On Parametric Border Bases

Abstract

We study several properties of border bases of parametric polynomial ideals and introduce a notion of a minimal parametric border basis. It is especially important for improving the quantifier elimination algorithm based on the computation of comprehensive Gröbner systems.

1 Introduction

We study properties of border bases of zero-dimensional parametric polynomial ideals. Main motivation of our work is to improve the CGS-QE algorithm introduced in [1]. It is a special type of a quantifier elimination (QE) algorithm which has a great effect on QE of a first order formula containing many equalities. The most essential part of the algorithm is to eliminate all existential quantifiers $\exists \bar{X}$ from the following basic first order formula:

$$\begin{aligned} \phi (\bar{A})\wedge \exists \bar{X} \; (\bigwedge _{1 \le i \le s} f_i(\bar{A},\bar{X})=0 \wedge \bigwedge _{1 \le i \le t} h_i(\bar{A},\bar{X}) \ge 0) \end{aligned}$$

(1)

with polynomials $f_1,\ldots ,f_s, h_1,\ldots ,h_t$ in $\mathbb Q[\bar{A},\bar{X}]$ such that the parametric ideal $I=\langle f_1,\ldots ,f_s\rangle$ is zero-dimensional in $\mathbb C[\bar{X}]$ for any specialization of the parameters $\bar{A}=A_1,\ldots ,A_m$ satisfying $\phi (\bar{A})$, where $\phi (\bar{A})$ is a quantifier free formula consisting only of equality $=$ and disequality $\not =$. The algorithm computes a reduced comprehensive Gröbner system (CGS) $\mathcal G = \{(\mathcal S_1,G_1),\ldots , (\mathcal S_r, G_r)\}$ of the parametric ideal I on the algebraically constructible set $\mathcal S =\{\bar{a} \in \mathbb C^m|\phi (\bar{a})\}$, then applies the method of [9] with several improvements of [2,3,4, 7]. One of the most important properties of the reduced CGS is that $\mathbb C[\bar{X}]/\langle f_1(\bar{X},\bar{a})\ldots ,f_s(\bar{X},\bar{a})\rangle$ has an invariant basis $\{t\in T(\bar{X}): t\not \mid LT(g)\, \mathrm{for}\; \mathrm{any}\; g\in G_i\}$ as a $\mathbb C$-vector space for every $\bar{a}\in \mathcal S_i$. It enables us to perform several uniform computations with parameters $\bar{A}$ for every $\bar{a}\in \mathcal S_i$. (More detailed descriptions can be found in [1].) In order to obtain a simple quantifier free formula, a compact representation of a reduced CGS of I is desirable, minimizing the number r of the partition $\mathcal S_1,\ldots ,\mathcal S_r$ of $\mathcal S$ is particularly important. Border bases are alternative tools for handling zero-dimensional ideals [5]. We have observed that the reduced CGS can be replaced with a parametric border basis in our algorithm. Since border bases have several nice properties which Gröbner bases do not possess, we can obtain a simpler quantifier free formula using a parametric border basis.

In this paper, we study border bases in parametric polynomial rings. We give a formal definition of a parametric border basis and show several properties which are important for improving the CGS-QE algorithm. Since our work is still on going and the paper is a short paper, we do not get deeply involved in the application of parametric border bases to QE.

The paper is organized as follows. In Sect. 2, we first give a quick review of a CGS for understanding the merit of our work, then give a formal definition of a parametric border basis. In Sect. 3, we introduce our main results together with a rather simple example for understanding our work. Numerical stability is one of the most important properties of border bases. In Sect. 4, we study this property in our setting. We follow the book [5] for the terminologies and notations concerning border bases.

