Behaviour Analysis, Complexity and Control of Networked Dynamical SystemsView this Special Issue
The Extension of the GVW Algorithm to Valuation Domains
The GVW algorithm is an effective algorithm to compute Gröbner bases for polynomial ideals over a field. Combined with properties of valuation domains and the idea of the GVW algorithm, we propose a new algorithm to compute Gröbner bases for polynomial ideals over valuation domains in this study. Furthermore, we use an example to demonstrate the improvement of our algorithm.
The notion of Gröbner basis was first put forward by Buchberger . The theory of Gröbner has been widely applied in numerous fields such as engineering, signal processing, neuroscience, coding theory, complexity, and control of networked dynamical systems and so on. For example, in the theory of symbolic dynamic systems, the problems of determining whether there is a shift equivalence of lag from one nonnegative matrix to another can be transferred into solving large-scale equations, while the latter can be solved by the Gröbner basis theory [2–13].
Seeking more efficient algorithms for the computation of Gröbner bases is a problem in which many researchers cared about extremely [14–19]. Faugère  proposed a fast algorithm called F5 for computing Gröbner bases. In this algorithm, he introduced two notions of rewriting and signatures, which allow them to filter the useless S-polynomials in a rather convenient way. A new algorithm named G2V  for computing Gröbner bases is presented by Gao et al., which is an algorithm of incremental signature and based on a simple theory. A few months later, they gave an extend version named the GVW algorithm . We are particularly interested in the GVW algorithm which not only matches the original algorithm given by Buchberger in simplicity but also more effective than F5 under some term orders. The algorithms mentioned above are applied to polynomial ideals over fields.
In this study, we aim to extend the GVW algorithm to valuation domains and present a signature-based algorithm to compute Gröbner bases for ideals in , where is a valuation domain. In this algorithm, we study relations between J pairs and propose a new concept named factor, which allows us to filter the useless J pairs in a rather convenient way.
The structure of the study is arranged as follows: some basic concepts of the Gröbner basis theory is given in Section 2. In Section 3, we propose theory for the GVW algorithm over valuation domains and obtain main results of this study. Then, we present the new algorithm and demonstrate the improvement clearly by an example in Section 4.
Let be a valuation ring. For any two nonzero elements , there always exists or . The term order for monomials in is arbitrary throughout this section. The form of the monomial in iswhere . The definition of the leading monomial (abbreviated as ), the leading term (abbreviated as ), and the leading coefficient (abbreviated as ) of a given polynomial is as usual.
A nonzero polynomial set in an ideal is named as (weak) Gröbner basis for if
This does not imply that, for each , there exists some polynomial , so that . For example, has a Gröbner basis , and , but is not divisible by nor in .
Definition 1. A set is called a strong Gröbner basis for if , and there is a polynomial , so that , where is formed by the nonzero polynomials from .
For the above example, is a strong Gröbner basis of the ideal , but the set is not. This shows already some difference when dealing with polynomials over rings from those over fields.
Proposition 1. Suppose is a valuation domain. Then, every ideal in has a strong Gröbner basis.
Proof. It can be easily obtained by properties of valuation ring.
We now follow the notations in . Let , where denotes a valuation domain, and are the polynomials. Letbe an ideal, and its Gröbner basis is what we want to obtain. Vectors in are denoted by bold letters, for example, . Let be the th unit vector in for . Define an submodule of :Note that, as an module, is generated by is defined to be the syzygy module of . We shall see that the big module allows us to get the Gröbner bases for and in the same time and allows us to develop a criterion to detect useless polynomials.
We define quasiordering in : for arbitrary , we say if . Following this definition, then a term order (throughout this study, by a monomial order, we mean a global ordering ) on is defined by iffWe assume has a term order that is compatible with that of . We refer the readers to  for several examples on how term order of can be extended to . Note that a term in is the form asfor some and . For any nonzero ,where is the leading monomial of , and is the leading coefficient of .
3. Theory of the Algorithm
In this section, we present the theory of our algorithm. First, some basic definitions are needed.
Definition 2. is said to be the signature of , where .
Definition 3. We say can be top-reduced by when they meet the following two conditions:(i) divides (i.e., is top-divisible by ) and(ii), where Then, the relevant top-reduction isHence, we can divide this reduction into two types; one is called regular whenand the other is called super ifBesides, there is another super top-reduction, that is, when , we say that is super top-reduced by if and are both nonzero and divides . So when a pair can be reduced by , we just reduce the signature of but without increasing (even if ). What attracts more of our attention is that is never top-reduced by when .
Similar to Definition 1, we give the definition of strong Gröbner basis for in the following.
