\prime }$ are called the “old” and the “new” MSQISs, respectively.

If the algorithm terminates in finite steps, then we denote by $\widehat{U}$, $\widehat{F}$ and ${\prec }_{\widehat{U},\widehat{V}}^{grl}$ the final indeterminate set, polynomial set, and block monomial order, respectively. It can be proved that $\widehat{U}$ is a maximal independent set modulo I, and $\widehat{F}$ is a Gröbner basis of I with respect to ${\prec }_{\widehat{U},\widehat{V}}^{grl}$. The algorithm is referred to as “the main algorithm”.

The above procedure shows that the idea of the main algorithm is adjusting MSQISs continuously according to the results of the S-polynomial calculations.

Note that the key issue of the main algorithm is computing U$\left(U^{\prime }\right)\subset X$, such that there exists a block monomial order, ${\prec }_{U,V}$ $\left({\prec }_{U^{\prime },V^{\prime }}\right)$, with respect to which U$\left(U^{\prime }\right)$ is an MSQIS modulo F$\left(F^{\prime }\right)$. Next we give an algorithm for resolving this issue.

3.2. Computing an MSQIS U with Respect to Each Block Monomial Order with $U\ll V$

According to Proposition 4, we present Algorithm 2 for computing U, which is an MSQIS modulo F, with respect to each block monomial order with $U\ll V$.

Algorithm 2 (MSQIS_Alg 1)

Input: $F\subset I$ a polynomial set, where $I\subset k\left[X\right]$ is a positive-dimensional ideal Output: $U\subset X$ an MSQIS modulo F w.r.t. each block monomial order with $U\ll V$ Line 1: Compute $Lb{l}_{X,F}$, and reorder X in descending order and $Lb{l}_{X,F}$ on the order of X. Line 2: $V:=\varnothing$, and $Vlbl:=\varnothing$. Line 3: while  $Vlbl\ne F$  do Line 4:              $V:=V\cup \left\{X_1\right\}$, $X:=X\setminus \left\{X_1\right\}$ and $Vlbl:=Vlbl\cup {\left(Lb{l}_{X,F}\right)}_1$. Line 5:              Remove ${\left(Lb{l}_{X,F}\right)}_1$ from $Lb{l}_{X,F}$. Line 6: $d:=♯V$. Line 7: for i from $d-1$ to 1 by $-1$ do Line 8:      if ${\displaystyle \bigcup _{v\in V,v\ne V_i}}lb{l}_{v,F}=F$ then Line 9:             $V:=V\setminus \left\{V_i\right\}$, and turn back to Line 6. Line 10: $U:=X\setminus V$, and return U.

Theorem 2. 

Let $I\subset k\left[X\right]$ be a positive-dimensional ideal and $F\subset I$ be a polynomial set. Then Algorithm 2 computes an MSQIS U modulo F, with respect to each block monomial order with $U\ll V$.

Proof. Termination: 

According to Definition 5, ${\displaystyle \bigcup _{x\in X}}lb{l}_{x,F}\subset F$. Furthermore, for each $f\in F$, since $f\ne 0$, there exists $x_i$ such that f contains $x_i$, i.e., $f\in lb{l}_{x_i,F}$. So ${\displaystyle \bigcup _{x\in X}}lb{l}_{x,F}=F$, and Lines 3–5 terminate. Moreover, the repetitive running of Line 9 gives rise to a strictly descending chain of sets, so the algorithm terminates.

Correctness: If the algorithm terminates, we have ${\displaystyle \bigcup _{v\in V}}lb{l}_{v,F}=F$ and ${\displaystyle \bigcup _{v\in V,v\ne V_i}}lb{l}_{v,F}\ne F$, $i=1,2,\cdots ,♯V$. We prove that $V\ne X$ by contradiction. Suppose that $V=X$. Then ${\displaystyle \bigcup _{x\in X,x\ne x_i}}lb{l}_{x,F}\ne F$, $i=1,2,\cdots ,n$. So for each $x_i\in X$, there exists $f_i\in F$, such that $f_i\in lb{l}_{x_i,F}$ and $f_i\notin lb{l}_{x_j,F}$ with $j\ne i$. This means that $f_i\in k\left[x_i\right]$, $i=1,2,\cdots ,n$. Then I is zero-dimensional, which contradicts with the condition, and thus $V\ne X$, and $U\ne \varnothing$. So according to Proposition 4, U is an MSQIS modulo F, with respect to each block monomial order with $U\ll V$.    □