2 Preliminary

In the rest of the paper, let $\mathbb Q$ and $\mathbb C$ denote the field of rational numbers and complex numbers, $\bar{X}$ and $\bar{A}$ denote some variables $X_1,\ldots ,X_n$ and $A_1,\ldots ,A_m$, $T(\bar{X})$ denote a set of terms in $\bar{X}$. For $t_1,t_2\in T(\bar{X})$, $t_1\mid t_2$ and $t_1\not \mid t_2$ denote that “$t_2$ is divisible by $t_1$” and “$t_2$ is not divisible by $t_1$” respectively. For a polynomial $f\in \mathbb C[\bar{A},\bar{X}]$, regarding f as a member of a polynomial ring $\mathbb C[\bar{A}][\bar{X}]$ over the coefficient ring $\mathbb C[\bar{A}]$, its leading term and coefficient w.r.t. an admissible term order $\succ$ of $T(\bar{X})$ are denoted by $LT_{\succ }(f)$ and $LC_{\succ }(f)$ respectively. When $\succ$ is clear from context, they are simply denoted by LT(f) and LC(f).

2.1 Comprehensive Gröbner System

Definition 1

For an algebraically constructible subset (ACS in short) $\mathcal S$ of $\mathbb {C}^m$, a finite set $\{ \mathcal S_1,\ldots ,\mathcal S_r \}$ of ACSs of $\mathbb {C}^m$ which satisfies $\cup _{i=1}^r \mathcal S_i=\mathcal S$ and $\mathcal S_i \cap \mathcal S_j = \emptyset (i \not = j)$ is called an algebraic partition of $\mathcal S$. Each $\mathcal S_i$ is called a segment.

Definition 2

Fix an admissible term order on $T(\bar{X})$. For a finite set $F\subset \mathbb {Q}[\bar{A},\bar{X}]$ and an ACS $\mathcal S$ of $\mathbb {C}^m$, a finite set of pairs $\mathcal G = \{ (G_1,\mathcal S_1),\ldots , (G_r,\mathcal S_r)\}$ with finite sets $G_1,\ldots ,G_r$ of $\mathbb {Q}[\bar{A},\bar{X}]$ satisfying the following properties is called a reduced comprehensive Gröbner system (CGS) of $\langle F\rangle$ on $\mathcal S$ with parameters $\bar{A}$.

(When $\mathcal S$ is the whole space $\mathbb C^m$, “on $\mathbb C^m$” is usually omitted.)

• 1. $\{ \mathcal S_1,\ldots ,\mathcal S_r \}$ is an algebraic partition of $\mathcal S$.

• 2. For each i and $\bar{a} \in \mathcal S_i$, $G_i(\bar{a})$ is a reduced Gröbner basis of $\langle F(\bar{a})\rangle \subset \mathbb {C}[\bar{X}]$, where $G_i(\bar{a})= \{ g(\bar{a},\bar{X}) | g(\bar{A},\bar{X}) \in G_i\}$ and $F(\bar{a})=\{f(\bar{a},\bar{X})|f(\bar{A},\bar{X})\in F\}$.

• 3. For each i, $\mathrm {LC}(g)(\bar{a}) \not = 0$ for every $g \in G_i$ and $\bar{a} \in \mathcal S_i$.

Remark 3

The set of leading terms of all polynomials of $G_i(\bar{a})$ is invariant for each $\bar{a}$ $\in$ $\mathcal S_i$. Hence, not only the dimension of the ideal $\langle G_i(\bar{a})\rangle$ is invariant but also the $\mathbb C$-vector space $\mathbb C[\bar{X}]/\langle F(\bar{a})\rangle$ has the same finite basis $\{ t\in T(\bar{X}): t\not \mid LT(g)\, \mathrm{for}\; \mathrm{any}\; g\in G_i\}$ for every $\bar{a}\in \mathcal S_i$ when $\langle F(\bar{a})\rangle$ is zero-dimensional.