Definition 4. Suppose is a subset of , is said to be a strong Gröbner basis of , if every pair can be top-reduced by at least one pair in .
It is easy to draw a conclusion from this definition, that is, every pair in can be top-reduced to 0 by its strong Gröbner basis.
Lemma 1. If is a strong Gröbner basis of , where , then(1)A strong Gröbner basis for the syzygy module of exists, which is (2)The strong Gröbner basis for also exists, which is
Proof. Assume is an element from the syzygy module of , then , and there must exist some pair in that can top-reduce with . Thus, and can be reduced by . This tell us that is the set we need, which is a Gröbner basis for the syzygy module of .
For an arbitrary nonzero polynomial , there is , such that according to the definition of and then . Among all such , we choose the minimum . By our assumption, there exists at least one pair in which can top-reduce .
If , then can be reduced by and get , but , which contradicts the minimality of as . So there exists with and . Hence, is a Gröbner basis of .
Definition 5. For any pair, , and . Let(i)If , define(ii)If , define
Remark 1. With notations as above, we do not define J pair for and when one of and is zero nor when .
In order to study the relation between J pairs, we propose the following conception:
Definition 6. Suppose are the J pairs formed from which is a (finite) subset of , and is called a factor of iffor some monomial .
The next result is very useful for reduction.
Lemma 2. Assume can be regular top-reduced by with neither of is zero, where is an element from and is a monomial from R; then, is the J pair of and , wherewhere is a divisor of , , and can regular top-reduce .
Proof. By our assumption, there exist and a monomial say , so thatSetwe have thatThen, some and monomial exist, such thatThus,Then, we have thatHence, as . Thus, . And the J pair of and is by Definition 5. Note thatand then,Therefore,We see that can regular top-reduce .
Suppose is a set formed by the pairs in , we say can regular (super) top-reduce the pair , if there is at least one pair in which can regular (super) top-reduce the pair . Furthermore, we implement a series of such reductions to until it cannot be regular top-reduced by this set anymore, but can be super top-reduced by , and this reduction defined as is eventually super top-reduced by .
Theorem 1. Suppose is any term in , and there always exists a pair in , a monomial and , such that , where is a subset of . Then, is a strong Gröbner basis for if and only if, for the J pairs formed from such as , there always exists a pair , so that and , where .
Proof. Necessity: let be an arbitrary J pair formed from ; then, is in , and it is top-reduced by as the set is a strong Gröbner basis for the module . We can do the regular top-reductions to as much as possible until cannot be regular top-reduced any more, say to get . And can be top-reduced by as it is still in ; but now, the reduction can only be super reduction, say can be super reduced by .(1)If , then , and is smaller than , and the conclusion is true.(2)If ,Combined with the definition of quasiordering given, we have thatand then, .
Since is obtained by performed regular top-reduction to , then and ; the latter shows thatSo, and .
Sufficiency: suppose there are some pairs in which cannot be top-reduced the set , say is such a pair. We prove that such a pair does not exist. Select the minimal signature from all such pairs with , and we choose a pair from , so that(i) with and is a monomial in , and(ii) is the smallest one among all satisfying (i)In the following, we prove that should not be regular top-reduced by . Suppose that could be regular top-reduced by a pair , , then . What we expect is to get a conclusion that contradicts condition (ii). From Lemma 2, we have that is the J pair of and ; moreover, it can still be regular top-reduced by , wherefor some and monomial .
Case 1. If , then , and the J pair of and is . Note that is a J pair formed from , so there must exists a pair , so that and with . Since , we set , and then,This contradicts the condition (ii) for the selection of in .
Case 2. If , then , and the J pair of and is . Note that is a J pair formed from , and there must exist some pair say , such that and with . Since , set , and then,This contradicts the condition (ii) for the selection of in .
Letand then, and . Otherwise, would be top-reduced by which contradicts with the selection of . So, and can be top-reduced by as and , , say it is top-reduced by . If , we use this type of pairs to reduce as much as possible and obtain a new pair finally, which cannot be top-reduced by the same type of pairs (here, it refers the pairs whose f-part is zero) in . Since and , , and then can be top-reduced by , say by , . For , there are the following three cases that need to be considered:(i)If , then ; but , so there must exist some pairs in that can regular top-reduce ; assume the pair in is . This is impossible as cannot be regular top-reduced by any pair in .(ii)If , then , and can be regular top-reduced by ; this contradicts the choice of .(iii)If and , is top-reduced by , and then, ; this means that and .From the property of valuation ring, we consider the relation between and in the following three cases:(a)If and is not a nonzero unit, note that a valuation ring is a local ring, and is in the unique maximal ideal. Since and is invertible, then . Note that and are regular top-reduced by ; this contradicts the choice of .(b)If and is not a nonzero unit, note that a valuation ring is a local ring, and is in the unique maximal ideal. Since and is invertible, then . Note that , then is regular top-reduced by , and this case is impossible.(c)If and is a nonzero unit, then and . Note that ; combining the definition of the order, we have . Therefore, can be top-reduced by , and this contradicts the choice of .Thus, such pairs like cannot exist in at all; hence, all pairs of can be top-reduced by , and is a strong Gröbner basis for .