Although Algorithm 2 computes an MSQIS, U, with respect to each block monomial order with $U\ll V$, which certainly means that there exists ${\prec }_{U,V}$, with respect to which U is an MSQIS; experiments show that the cardinality of the “new” MSQIS obtained with Algorithm 2 may be not smaller than the “old” one, and it affects the efficiency of the main algorithm: Regarding the procedure of the main algorithm, it contains two aspects of calculations: a series of MSQIS’s and corresponding S-polynomial computations. Moreover, experiments show that the cost of the latter is dominant. By Theorem 1, if U is verified to be an MSQIS with respect to ${\prec }_{U,V}$, then we exclude all the dependent set $\overline{U}⊋U$ modulo I. In this sense, to avoid redundant S-polynomial computations corresponding to $\overline{U}$, MSQISs with smaller cardinalities are preferred. To make up for such defect of Algorithm 2, we present another algorithm for computing a new MSQIS with strictly smaller cardinality than the old MSQIS.

3.3. Computing a New MSQIS with Smaller Cardinality than the Old One

Let $U_0$ be an MSQIS modulo $F_0$, with respect to ${\prec }_{U_0,V_0}$, H be the normal form set obtained by executing the Buchberger operations on $F_0$ with respect to ${\prec }_{U_0,V_0}^{grl}$, and $F:=F_0\cup H$. Suppose that the altering condition $H\cap k\left[U_0\right]\ne \varnothing$ holds. Now we think about computing a new MSQIS from a different point of view. Let $U_{max}:=\{\tilde{U}\subset U_0|♯\tilde{U}=♯U_0-1\}$. Then we try to find U from $U_{max}$, such that there exists ${\prec }_{U,V}$, with respect to which U is an MSQIS modulo F.

Proposition 6. 

Let $F\subset k\left[X\right]$ be a polynomial set, $U\subset X$, $V:=X\setminus U$, and $F\cap k\left[U\right]=\varnothing$. There exists ${\prec }_{U,V}$ on $T\left(X\right)$, such that U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$, if there exists ${\prec }_V$ on $T\left(V\right)$, such that $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$ for each $v\in V$.

Proof. Necessity: 

Since U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$, by Definition 4, for each $v\in V$, we have $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$ and $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T(U,v)\ne \varnothing$, i.e., there exists $f\in F$, such that $L{M}_{{\prec }_{U,V}}\left(f\right)\notin T\left(U\right)$ and $L{M}_{{\prec }_{U,V}}\left(f\right)\in T(U,v)$. According to Proposition 2, we have $f\notin k\left[U\right]$ and $f\in k[U,v]$. Thus $f_V\in k\left[v\right]$. Let ${\prec }_V$ be the restriction of ${\prec }_{U,V}$ to $T\left(V\right)$. Then $L{M}_{{\prec }_V}\left(f_V\right)\in T\left(v\right)$, i.e., $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$ for each $v\in V$.

Sufficiency: Let ${\prec }_{U,V}$ be a block monomial order of which the restriction to $T\left(V\right)$ is ${\prec }_V$. Since $F\cap k\left[U\right]=\varnothing$, $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T\left(U\right)=\varnothing$ by Proposition 2. So U is an SQIS modulo F, with respect to ${\prec }_{U,V}$. Since $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$, there exists $f_V\in F_V$, such that $L{M}_{{\prec }_V}\left(f_V\right)\in T\left(v\right)$ for each $v\in V$. We write $L{M}_{{\prec }_{U,V}}\left(f\right)=t_V{t}_U$, where $t_V\in T\left(V\right)$, $t_U\in T\left(U\right)$. By Definition 3, $t_V=L{M}_{{\prec }_V}\left(f_V\right)$. Thus $t_V\in T\left(v\right)$, and $L{M}_{{\prec }_{U,V}}\left(f\right)\in T(U,v)$, i.e., $L{M}_{{\prec }_{U,V}}\left(F\right)\cap T(U,v)\ne \varnothing$ for each $v\in V$. So U is an MSQIS modulo F, with respect to ${\prec }_{U,V}$.    □