2.2 Border Bases in Parametric Polynomial Rings

Definition 4

For a finite set $F \subset \mathbb {Q}[\bar{A},\bar{X}]$ and an ACS $\mathcal S$ of $\mathbb {C}^m$ such that the ideal $\langle F(\bar{a})\rangle$ is zero-dimensional for each $\bar{a}\in \mathcal S$, a finite set of triples $\mathcal B = \{ (B_1,\mathcal S_1,\mathcal O_1),\ldots , (B_r, \mathcal S_r,\mathcal O_r)\}$ with a finite set $B_i$ of $\mathbb {Q}(\bar{A})[\bar{X}]$ and an order ideal $\mathcal O_i$ of $T(\bar{X})$ for each i satisfying the following properties is called a parametric border basis (PBB) of $\langle F\rangle$ on $\mathcal S$ with parameters $\bar{A}$.

(When $\mathcal S$ is the whole space $\mathbb C^m$, “on $\mathbb C^m$” is usually omitted.)

• 1. $\{ \mathcal S_1,\ldots ,\mathcal S_r \}$ is an algebraic partition of $\mathcal S$.

• 2. For each i, any denominator of a coefficient of an element of $B_i$ does not vanish on $\mathcal S_i$.

• 3. For each i and $\bar{a} \in \mathcal S_i$, $B_i(\bar{a})$ is a $\mathcal O_i$-border basis of $\langle F(\bar{a})\rangle \subset \mathbb {C}[\bar{X}]$.

3 Properties of Parametric Border Bases

Consider the set $F=\{ X^2+\frac{1}{4}Y^2-AXY+B-1,\frac{1}{4}X^2+Y^2-BXY+A-1\}$ of parametric polynomials in $\mathbb Q[A,B,X,Y]$ with parameters A and B, which is a similar but a little bit more complicated example than the one discussed in the book [5]. $\langle F(a,b)\rangle$ is zero-dimensional for every $(a,b)\in \mathbb C^2$. It has the following reduced CGS $\mathcal G=\{ (G_1,\mathcal S_1),\ldots ,(G_7,\mathcal S_7)\}$ w.r.t. the lexicographic term order such that $X\succ Y$.

$$\begin{aligned} \begin{array}{l} G_1=\{ -5X^2+20BYX+4,-5Y^2-20B+4\}, \mathcal S_1=\mathbb V(A-4B),\\ G_2=\{ 5X^2-4YX+5B-5,(5B-1)YX-5Y^2,(20B-29)Y^3+(-25B^3+35B^2-11B+1)Y\},\\ \mathcal S_2=\mathbb V(4A-B-3)\setminus \{(\frac{4}{5},\frac{1}{5}),(\frac{89}{80},\frac{29}{20})\},\\ G_3=\{ 16(A-4B)(4A-B-3)X+(-64A^2+272AB-64B^2-225)Y^3+ (-64A^3+(256B+64)A^2+\\ (64B^2-320B-240)A-256B^3+256B^2+60B+180)Y, (-64A^2+272AB-64B^2-225)Y^4+(-64A^3+\\ (256B+64)A^2+ (64B^2-320B-480)A -256B^3+256B^2+120B+360)Y^2 -16(4A-B-3)^2\},\\ S_3=\mathbb C^2\setminus \mathcal S_1\cup \mathcal S_2\cup \mathcal S_4\cup \cdots \cup \mathcal S_7 =\mathbb C^2\setminus \mathbb V((A-4B)(4A-B-3)(64A^2-272AB+64B^2+225)),\\ G_4=\{ 20X^2+9,Y\}, \mathcal S_4=\{(\frac{89}{80},\frac{29}{20})\},\\ G_5=\{58Y^2+245, 35X-Y\}, \mathcal S_5=\{(\frac{101}{20}, \frac{29}{20})\}, \\ G_6=\{60(20B-29)X+((400B^2-400B-36)A-1600B^3+1600B^2+519B-375)Y, 15((400B-\\ 64)A-64B-425)Y^2+128((200B^2-80B-42)A-50B^3+20B^2-177B+75) \}, \\ \mathcal S_6= \mathbb V(64A^2-272AB+64B^2+225)\setminus \mathcal S_3\cup \mathcal S_4\cup \mathcal S_5, \\ G_7=\{ 1\}, \mathcal S_7= \mathbb V(-10881A-10000B^3+8400B^2+9744B+3925,25A^2-17A+25B^2-17B-\\ 25, (400B-64)A-64B-425) = \{ (\alpha _1+\beta _1 \mathrm {i},\alpha _1-\beta _1 \mathrm {i}), (\alpha _1-\beta _1 \mathrm {i},\alpha _1+\beta _1 \mathrm {i}), (-\alpha _2-\beta _2 \mathrm {i},-\alpha _2+\\ \beta _2 \mathrm {i}), (\alpha _2+\beta _2 \mathrm {i},-\alpha _2-\beta _2 \mathrm {i})\} with \alpha _1\fallingdotseq 1.16856,\beta _1\fallingdotseq 0.266288, \alpha _2\fallingdotseq 0.668559,\beta _2\fallingdotseq 0.633712. \end{array} \end{aligned}$$

