Existence and Construction of a Gröbner Basis for a Polynomial Ideal^***Abstract of a paper presented at the workshop, Combinatorial Algorithms in Algebraic Structures, Europäische Akademie, Otzenhausen, West Germany, Sept. 30 - Oct. 4, 1985.

We show the existence of a Gröbner basis for a polynomial ideal over $R[x_{1},\dots,x_{n}]$ assuming the existence of a Gröbner basis for every ideal over the underlying ring $R$. We also give an algorithm for computing a Gröbner basis assuming there is an algorithm to compute a Gröbner basis for every ideal over $R$. We show how most of the properties that hold for Gröbner bases for ideals over $R$ extend to Gröbner bases for polynomial ideas over $R[x_{1},\dots,x_{n}]$. These results are structurally similar to Hilbert’s Basis Theorem.

A Gröbner basis of a polynomial ideal I can be defined in two ways: either as (a) any basis $B$ of $I$ such that every polynomial $p$ in $I$ reduces to 0 with respect to $B$, or as (b) any basis $B$ of $I$ such that every polynomial in the ring has a unique normal form with respect to $B$. We call bases satisfying (a) weak Gröbner bases and those satisfying (b) strong Gröbner bases. When Buchberger introduced the concept of Gröbner basis [1] for the case when $R$ is a field, he defined them to satisfy (a). Later [2] he showed that they also satisfy condition (b). Weak Gröbner bases for various kinds of rings have since been studied in [6, 7, 8, 9, 13]. In [10] we defined weak Gröbner bases over first-order rings and showed their importance in theorem proving in first-order predicate calculus. Algorithms for computing strong Gröbner bases have been given in [4] (for reduction rings) and [5] (for Euclidean rings). In this paper we unify these two streams and show how to construct weak (strong) Gröbner bases over $R[x_{1},\dots,x_{n}]$ provided weak (strong) Gröbner bases can be constructed over the underlying structure $R$.

Let $R$ be a commutative ring with 1. We assume that for every ideal over $R$, there exists a Gröbner basis for the ideal. Let $\to$ stand for a reduction relation induced by a basis on $R$. We will assume the following properties of $\to$:

Definition:  A basis $B$ of an ideal $I$ over $R$ is a weak Gröbner basis iff every element in the ideal has 0 as a normal form using $B$.

Definition:  A basis $B$ of an ideal $I$ over $R$ is a strong Gröbner basis iff every element in $R$ has a unique normal form using $B$.

A well-founded reduction relation $\to$ induces a well-founded ordering on $R$ as follows: $a<c$ iff there exists a reduction sequence $c\to c_{1}\to\dots\to a$. Elements of $R$ in normal form wrt $\to$ are the minimal elements wrt $<$.

We will assume that we know whether the definition of reduction of elements of $R$ wrt other elements in $R$ admits weak Gröbner bases or strong Gröbner bases. Let Gröbner be a function which takes a finite basis $B$ of an ideal over $R$ as the input and gives a Gröbner basis of the ideal as output; obviously (Gröbner(B))=(B).

Let $R[x_{1},\dots,x_{n}]$ be a polynomial ring over $R$. We assume a total ordering on indeterminates which can be extended in many different ways to a total ordering on terms [4,5,12]. The ordering on $R$ induced by $\to$ and the ordering on terms can be used to define an ordering on monomials and polynomials. Given a polynomial $p$, let $HD(p),HT(p),\text{ and }HC(p)$ stand for its head-monomial, head-term, and head-coefficient, respectively. Let $Rest(p)$ be $p-HD(p)$. We assume that the functions $HD,HT,\text{ and }HC$ can be used on a set of polynomials also; for instance, $HD(I)=\{m\;|\;m\text{ is a head-monomial of a polynomial in }I\}$. Let $B$ be a finite set of polynomials in $R[x_{1},\dots,x_{n}];B=\{p_{1},\dots,p_{m}\}$. Let $I=(p_{1},\dots,p_{m})$.