According to Proposition 6, determining the existence of ${\prec }_{U,V}$, with respect to which U is an MSQIS, is transformed to determining the existence of ${\prec }_V$, such that $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$ for each $v\in V$. It is sufficient to select $n-r$ polynomials, $\{f_{V,1},f_{V,2},\cdots ,f_{V,n-r}\}$, from $F_V$, such that $L{M}_{{\prec }_V}\left(f_{V,i}\right)={v_i}^{a_i}$, i.e., $n-r$ polynomials of the form

$$\begin{array}{cc}\hfill f_{V,1}& :=c_1\left(U\right)·v_1^{a_1}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{\prec }_V,\hfill \\ \hfill f_{V,2}& :=c_2\left(U\right)·v_2^{a_2}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{\prec }_V,\hfill \\ \hfill & ⋮\hfill \\ \hfill f_{V,n-r}& :=c_{n-r}\left(U\right)·v_{n-r}^{a_{n-r}}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{\prec }_V,\hfill \end{array}$$

(2)

where $c_i\left(U\right)\in k\left[U\right]$ and $a_i\in {\mathbb{N}}_{+}$, $i=1,2,\cdots ,n-r$. This implies two things: (1) There is a power of $v_i$ in $f_{V,i}$. (2) The power of $v_i$ is greater than other terms of $f_{V,i}$ with respect to ${\prec }_V$.

As for (2), to compare two terms, the coefficients are useless, so we “assign coefficients to 1”. Moreover, the term which is a factor of another is unnecessary to be compared. So we “drop the term which is a factor of another”. Let $f:=u^{2}v+u^{3}v$. Then $u^{2}v$ and $u^{3}v$ are different monomials. However, $f_V:=(u^2+u^3)v$, and $u^{2}v$ and $u^{3}v$ are similar monomials, and they need to be collected when they are regarded as polynomials in $k\left(U\right)\left[V\right]$. All these are called the simplification of $F_V$: For $f_V\in F_V$, we collect similar monomials, assign coefficients to 1, and drop the monomials that are factors of others. Let $V:=[v_1,v_2,\cdots ,v_{n-r}]$ and $\gamma :=({\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_{n-r})$. Then $V^{\gamma }$ is defined as $v_1^{{\gamma }_1}{v}_2^{{\gamma }_2}\cdots v_{n-r}^{{\gamma }_{n-r}}$. We check the existence of ${\prec }_V$ in two steps. (1) Check whether there exists $\tilde{F}\subset F_V$ (maybe not unique) consisting of $n-r$ polynomials of the form

$$f_{V,i}=V^{\alpha \left(i\right)}+\sum _{\beta }{V}^{\beta },$$

(3)

where $\alpha \left(i\right)=m_i$$e_i$, $m_i\in {\mathbb{N}}_{+}$, and $\beta \in {\left(\mathbb{N}\right)}^{n-r}$, $i=1,2,\cdots ,n-r$. If $\tilde{F}$ does not exist, there does not exist ${\prec }_V$, such that $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$ for each $v\in V$. If $\tilde{F}$ exists, we go to the next step. 2) For each $\tilde{F}$, we check whether there exists ${\prec }_V$, such that $L{M}_{{\prec }_V}\left(f_{V,i}\right)=V^{\alpha \left(i\right)}$, $i=1,2,\cdots ,$ $n-r$. If ${\prec }_V$ exists, then $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$ for each $v\in V$. If ${\prec }_V$ does not exist for all $\tilde{F}$, then there does not exist ${\prec }_V$, such that $L{M}_{{\prec }_V}\left(F_V\right)\cap T\left(v\right)\ne \varnothing$ for each $v\in V$.

However, it is difficult to check the existence of such a monomial order. According to the relation between a monomial order and a regular matrix in [15], we give a sufficient condition for that.

Proposition 7. 

Let $U\subset X$, $V:=X\setminus U:=[v_1,v_2,\cdots ,v_{n-r}]$, $\tilde{F}\subset k\left(U\right)\left[V\right]$ be a polynomial set of the form (3) and  w  $\in {\left({\mathbb{R}}_{+}\right)}^{n-r}$ be a vector. Define

$$\begin{array}{c}\hfill C_{\tilde{F}}:=\{\mathbf{w}\in {\left({\mathbb{R}}_{+}\right)}^{n-r}|(\alpha \left(i\right)-\beta )·\mathbf{w}>0,i=1,2,\cdots ,n-r\}.\end{array}$$

If $C_{\tilde{F}}\ne \varnothing$, then there exists ${\prec }_V$ on $T\left(V\right)$, such that $L{M}_{{\prec }_V}\left(f_{V,i}\right)=V^{\alpha \left(i\right)}$, $i=1,2,$ $\cdots ,n-r$.