Note that the $\mathbb C$-vector space $\mathbb C[X,Y]/\langle F(a,b)\rangle$ has dimension 4, 4, 4, 2, 2, 2 and 1 for $(a,b)\in \mathcal S_1,\mathcal S_2,\mathcal S_3,\mathcal S_4 ,\mathcal S_5,\mathcal S_6$ and $\mathcal S_7$ respectively. Even though $\mathcal S_1,\mathcal S_2$ and $\mathcal S_3$ are connected and the $\mathbb C$-vector space $\mathbb C[X,Y]/\langle F(a,b)\rangle$ has the same dimension 4 on $\mathcal S_1,\,\mathcal S_2$ and $\mathcal S_3$, we cannot glue them into a single segment as long as we use a reduced CGS. On the other hand, we can glue them into a single segment with the following PBB $\mathcal B = \{ (B_1,\mathcal S_1',\mathcal O_1),\ldots ,(B_5,\mathcal S_5',\mathcal O_5)\}$.

$$\begin{aligned} \begin{array}{l} B_1=Y^2+\frac{4(A-4B)}{15}XY+\frac{4}{15}(4A-B-3), XY^2+\frac{16(A-4B)(A-4B+3)}{64A^2-272AB+64B^2+225}Y\\ \qquad \quad + \frac{60(4A-B-3)}{64A^2-272AB+64B^2+225}X,\\ X^2+\frac{4(B-4A)}{15}XY+\frac{4}{15}(4B-A-3), X^2Y+\!\frac{16(B-4A)(B-4A+3)}{64A^2-272AB+64B^2+225}X+ \frac{60(4B-A-3)}{64A^2-272AB+64B^2+225}Y, \\ \mathcal S_1'= \mathcal S_1\cup \mathcal S_2\cup \mathcal S_3 =\mathbb C^2\setminus \mathbb V(64A^2-272AB+64B^2+225), \mathcal O_1=\{ 1,X,Y,XY\}, \\ B_2=\{ X^2+\frac{9}{20},Y,XY\}, \mathcal S_2'=\mathcal S_4, \mathcal O_2=\{ 1,X\}, \\ B_3=\{ X-\frac{1}{35}Y, XY+\frac{7}{58}, Y^2+\frac{245}{58}, \}, \mathcal S_3'=\mathcal S_5\mathcal O_3=\{ 1,Y\}, \\ B_4=\{ X+\frac{(400B^2-400B-36)A-1600B^3+1600B^2+519B-375}{60(20B-29)}Y, \\ XY-\frac{32((400B^2-400B-36)A-1600B^3+1600B^2+519B-375)((200B^2-80B-42)A-50B^3+20B^2-177B+75)}{225(20B-29)((400B-64)A-64B-425)}, \\ Y^2+\frac{128((200B^2-80B-42)A-50B^3+20B^2-177B+75)}{15((400B-64)A-64B-425)} \}, \mathcal S_4'=\mathcal S_6, \mathcal O_4=\{ 1,Y\}, \\ B_5=\{ 1\}, \mathcal S_5'=\mathcal S_7, \mathcal O_5=\emptyset . \end{array} \end{aligned}$$

