Abstract
The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.
1. Introduction
The concept of H-bases, introduced long ago by Macaulay [1], is based solely on homogeneous terms of a polynomial. In [2], an extension of Buchberger’s algorithm is presented to construct H-bases algorithmically. Some applications of H-bases are given in [3]; in addition, many of the problems in applications which can be solved by the Gröbner technique can also be treated successfully with H-bases. The concept of H-basis for ideals of a polynomial ring over a field can be adopted in a natural way to -subalgebras of a polynomial ring. In [4], SH-basis (Subalgebra Analogue to H-basis for Ideals) for the -subalgebra of is defined. The properties of SH-bases are typically similar to H-basis results [3]. Like H-bases, the concept of SH-basis is also tied to homogeneous polynomials. In this paper, we will present an analogue to H-bases for ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.”
The paper is organized as follows. In Section 2, we briefly describe the underlying concept of grading which leads to SAGBI-Gröbner bases and HSG-basis. Then, we give the notion of -reduction, which is one of the key ingredients for the characterization and construction of HSG-basis. After setting up the necessary notation, we present the -reduction algorithm (see Algorithm 1). Also, here we present some properties characterizing HSG-basis (Theorem 1). In Section 3, we present a criterion through which we can check that the given system of polynomials is an HSG-basis of the subalgebra it generates (Theorem 2), and further on the basis of this theorem, we present an algorithm for the construction of HSG-basis (Algorithm 2).
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2. HSG-Bases and SAGBI-Gröbner Bases
Here and in the following sections we consider polynomials in variables with coefficients from a field . For short, we write
If is a subset of subalgebra in , then the setis the ideal of in generated by and we write it shortly as . In this section, we want to introduce HSG-bases and discuss some of their properties. This concept is very similar to the concept of SAGBI-Gröbner bases. Therefore, we will briefly explain the underlying common structure. Let denote an ordered monoid, i.e., an abelian semigroup under an operation , equipped with a total ordering such that, for all ,
A direct sum,is called grading (induced by ) or briefly a -grading if for all ,
Since the decomposition above is a direct sum, each polynomial has a unique representation.
Assuming that , the -homogeneous term is called the maximal part of , denoted by , and is called the d-reductum of . For , .
There are two major examples of gradings. The first one is grading by degrees:
Here, with the natural total ordering. This grading is called the -grading because of the homogeneous polynomials. Therefore, we also write in place of this . The space of all polynomials of degree at most can now be written as
The maximal part of a polynomial is its homogeneous form of highest degree, . For simplicity, let .
Definition 1. A subset (subalgebra) is called HSG-basis for the ideal , if for all ,The representation for in (9) is also called its HSG representation with respect to .
Note that HSG-basis for ideal in a subalgebra is also a generating set of it. To obtain more insights into HSG-bases, we will give some equivalent definitions. First, we need a more technical notion.
Definition 2. For given , we say that reduces to with respect to in ifholds with polynomials satisfying . We write it as By we denote the transitive closure of the binary relation 1.
The concept of reduction plays an important role in the characterization and construction of HSG-basis. For and , the following algorithm computes such that (i.e., reduces to completely).
We note that such an element in the subalgebra can easily be determined as in the case of reduction in polynomial ring. We also note that is strictly smaller than the (by the choice of ). This shows that Algorithm 1 always terminates.
Theorem 1. Let (subset of subalgebra ) and be an ideal of . Then, the following conditions are equivalent:(1) is an -basis for the ideal .(2).(3)For all .
Proof. . Let for some . Since is an HSG-basis, by (9), there are some so that and , where . . Let . By using Algorithm 1, we get , where is reduced any further with respect to . implies ; then, . If we follow the above process inductively, then . . Letwhere . Then,Note thatandHence,(11) and (15) give the HSG representation.