Proof. 

According to [15], a regular $n\times n$ matrix A over ${\mathbb{R}}_{+}$ determines a monomial order ≺ as follows. Each row $A_i$ of A is actually a weight vector, $i=1,2,\cdots ,n$, and monomials $t_1\prec t_2$ are determined by comparing the $A_i$ degrees of them until the first inequality is found. So if $C_{\tilde{F}}\ne \varnothing$, there exists ${\prec }_V$ on $T\left(V\right)$ determined by an $(n-r)\times (n-r)$ matrix of which the first row is a vector w $\in C_{\tilde{F}}$, such that $L{M}_{{\prec }_V}\left(f_{V,i}\right)=V^{\alpha \left(i\right)}$, $i=1,2,\cdots ,n-r$.    □

Moreover, $C_{\tilde{F}}$ is regarded as the feasible region of a linear programming problem, and the work is done at a small cost. Details are referred to in [16].

We next present Algorithm 3 for seeking U from $U_{max}$, such that there exists ${\prec }_{U,V}$, with respect to which U is an MSQIS modulo F.

Algorithm 3 (MSQIS_Alg 2)

Input: $F\subset k\left[X\right]$ a polynomial set            $U_0\subset X$ an indeterminate set Output: $U\ne \varnothing$ there exists ${\prec }_{U,V}$, w.r.t. which U is an MSQIS modulo F or               ∅ if we find none Line 1: $U_{max}:=\{\tilde{U}\subset U_0|♯\tilde{U}=♯U_0-1\}$. Line 2: for i from 1 to $♯U_{max}$ do Line 3:       if $F\cap k\left[{\left(U_{max}\right)}_i\right]=\varnothing$ then Line 4:           $V:=X\setminus {\left(U_{max}\right)}_i$, and for each $f_V\in F_V$, collect similar monomials, assign                        coefficients to 1 and drop the monomials that are factors of others. Line 5:           $F_{\left(2\right)}:=\{\tilde{F}\subset F_V|\tilde{F}\mathrm{is}\mathrm{a}\mathrm{polynomial}\mathrm{set}\mathrm{of}\mathrm{the}\mathrm{form}\left(3\right)\}$. Line 6:           if $F_{\left(2\right)}\ne \varnothing$ then Line 7:               $\tilde{F}:={\left(F_{\left(2\right)}\right)}_1$, and $F_{\left(2\right)}:=F_{\left(2\right)}\setminus \{\tilde{F}\}$. Line 8:               if $C_{\tilde{F}}\ne \varnothing$ (through linear programming theory) then Line 9:                   return ${\left(U_{max}\right)}_i$. Line 10:              else Line 11:                 Turn back to Line 6. Line 12: return ∅.

The case that Algorithm 3 fails to obtain an MSQIS will be illustrated by Example 1. Furthermore, the finiteness of $U_{max}$ and $F_{\left(2\right)}$ plus the linear programming theory guarantee the termination of Algorithm 3, and the correctness of it follows from Proposition 7.

Example 1. 

Consider the polynomial set $F_0=\{f_1=x{y}^2-x{z}^2,f_2=yz-uv,f_3=y^{2}w+zw,$ $f_4=u^{2}v-u{w}^2\}\subset k[x,y,z,u,v,w]$.

Let ${\prec }_{U_0,V_0}:={\prec }_{[u,z,y],[x,w,v]}$ with ${\prec }_{U_0}=grlex(u,z,y)$ and ${\prec }_{V_0}=plex(x,w,v)$. Then $U_0=\{u,z,y\}$ is an MSQIS modulo $F_0$, with respect to ${\prec }_{U_0,V_0}$.

We start to execute the Buchberger operations on $F_0$ with respect to ${\prec }_{[u,z,y],[x,w,v]}^{grl}$. The critical pair set $S:=\left\{(f_i,f_j)\right|1\le i,j\le 4,i<j\}$. Let $\tilde{S}:=\left\{(f_3,f_4)\right\}$ be a prior subset of S. The normal form of the S-polynomial of the $(f_3,f_4)$ modulo $F_0$, with respect to ${\prec }_{[u,z,y],[x,w,v]}^{grl}$ is $f_5=uz{y}^3+u{z}^{2}y$, and $H:=\left\{f_5\right\}$.