The pair is covered by when there exists at least one pair such as in , such that and , where .
Theorem 2. Suppose is a special subset of , whose particularity is reflected in for any term , there always exists a pair and monomial and , such that . Then, is a strong Gröbner basis for if the factors of the J pairs formed by can always be eventually super top-reduced by the set .
Proof. Assume that are the two J pairs formed by , and is a factor of . Then, can be eventually super top-reducible by , that is, after doing a series of regular top-reduction to , say it to get where ; besides, can be super top-reduced by , say ; then, , where . Clearly, can be covered by . It is also correct for the rest of J pairs and their factors. By Theorem 1, we have that is a strong Gröbner basis for .
According to Theorems 1 and 2, we can discard the J pairs which can be covered by without doing any regular nor super top-reductions. As a consequence, there are four criteria for discarding redundant J pairs.
Corollary 1. (Covered criterion) For any J pair of , it can be discard if is covered by .
Corollary 2. (Syzygy criterion) If a J pair can be top-reduced by a syzygy, then it can be discarded.
Corollary 3. (Signature criterion) As for the J pairs with the same signature, we only need to keep the one whose -part is minimal.
Corollary 4. (Factor criterion) As for the J pair which has a factor, we just need to keep the factor.
4. Algorithm and Example
According to the theorems and corollaries in Section 3, we can get an algorithm for computing Gröbner bases for the polynomial ideals over valuation domains. We call the algorithm as . The main idea of is analogue to the GVW algorithm of principal ideal domain . First, we form J pairs by the initial pairs. By Theorems 1 and 2, we just store the J pairs with different signatures. We only consider the J pairs that we store, choose any one of them, denoted as , and then check whether it satisfies Corollary 1, that is, whether it is covered by . If so, discard it. Otherwise, delete all the J pairs whose factor is and perform regular top-reductions to it repeatedly until it cannot be regular top-reduced any more, say to get finally. If , then is a syzygy in . We add it to and delete the pairs whose signature is divisible by . Otherwise, adds to the set , and we will not stop the process until the set of J pair is empty. In the while-loop, all the J pairs formed from will be top-reduced by the set . We describe the algorithm in more detail and accurately with Figure 1. is used to store the leading terms of syzygies; the Gröbner basis we get is a list of pairs , where for . We store this list as
So the whole list is represented by .
Theorem 3. Assume the term orders in are compatible with which in ; then, the algorithm shown in Figure 1 will terminate after a finite number of steps and get a strong Gröbner basis for .
Proof. The correctness of our algorithm is obviously according to Theorems 1 and 2. As for the termination of the algorithm, we can refer to the Theorem 1 in .
Next, we present our algorithm and use a concrete example to demonstrate the improvement clearly.
Example 1. Let , and we consider the Gröbner basis of the ideal , whereand is a discrete valuation ring, for each prime (here is 3); set is a function given by if are integers relatively prime to .
The term order we set on is the lexicographical ordering, which is defined by . Besides, the term order on is if or and .
First, let ; then, , which is a set that stands for leading term of principle syzygy.
Choose from ; then, . Hence, , and pair of and is , whose signature cannot be reduced by , and we store it.
Doing the same process, the -pairs set of other pairs in is , but is a factor of , so delate. Then,Selecting from , we get after a series of regular top-reduction, add it to , and recalculate the J pairs andUsing the syzygy criterion and factor criterion, we obtainSelect from ; it can be regular top-reduced to and add to .
Select from ; it can be regular top-reduced to and add to .
Now, . So is a strong Gröbner basis for in .
In this study, we have generalized the GVW algorithm and presented an algorithm to compute Gröbner bases for polynomial ideals over valuation domains. We have also given an example to illustrate our method. All of these could provide useful information for engineers to solve large linear systems and networked dynamical systems. Valuation domains are special kinds of valuation rings which may have zero divisors. In the future, we want to consider new algorithms for Gröbner bases of ideals over general valuation rings. And we also hope to establish a dynamical Gröbner basis algorithm which combined with the algorithm in the study.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
This research was supported by the National Natural Science Foundation of China (11871207).
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