Note also that $\mathbb C[X,Y]/\langle F(a,b)\rangle$ has the same dimension 2 on $\mathcal S_2', \mathcal S_3'$ and $\mathcal S_4'$. Even though $\mathcal S_2'$, $\mathcal S_4'$ and $\mathcal S_3'$, $\mathcal S_4'$ are connected, however, we cannot glue them into a single segment for both of them. The reason for $\mathcal S_2'$, $\mathcal S_4'$ is that $\langle F(a,b)\rangle$ has the only one order ideal $\mathcal O_2$ on $(a,b)\in S_2'$ (i.e., $(a,b)=(\frac{89}{80},\frac{29}{20})$), while $\mathcal S_4'$ contains a point $(\frac{29}{20},\frac{89}{80})$ such that $\langle F(\frac{29}{20},\frac{89}{80})\rangle$ has the only one order ideal $\mathcal O_4$ different from $\mathcal O_2$. The reason for $\mathcal S_3'$, $\mathcal S_4'$ is rather subtle. We cannot have a uniform parametric representation for both of $B_3$ and $B_4$. Those observations lead us to the following definition of a minimal PBB.

Definition 5

A PBB $\mathcal B = \{ (B_1,\mathcal S_1,\mathcal O_1), \ldots , (B_r, \mathcal S_r,\mathcal O_r)\}$ of $\langle F\rangle$ is said to be minimal if for any pair $(\mathcal S_i$, $\mathcal S_j)$ of connected segments such that $\mathbb C[\bar{X}]/\langle F(\bar{a},\bar{X})\rangle$ has the same dimension on them it satisfies either of the following:

• 1. $\mathcal O_i\not =\mathcal O_j$, but also $\langle F(\bar{a})\rangle$ does not possess a common order ideal on $\mathcal S_i\cup \mathcal S_j$.

• 2. $\mathcal O_i=\mathcal O_j$ and there exist no uniform parametric representation for both of $B_i$ and $B_j$ on $\mathcal S_i\cup \mathcal S_j$.

Where “$\mathcal S_i$ and $\mathcal S_j$ are connected” means that $\overline{S_i}\cap \overline{S_j}\cap (S_i\cup S_j)\not =\emptyset$, $\overline{X}$ denotes the Zariski closure of X. Intuitively, $\mathcal S_i$ and $\mathcal S_j$ are connected if and only if there exist two points $\bar{a}_i\in \mathcal S_i$ and $\bar{a}_j\in \mathcal S_j$ which are connected by a continuous path in $\mathcal S_i\cup \mathcal S_j$.

Note that a Gröbner basis can be considered as a border basis with the naturally induced order ideal, we can convert a reduced CGS into a PBB using uniform parametric monomial reductions on each segment. Hence, we can compute a PBB of any given $\langle F\rangle$. Existence of a minimal PBB is also obvious, however, we have not obtained an effective algorithm yet. The reason is that we do not have an algorithm to decide whether the property 2 holds yet, while it is easy to check the property 1 using the (parametric) border division algorithm by $B_i$ on $\mathcal S_i$ and by $B_j$ on $\mathcal S_j$. At this time, we have obtained the following results.