Let $F:=F_0\cup H:=\{f_1,f_2,\cdots ,f_5\}$. Since $H\cap k\left[U_0\right]\ne \varnothing$, we try to find U from $U_{max}$ with Algorithm 3, where $U_{max}:=\{\{z,y\},\{u,y\},\{u,z\}\}$, such that there exists ${\prec }_{U,V}$, with respect to which U is an MSQIS modulo F.

Let $U:=\{z,y\}$. Then $V:=\{x,w,v,u\}$, and the simplified $F_{[x,w,v,u]}:=\{x,vu,w,v{u}^2+w^{2}u,u\}$. Note that there does not exist a power of v in $F_{[x,w,v,u]}$. Thus there do not exist four polynomials of the form (3), and we need to check the cases when $U:=\{u,y\}$ and $\{u,z\}$. These two cases are similar to the case when $U:=\{z,y\}$, and there do not exist four polynomials of the form (3), either.

In this case, Algorithm 3 fails to return $U\ne \varnothing$, such that there exists ${\prec }_{U,V}$, with respect to which U is an MSQIS modulo F.

Example 1 shows that Algorithm 3 does not always work, even though it returns a new MSQIS with smaller cardinality than the old if it succeeds. We can now describe the alternative application of Algorithms 2 and 3 to obtain a positive MSQIS. At the start of the main algorithm, since we have no MSQIS, we need to compute one with Algorithm 2. In other cases, we will try with Algorithm 3 at first if the altering condition holds, and then we can compute an MSQIS with Algorithm 2 if Algorithm 3 fails. In 96% of more than 100 examples we experimented with, MSQISs were obtained with Algorithm 3. In the rest, Algorithm 3 failed, and MSQISs were computed with Algorithm 2.

Furthermore, we need to make some explanation for the choice of $U_{max}$. First, since $H\cap k\left[U_0\right]\ne \varnothing$, $U_0$ is dependent modulo I, this means that at least one indeterminate of $U_0$ has to be moved into $V_0$. So we give priority to $U⊊U_0$ rather than other indeterminate sets with smaller cardinalities than $U_0$. Second, generally speaking, the greater the cardinality of U is, the greater the possibility of U becoming an MSQIS is, so we only check the proper subsets of maximal cardinality of $U_0$.

3.4. Strategy for Selecting a Prior Subset of the Critical Pair Set

Weispfenning proposed the normal strategy for selecting prior critical pairs from the critical pair set for the normal form computation in [3]. Similarly, we present a strategy for our problem, in order to compute fewer S-polynomials to arrive at the altering condition and, furthermore, reduce the number of critical pairs calculated before we obtain a maximal independent set.

From the procedure of the main algorithm, if the altering condition $H\cap k\left[U\right]\ne \varnothing$ is true, we will compute a new MSQIS, $U^{\prime }$, and switch to execute the Buchberger operations with respect to ${\prec }_{U^{\prime },V^{\prime }}^{grl}$. So the Buchberger operations with respect to one block monomial order can be regarded as eliminating the indeterminates of V. To compute the S-polynomial of $(f,g)$, f and g will be multiplied by s and t, respectively. Generally speaking, the smaller the degrees of s and t with respect to V are, the sooner the elements of V will be eliminated. To accelerate the elimination, we give priority to such critical pairs. Note that in Algorithm 4, for a monomial m and $U\subset X$, $Coeff(m,U)$ returns the coefficient of m with respect to U and $Degree\left(m\right)$ computes the total degree of m with respect to X.

Algorithm 4 (SelCritPairs)

Input: P a critical pair set             ${\prec }_{U,V}$ a block monomial order Output: $P_{pri}$ a critical pair set Line 1: $DegV:=\left[\right]$. Line 2: for i from 1 to $♯P$ do Line 3:       $LcmP:=lcm(L{M}_{{\prec }_{U,V}}\left({\left(P_i\right)}_1\right),L{M}_{{\prec }_{U,V}}\left({\left(P_i\right)}_2\right))$. Line 4:       $q_1:=LcmP/L{M}_{{\prec }_{U,V}}\left({\left(P_i\right)}_1\right)$, $q_2:=LcmP/L{M}_{{\prec }_{U,V}}\left({\left(P_i\right)}_2\right)$. Line 5:       $DegP:=Degree\left(Coeff(q_1,U)\right)+Degree\left(Coeff(q_2,U)\right)$. Line 6:       Append $DegP$ to $DegV$. Line 7: $P_{pri}:=\{P_i\in P|{\left(DegV\right)}_i=min\left(DegV\right)\}$. Line 8: return $P_{pri}$.