Using a well-founded relation $\to$ on $R$, we can define a well-founded reduction relation on polynomials in $R[x_{1},\dots,x_{n}]$; we will denote it also by $\to$. A polynomial $p\to q$ using another polynomial $p_{1}$ iff:

• (i)

  There is a monomial $ct$ in $p$ i.e., $p=ct+p^{\prime}$, such that $HT(p_{1})$ divides $t$, i.e., $t=t^{\prime}\cdot HT(p_{1})$, and

• (ii)

  There exists a $d$ such that $c\to d$ using $HC(p_{1})$ and $q=p-k\cdot t^{\prime}\cdot p_{1}$, where $c=k\cdot HC(p_{1})+d$.

The above definition extends to the case when more than one polynomial is used for reduction. Weak and strong Gröbner bases can be defined for polynomial ideals in the same way as definitions given above.

Let $MinMon(M)=\{m\;|\;m$ is a monomial in $M$ and there is no other monomial $m^{\prime}$ in $M$ that divides $m\}.$ Similarly, let $MinTerm(T)=\{t\;|\;t$ is a term in $T$ and there is no $t^{\prime}$ in $T$ that divides $t\}.$ It follows easily from Hilbert’s Basis Theorem as well as from Dickson’s Lemma that for any set of terms $T$, $MinTerm(T)$ is finite.

Given a term $t$, let $C(t,I)=\{c\;|\;ct\text{ in }HD(I)\}$. It is easy to see that $C(t,I)$ is an ideal over $R$; further, for any two distinct terms $t_{1}$ and $t_{2}$, such that $t_{1}$ divides $t_{2}$, $C(t_{1},I)\subseteq C(t_{2},I)$. For every term $t$ appearing in some monomial in $MinMon(HD(I))$, i.e., $t$ in $HT(MinMon(HD(I)))$, let $D(t,I)$ be the set of all coefficients of $t$ in $MinMon(HD(I))$. Let $Divisors(t)$ be the set of all terms including $t$ which can divide $t$. Note that for some $t$, $C(t,I)$ may not be empty, whereas $D(t,I)$ may be empty.

Given an ideal $I$, we construct its Gröbner basis $GB$ as follows: For each $t$ in $HT(MinMon(HD(I)))$, let $G(t,I)$ be a Gröbner basis of the ideal generated by $D(t,I)$ over $R$; i.e., $G(t,I)=\text{Gröbner}(D(t,I))$; for each element $g$ in $G(t,I)$, include in $GB$ a minimal polynomial in $I$ with $g\cdot t$ as its head-monomial.

Is such a $GB$ unique for an ideal $I$? No, because Gröbner bases for the same ideal may differ modulo units. However, if we assume that $R$ admits strong Gröbner bases and such a Gröbner basis is unique for every ideal, then $GB$ is unique for a particular term ordering.

We assume that (i) there is a way to compute a Gröbner basis $G$ from a basis $(c_{1},\dots,c_{k})$ over $R$; furthermore, it is also possible to compute representations of $c_{1},\dots,c_{k}$ in terms of the elements of $G$ as well as representations of every element $g$ in $G$ in terms of $c_{1},\dots,c_{k}$; these constructions are indeed possible from the construction of a Gröbner basis as illustrated by Buchberger in [12]. In addition, (ii) it should also be possible to generate a finite basis for a module of solutions to linear homogenous equations over $R$.

6.1 Critical Pairs

For any finite subset $F$ of $B$, we define, two types of critical pairs among polynomials in the basis. called G-polynomials and M-polynomials, (to stand for Gröbner-polynomials and Module-polynomials, respectively) as follows: Let F consist of the polynomials

$$p_{1}=c_{1}\cdot t_{1}+r_{1}\;,\;p_{2}=c_{2}\cdot t_{2}+r_{2},\;\dots\;,\;p_{j}=c_{j}\cdot t_{j}+r_{j}$$