The second major example of gradings leads to the SAGBI-Gröbner basis concept. Here, with component-wise addition equipped with a total ordering satisfying (11). In addition, . For arbitrary , the space is a vector space of dimension 1, namely,
The maximal part of a polynomial f is a product of a leading coefficient and a leading monomial , that is . The si-reduction is defined if there exists a polynomial and such that and then we set . The relation is constructed as above.
A SAGBI-Gröbner basis (with respect to a given monomial ordering and a given ideal in a subalgebra ) is a set of polynomials generating and satisfying one of the following equivalent conditions:(i)Every has a representation: where and .(ii).(iii)Every -reduces to 0 with respect to .
The proof of this equivalence and many other equivalent conditions can be found in [5]. If a monomial ordering is compatible with the semiordering by degrees,then any SAGBI-Gröbner representation as given in (i) is an HSG representation; in other words, a SAGBI-Gröbner basis with respect to a degree compatible ordering is an HSG-basis as well. The converse is false, as the following example shows.
Example 1. Let , , . These polynomials belong to the subalgebra . Then, we can see that , and already constitute an HSG-basis for ideal in . If we order the monomials by degree lexicographical ordering, then . Every SAGBI-Gröbner basis with respect to this ordering contains at least four elements, for instance, with , , , and . Obviously, this SAGBI-Gröbner basis is an HSG-basis as well.
3. Construction of HSG-Bases
In this section, we present an HSG-basis criterion, through which we can construct HSG-basis. For this purpose, we fix some notations which are necessary for this construction. Let be a subalgebra of .(i)We denote by .(ii)For a subset , we denote by .
Definition 3. For subalgebra of and a subset ,(1). We call an element of an syzygy of .(2)For , let represent the vector .
Definition 4. We call a subset a -generating set for if generates the -module , i.e., for , there are some such that
In the case of SAGBI-Gröbner bases, there is an algorithm for computing SAGBI-Gröbner bases by means of syzygies (see [6]) where syzygies and their connection to SAGBI-Gröbner bases are studied in detail. The analogue for constructing HSG-bases by means of syzygies is connected to the following result [7].
Theorem 2. (HSG-basis criterion). Let be the subset of a subalgebra . Let be generating set for the . Then, is an HSG-basis for if and only if for every , we have .
Proof. : The statement is a direct result of Theorem 1. : Take . We need to show that . For this, we write such that (degree wise) is minimal among all such representations of . We have . Suppose that . Assume that are contributing to , i.e., . If we set , we can see that . This implies that there are and such that . We may assume that for each by homogeneity of the syzygies. Now,where is an HSG representation for since . If we define , thenbecause .
Consider the first sum of equation (20). For , we have , so by the cancellation of highest terms,For and implies thatSinceSo, first sum of equation (20) is less than p0. For the second sum of equation (20), we haveHence, equation (20) does provide a new representation for f such that , a contradiction. Therefore, and .
On the basis of Theorem 2, now we present an algorithm which computes HSG-basis from a given set of generators. This algorithm is not necessarily terminating but does terminate, if and only if, the considered ideal in the subalgebra has a finite HSG-basis.
Now we present some examples which show the computation of HSG-basis through Algorithm 2.
Example 2. Let the subalgebra and . Consider ; then, . First pass through the while loop:(i) implies . Then, gives .(ii)As is si-reduced with respect to , red.(iii)Define: Old. As Old, we repeat the whole process. Now we have . Second pass through the while loop:(i) implies . Therefore, . Then, gives .(ii)Now, red.Since Old, we stop here. The HSG-basis for is .
Example 3. Let and . Consider ; then, . First pass through the while loop:(i) gives . Then, from ,(ii)red.(iii)Define: Old.As Old, we repeat the whole process. Now we have . Second pass through the while loop:(i)From the equation , we have . We can compute from .(ii)Now, red.Since Old, we stop here. The HSG-basis for is .
4. Conclusion
In this paper, we presented the theory of HSG-bases, which are a good basis of an ideal in a subalgebra of a polynomial ring. We can further develop this theory for an arbitrary grading for which HSG-bases would be a special case for degree-based grading.
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.