Lemma 6

Let $(B,\mathcal S,\mathcal O)$ be a member of a PBB $\mathcal B$ of $\langle F\rangle$ such that $\mathcal S=\mathbb C^m\setminus \mathbb V(I)$ for some ideal $I\subset \mathbb Q[\bar{A}]$. If there are other members $(B_{n_1},\mathcal S_{n_1},\mathcal O_{n_1})$ $,\ldots ,$ $(B_{n_k},\mathcal S_{n_k},\mathcal O_{n_k})$ of $\mathcal B$ such that $\mathbb C[\bar{X}]/\langle F(\bar{a},\bar{X})\rangle$ has the same dimension on $\mathcal S\cup \mathcal S_{n_1}\cup \cdots \cup \mathcal S_{n_k}$ and $\langle F(\bar{a},\bar{X})\rangle$ also has a unique order ideal $\mathcal O'$ on every $\bar{a}\in \mathcal S\cup \mathcal S_{n_1}\cup \cdots \cup \mathcal S_{n_k}$, then we can compute a finite subset $B'$ of $\mathbb Q(\bar{A})[\bar{X}]$ such that $B'(\bar{a})$ is a $\mathcal O'$-border basis of $\langle F(\bar{a})\rangle$ on $\mathcal S\cup \mathcal S_{n_1}\cup \cdots \cup \mathcal S_{n_k}$.

In the above example, by this lemma, we can glue $(G_1,S_1),(G_2,S_2),(G_3,S_3)$ into $(B_1,\mathcal S_1',\mathcal O_1)$ with $\mathcal S_1'=\mathcal S_1\cup \mathcal S_2\cup \mathcal S_3$ and the order ideal $\mathcal O_1$ induced from $(G_1,S_1)$.

Lemma 7

Let $(B_i,\mathcal S_i,\mathcal O)$ and $(B_j,\mathcal S_j,\mathcal O)$ be members of a PBB. If there exists $\bar{a}$ $\in$ $\mathcal S_i\cap \overline{\mathcal S_j}$ such that we cannot specialize some $t+h(\bar{A},\bar{X})\in B_j$ with $t \in \partial \mathcal O$ and $\bar{A}=\bar{a}$, then there exists no uniform parametric representation for $B_i$ and $B_j$ on $\mathcal S_i\cup \mathcal S_j$.

In the above example, $B_3$ and $B_4$ do not have a uniform parametric representation since the denominator $60(20B-29)$ of a coefficient of a polynomial in $B_4$ vanishes for $(A,B)=(\frac{101}{20},\frac{29}{20})\in \mathcal S_3'\cap \overline{\mathcal S_4'}$.

4 Stability of Parametric Border Basis

Numerical stability is one of the most important properties of border bases. We give a precise definition of the stability of a border basis of a parametric ideal as follows.

Definition 8

Let F be a finite subset of $\mathbb {Q}[\bar{A},\bar{X}]$ and $\mathcal S$ be a subset (not necessary to be algebraically constructible) of $\mathbb C^m$ such that the $\mathbb C$-vector space $\mathbb C[\bar{X}]/\langle F(\bar{a},\bar{X})\rangle$ has an invariant finite dimension for every $\bar{a}\in \mathcal S$. For $\bar{a}\in \mathcal S$ which is not an isolated point of $\mathcal S$, let $\langle F(\bar{a},\bar{X})\rangle$ have a $\mathcal O$-border basis $B=\{ t_1+g_1,\ldots ,t_l+g_l\}$ with $\{ t_1,\ldots ,t_l\}= \partial \mathcal O$ and $g_1,\ldots ,g_l\in \mathbb C[\bar{X}]$ for some order ideal $\mathcal O=\{ s_1,\ldots ,s_k\}$. If there exists an open neighborhood $\mathcal S'\subset \mathcal S$ of $\bar{a}$ such that $\langle F(\bar{c},\bar{X})\rangle$ has an invariant order ideal $\mathcal O$ together with a $\mathcal O$-border basis $\{ t_1+\phi ^1_1(\bar{c})s_1+\cdots +\phi ^1_k(\bar{c})s_k, \ldots , t_l+\phi ^l_1(\bar{c})s_1+\cdots +\phi ^l_k(\bar{c})s_k\}$ for each $\bar{c}\in \mathcal S'$ with mappings $\phi ^i_j$ from $\mathcal S'$ to $\mathbb C$. (Note that it is uniquely determined.) In addition, if these mappings are continuous at $\bar{A}=\bar{a}$ that is $\lim _{\bar{c} \rightarrow \bar{a}} \phi ^i_1(\bar{c})s_1+\cdots +\phi ^i_k(\bar{c})s_k = g_i$ for each $i=1,\ldots ,l$, then we say B is stable at $\bar{A} =\bar{a}$ in $\mathcal S$.