where $HD(p_{i})=c_{i}\cdot t_{i},\;HT(p_{i})=t_{i},\;HC(p_{i})=c_{i},$ and $Rest(p_{i})=r_{i}$.
Let $t=lcm(t_{1},t_{2},\dots,t_{j})=s_{1}\cdot t_{1}=s_{2}\cdot t_{2}=\dots=s_{j}\cdot t_{j}$ for some terms $s_{1},\dots,s_{j}$. G-Polynomial: Let $\{g_{1},\dots,g_{m}\}$ be a Gröbner basis of $\{c_{1},\dots,c_{j}\}$; each $g_{i}$ can be written (represented) as $g_{i}=h_{i,1}\cdot c_{1}+h_{i,2}\cdot c_{2}+\dots+h_{i,j}\cdot c_{j}$ for some $h_{i,1},\dots,h_{i,j}$. Then, for each $i$, $q_{i}=h_{i,1}\cdot s_{1}\cdot p_{1}+h_{i,2}\cdot s_{2}\cdot p_{2}+\dots+h_{i,j}\cdot s_{j}\cdot p_{j}$ is a G-polynomial. Note that the head-monomial of $q_{i}$ is $g_{i}\cdot t$.

M-Polynomials: Consider the module $M=\{\langle a_{1},\dots,a_{j}\rangle\;|\;a_{1}\cdot c_{1}+\dots+a_{j}\cdot c_{j}=0\}$. Let the vectors $A_{1},A_{2},\dots,A_{v}$ represent its basis, where for every $i$, $A_{i}=\langle b_{i,1},\dots,b_{i,j}\rangle$ for some $b_{i,1},\dots,b_{i,j}$ in $R$. Then, for each $i$,
$q_{i}=b_{i,1}\cdot s_{1}\cdot p_{1}+b_{i,2}\cdot s_{2}\cdot p_{2}+\dots+b_{i,j}\cdot s_{j}\cdot p_{j}$ is an M-polynomial. Note that the head-term of $q_{i}$ is less than $t$ in the term ordering, since the monomials involving $t$ get cancelled in the summation.

We say that q is a G-polynomial (respectively, M-polynomial) of B if $q$ is a G-polynomial (respectively, M-polynomial) of some finite subset $F$ of $B$.

Furthermore, if for every ideal in $R$, the function Gröbner gives a unique strong Gröbner basis, then from Theorem 5.4 and Theorems 6.1 and 6.2, it follows that $B$ is also a unique strong Gröbner basis.

The completion algorithm for obtaining Gröbner bases should be obvious by now: given a basis $B$, compute the G-polynomials and M-polynomials, and their normal forms. Add all non-zero normal forms to the basis and repeat this step until no new polynomails are added to the basis. This process of adding new polynomials will terminate because of the finite ascending chain property of Noetherian rings.

It can be shown that most of the known algorithms are instances of the above algorithm with additional properties assumed on the ring $R$. For instance, when $R$ is a PID, then a Gröbner basis of a basis over $R$ can be obtained as the gcd of the elements in the basis. This can be computed by considering pairs of elements at a time. Further, a module basis of a finite set of non-zero elements in a PID can also be computed by considering pairs of elements. For any two non-zero elements, $a$ and $b$, $\{\langle lcm(a,b)/a,-lcm(a,b)/b\rangle\}$ generates their module basis. It can be shown that if G-polynomials and M-polynomials from every pair of polynomials in a basis $B$ reduce to 0, then G-polynomials and M-polynomials for every non-empty subset $B^{\prime}$ of $B$ also reduce to 0. So, it is sufficient to consider pairs of polynomials at a time for critical-pair computation. These considerations also apply to algorithms over fields and Euclidean rings since both are instances of PIDs.

Zacharias [7], Trinks (as quoted in [8]), and Schaller [6] (as quoted in [4]) have given weak Gröbner basis algorithms for a ring $R$ on which the ideal membership problem is solvable as well as a finite set of generators for the solutions of linear equations in $R$ can be found algorithmically. These algorithms can also be viewed as special cases of our algorithm where only M-polynomials are considered.

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