Termination and correctness of Algorithm 4 are obvious, and we omit the proofs.

3.5. The Main Algorithm

Based on Algorithms 2–4, we present the main algorithm (Algorithm 1). We apply into it the Buchberger Criteria [3] and the $F_4$ technique [17]. Note that in Algorithm 1, the function $Update(G,P,H,\prec )$ plays a role in updating the ideal basis G and the critical pair set P with respect to a polynomial set H and a monomial order ≺. It also leads to the fact that Algorithm 1 computes the reduced Gröbner basis of I [3]. The function $F_{4}Reduce(P,G,\prec )$ computes the normal form set of a critical pair set P modulo G, with respect to ≺.

Theorem 3. 

Let $I\subset k\left[X\right]$ be a positive-dimensional ideal. Then Algorithm 1 computes a maximal independent set U modulo I, and a Gröbner basis of I with respect to ${\prec }_{U,V}^{grl}$.

Proof. Termination: 

According to Algorithms 2–4, the functions $Update$ and $F_{4}Reduce$ all terminate, and we only need to prove the termination of the while loop. First we consider Lines 6–8. Let $Y:=\{\tilde{X}\subset X|\tilde{X}\ne \varnothing \}$. If $H\cap k\left[U\right]\ne \varnothing$, then Lines 6–8 run, and they compute $U^{\prime }$ (distinguishing from U), such that there exists ${\prec }_{U^{\prime },V^{\prime }}$, with respect to which $U^{\prime }$ is an MSQIS modulo $F_{all}$. Furthermore, according to (2) of Proposition 5, U is not an MSQIS modulo $F_{all}$, with respect to any block monomial order with $U\ll V$, so $U^{\prime }\ne U$. Denote $Y_1:=$ $Y\setminus \left\{U\right\}$. The repetitive running of Lines 6–8 gives rise to a strictly descending chain of sets: $Y⊋Y_1⊋Y_2⊋\cdots$. Since Y is finite, Lines 6–8 terminate. Suppose that Lines 6–8 terminate with $U:=\widehat{U}$ and $V:=\widehat{V}$. Then after they terminate, the while loop is just the execution of the Buchberger operations on $F_0$ with respect to ${\prec }_{\widehat{U},\widehat{V}}^{grl}$. Algorithm 1 terminates since Buchberger’s algorithm terminates.

Correctness: If the algorithm terminates, it returns a Gröbner basis G of I with respect to ${\prec }_{\widehat{U},\widehat{V}}^{grl}$, where $\widehat{U}$ is an SQIS modulo $F_{all}$, with respect to ${\prec }_{\widehat{U},\widehat{V}}^{grl}$ according to Proposition 3. Since $F_{all}\supset G$, $F_{all}$ is also a Gröbner basis of I with respect to ${\prec }_{\widehat{U},\widehat{V}}^{grl}$. According to Proposition 1, $\widehat{U}$ is strongly independent modulo I, with respect to ${\prec }_{\widehat{U},\widehat{V}}^{grl}$. Thus $\widehat{U}$ is independent modulo I. Furthermore, according to Theorem 1, any $\overline{U}⊋\widehat{U}$ is a dependent modulo I, so $\widehat{U}$ is maximally independent modulo I.    □

The cyclic 4 problem illustrates Algorithm 1 and the advantage of Algorithm 3.

Example 2. 

Consider the ideal $I:=⟨F_0⟩\subset k[a,b,c,d]$, where $F_0=\{f_1=a+b+c+d,f_2=ab+bc+ad+cd,f_3=abc+abd+acd+bcd,f_4=abcd-1\}$.

Initialization:

$F_{all}=\{f_1,f_2,f_3,f_4\}$, $U=[c,b,a]$ and $V=\left[d\right]$.

$G=\left\{f_1\right\}$, $P_p=P=\{\{f_1,f_2\},\{f_1,f_3\},\{f_1,f_4\}\}$ and $P_r=\varnothing$.