Unfortunately, the stability property does not hold for some parametric ideal $\langle F(\bar{A},\bar{X})\rangle$.

Example 9

Let $F=\{ A(X-Y),A X^4+X^2+A-1,A Y^4+Y^2+A-1\}$. $\mathbb C[X,Y]/\langle F(a)\rangle$ has dimension 4 for any $a\in \mathcal S=\mathbb C$. Possible order ideals of $\langle F(a)\rangle$ are $\{ 1,X,X^2,X^3\}$ and $\{ 1,Y,Y^2,Y^3\}$ for $a\not =0$ but only $\{ 1,X,Y,XY\}$ for $a=0$. Hence, the $\{ 1,X,Y,XY\}$-border basis B of $\langle F(0)\rangle$ is not stable at $A=0$ in $\mathcal S$.

In case a parametric ideal has an invariant order ideal in some connected region $\mathcal S$ its border basis seems to be stable at any point of $\mathcal S$, although we have not proved it yet.

Example 10

For the example of the previous section, $\langle F(a,b,X,Y)\rangle$ has an order ideal $\{ 1,Y\}$ for every $(a,b)\in \mathcal S'_3\cup \mathcal S'_4$. As is mentioned at the end of previous section, we do not have a uniform parametric representation of the $\{ 1,Y\}$-border basis of $\langle F(a,b,X,Y)\rangle$ for every $(a,b)\in \mathcal S'_3\cup \mathcal S'_4$. It seems that the $\{ 1,Y\}$-border basis of $\langle F(a,b,X,Y)\rangle$ is not stable at $(a,b)=(\frac{101}{20},\frac{29}{20})$. But it is actually stable at $(A,B)=(\frac{101}{20},\frac{29}{20})$ in $\mathcal S'_3\cup \mathcal S'_4$. That is $\frac{(400B^2-400B-36)A-1600B^3+1600B^2+519B-375}{60(20B-29)}$, $\frac{32((400B^2-400B-36)A-1600B^3+1600B^2+519B-375)((200B^2-80B-42)A-50B^3+20B^2-177B+75)}{225(20B-29)((400B-64)A-64B-425)}$ and

$\frac{128((200B^2-80B-42)A-50B^3+20B^2-177B+75)}{15((400B-64)A-64B-425)}$ converge to $-\frac{1}{35},-\frac{7}{58}$ and $\frac{245}{58}$ as (A, B) $\rightarrow$ $(\frac{101}{20},\frac{29}{20})$ in $\mathcal S'_3\cup \mathcal S'_4$.

5 Conclusion and Remarks

A terrace introduced in [8] is an ideal algebraic structure for a canonical representation of a comprehensive Gröbner system. It is the smallest commutative von Neumann regular ring extending $\mathbb Q[\bar{A}]$, meanwhile $\mathbb Q(\bar{A})$ is the smallest field extending $\mathbb Q[\bar{A}]$. If we are allowed to use this structure to represent coefficients of parametric polynomials, we can also similarly define a PBB and a minimal PBB. For the definition of a minimal PBB, we do not need the property 2, that is we always have $\mathcal O_i\not =\mathcal O_j$. Furthermore the better thing is that we can always compute it, though we have not tried to use it yet since the implementation of the structure of terrace is not very straightforward.