The while loop runs for the first time:

$H=\{f_5=a^{2}bc+a{b}^{2}c+ab{c}^2+1,f_6=a^{2}b+a^{2}c+a{b}^2+2abc+a{c}^2+b^{2}c+b{c}^2,f_7=a^2+2ac+c^2\}$, and $F_{all}=\{f_1,f_2,\cdots ,f_7\}$. Since $H\cap k[c,b,a]\ne \varnothing$, $MSQIS\_Alg2(F_{all},[c,b,a])$ returns $U=[b,a]$. Then $V=[d,c]$. Back to Line 2, $G=\left\{f_1\right\}$, $P_p=P=\{\{f_1,f_2\},\{f_1,f_3\},$ $\{f_1,f_4\}\}$ and $P_r=\varnothing$.

If we remove Algorithm 3, $MSQIS\_Alg1$ will return $U=[d,b,a]$, $V=\left[c\right]$. However, since $\{b,a\}$ is an MSQIS modulo $F_{all}$, with respect to ${\prec }_{[b,a],[d,c]}^{grl}$, by Theorem 1, $\{d,b,a\}$ is dependent modulo I, and the computation of the S-polynomials with respect to ${\prec }_{[d,b,a],\left[c\right]}^{grl}$ will be redundant.

The while loop runs for the second time:

$H=\{f_5,f_6,f_7\}$ and $F_{all}$ remains unchanged. Since $H\cap k[b,a]=\varnothing$, Line 10 returns $G=\{f_1,f_7\}$, $P_p=P=\{\{f_5,f_6\},\{f_6,f_7\}\}$ and $P_r=\varnothing$.

The while loop runs for the third time:

$H=\{f_8=-a^4+a^{3}b-a^{3}c+a^2{b}^2+a^{2}bc-1,f_9=-a^3-a^{2}c+a{b}^2+b^{2}c\}$, and $F_{all}=$$\{f_1,f_2,\cdots ,f_9\}$. Since $H\cap k[b,a]=\varnothing$, Line 10 returns $G=\{f_1,f_7,f_8,f_9\}$, $P=\{\{f_7,f_8\},\{f_7,f_9\},\{f_8,f_9\}\}$, $P_p=\left\{\{f_8,f_9\}\right\}$ and $P_r=\{\{f_7,f_8\},\{f_7,f_9\}\}$.

The while loop runs for the fourth time:

$H=\{f_{10}=a^3{b}^2+a^2{b}^3-a-b\}$, and $F_{all}=\{f_1,f_2,\cdots ,f_{10}\}$. Since $H\cap k[b,a]\ne \varnothing$, $MSQIS\_Alg2(F_{all},[b,a])$ returns $U=\left[a\right]$; then $V=[d,b,c]$. We go back to Line 2; $G=\left\{f_1\right\}$, $P=\{\{f_1,f_2\},\{f_1,f_3\},\{f_1,f_4\}\}$, $P_p=\left\{\{f_1,f_2\}\right\}$ and $P_r=\{\{f_1,f_3\},\{f_1,f_4\}\}$.

In fact, $\left\{a\right\}$ is a maximal independent set modulo I. We skip remaining calculations and give the final results. After the while loop runs for the 10th time, $P_p=\varnothing$ and the algorithm terminates. It returns $U=\left\{a\right\}$ as a maximal independent set modulo I, and $G=\{a+b+c+d,a^2+2ac+c^2,-a^4+a^{3}b-a^{3}c+a^2{b}^2+a^{2}bc-1,-a^3-a^{2}c+a{b}^2+b^{2}c,a^5+a^{4}c-a-c\}$ as the reduced Gröbner basis of I with respect to ${\prec }_{\left[a\right],[d,b,c]}^{grl}$.

4. Experiments

We compared Algorithm 1 with the direct method and the W-W method, and both of them used Weispfenning’s algorithm to compute a maximal independent set from a Gröbner basis with respect to a graded reverse lexicographic monomial order. It should be mentioned that the Buchberger Criteria and the $F_4$ technique were also applied to these two methods. Table 1 shows the final maximal independent set (U), the numbers of critical pairs (Number), and the CPU times (Time) of 18 examples. All examples and the names of them are referred to in [18]. In the “U” column, for each example, the first indeterminate set was returned by the direct and the W-W methods, and the second indeterminate set was computed by our method. Experiments were conducted on a Dell notebook, which was configured with Intel Core i5-8250U, with 8 G of memory. The development environment was Windows 10 Professional, Maple 2020. The code is available upon request.

Table 1. Comparison with the direct method and the W-W method.

Example	U	Number	Time (s)
	Direct/W-W, Ours	Direct, W-W, Ours	Direct, W-W, Ours
Chemistry	$\{j,h,e\}$, $\{b,f,g\}$	168, 202, 81	7542.234, 354.578, 180.375
Sturmfels and Eisenbud	$\{z,y,x\}$, $\{b,u,t\}$	104, 158, 62	1.484, 2.906, 0.359
${\mathrm{Schimoyama}/\mathrm{Yokoyama}}_1$	$\{z,y\}$, $\left\{x\right\}$	35, 37, 17	0.203, 0.219, 0.125
${\mathrm{Schimoyama}/\mathrm{Yokoyama}}_2$	$\{x,t,z,b,w,u\}$, $\{y,w,u,s,v\}$	48, 72, 26	0.313, 0.641, 0.109
Horrocks	$\{d,e,f\}$, $\{e,f,c\}$	513, 1057, 285	78.140, 191.250, 18.281
Schwarz	$\left\{c\right\}$, $\left\{c\right\}$	80, 119, 90	63.735, 60.985, 6.125
Cyclic roots 4	$\left\{b\right\}$, $\left\{a\right\}$	27, 33, 17	0.141, 0.172, 0.078
Cyclic roots 5 homog	$\left\{h\right\}$, $\left\{a\right\}$	228, 334, 167	44.360, 65.453, 22.016
De Jong	$\{c,b,g,f,l,k,h,d\}$, $\{l,k,d,a,e,j\}$	326, 449, 62	191.406, 415.000, 5.266
mat32	$\{h,g,e,a\}$, $\{c,b,e,a\}$	131, 173, 100	7.687, 10.547, 5.937
${\mathrm{Schimoyama}/\mathrm{Yokoyama}}_3$	$\{b,c,d,e,f,g,h,j,l\}$, $\{b,c,d,e,f,g,h,j,l\}$	72, 108, 36	51.844, 45.719, 7.047
Riemenschneider	$\{y,p,q,z,s,w,t,v\}$, $\{p,w,t,x\}$	92, 97, 44	0.984, 0.765, 0.187
Mikro	$\{b,g,f,e,c\}$, $\{c,d\}$	848, 1114, 319	178,650.985, 625.438, 111.828
Buchberger	$\{x,c,t,d\}$, $\{c,a,t,d\}$	7, 11, 7	0.094, 0.172, 0.078
Lanconelli	$\{l,k,j,h,e,g,c\}$, $\{j,h,f,d,e,g,c\}$	5, 8, 2	0.719, 0.610, 0.078
Wang2	$\left\{y\right\}$, $\left\{z\right\}$	2, 9, 5	0.016, 0.079, 0.032
Siebert	$\{y,x\}$, $\left\{z\right\}$	239, 306, 398	101.640, 116.172, 34.422
Macaulay	$\{d,c\}$, $\{c,b\}$	55, 84, 40	0.656, 1.203, 0.297

The numbers in bold mean that our algorithm has obvious advantages than the direct or the W-W methods. For instance, in Example Chemistry, the numbers 7542.234 and 180.375 in bold mean that our algorithm is dozens of times faster than the direct method.

As is evident from Table 1, our algorithm generally computed fewer critical pairs and our CPU times were usually less than those of the other two methods. Precisely speaking, in 14 of 18 examples, our algorithm computed the fewest critical pairs, and especially in the Horrocks, De Jong, and Mikro examples, our algorithm computed critical pairs several times fewer than those computed by the other two methods. In 17 of 18 examples, except Wang2, our CPU times were all less than those of the other two approaches. Especially in the Mikro example, our algorithm was thousands of times faster than the direct method. In the De Jong example, our CPU time was dozens of times less than those of the other two methods.

5. Conclusions

In this paper, we have presented an efficient algorithm to compute a maximal independent set U modulo a positive-dimensional ideal I, and a Gröbner basis of I with respect to a block monomial order, ${\prec }_{U,V}$, simultaneously. It is based on a new concept we proposed: the maximal strongly quasi-independent set modulo a polynomial set. As a “local” concept, it reveals some information about the independent set modulo I in advance; thus our algorithm considerably reduces the number of critical pairs calculated compared with existing methods which compute the maximal independent set and the Gröbner basis separately. Experiments verify the good performance of our algorithm.