An Algorithm to Compute the H-Bases for Ideals of Subalgebras

,


Introduction
e 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. e concept of H-basis for ideals of a polynomial ring over a field K can be adopted in a natural way to K-subalgebras of a polynomial ring. In [4], SH-basis (Subalgebra Analogue to H-basis for Ideals) for the K-subalgebra of K[x 1 , . . . , x n ] is defined. e 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." e 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. en, we give the notion of si-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 si-reduction algorithm (see Algorithm 1). Also, here we present some properties characterizing HSG-basis ( eorem 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 ( eorem 2), and further on the basis of this theorem, we present an algorithm for the construction of HSG-basis (Algorithm 2).

HSG-Bases and SAGBI-Gröbner Bases
Here and in the following sections we consider polynomials in n variables x 1 , . . . , x n with coefficients from a field K. For short, we write If G is a subset of subalgebra A in K[x 1 , . . . , x n ], then the set I ≔ g∈G h g g|h g ∈ A and only finitely many is the ideal of A in P generated by G and we write it shortly as 〈G〉 A . In this section, we want to introduce HSG-bases and discuss some of their properties. is concept is very similar to the concept of SAGBI-Gröbner bases. erefore, we will briefly explain the underlying common structure. Let Γ denote an ordered monoid, i.e., an abelian semigroup under an operation +, equipped with a total ordering > such that, for all α, β, c ∈ Γ, α > β⇒α + c > β + c.
A direct sum, is called grading (induced by Γ) or briefly a Γ-grading if for all α, β ∈ Γ, Since the decomposition above is a direct sum, each polynomial f ≠ 0 has a unique representation.
Assuming that c 1 > c 2 > · · · > c s , the Γ-homogeneous term f c 1 is called the maximal part of f, denoted by ere are two major examples of gradings. e first one is grading by degrees: Here, Γ � N with the natural total ordering. is grading is called the H-grading because of the homogeneous polynomials. erefore, we also write H in place of this Γ. e space of all polynomials of degree at most d can now be written as e maximal part of a polynomial f ≠ 0 is its homogeneous form of highest degree, M (H) (f). For simplicity, let M (H) (0): � 0.
e representation for f in (9) is also called its HSG representation with respect to G.
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.
ALGORITHM 1: Algorithm to compute si-reduction Input: a subalgebra A and a finite subset G ⊂ A. Output: ALGORITHM 2: Algorithm for the construction of HSG basis. 2 Discrete Dynamics in Nature and Society e concept of si− reduction plays an important role in the characterization and construction of HSG-basis. For f ∈ A and G ⊂ A, the following algorithm computes h such that f ⟶ G A , * h (i.e., f reduces to h completely).
We note that such an element a i in the subalgebra A can easily be determined as in the case of reduction in polynomial ring. We also note that deg(h − i a i g i ) is strictly smaller than the deg(h) (by the choice of i a i g i ). is shows that Algorithm 1 always terminates.
. . , g s ⊂ A (subset of subalgebra A) and I A be an ideal of A. en, the following conditions are equivalent: (1) G is an HSG-basis for the ideal I A .
Since G is an HSG-basis, by (9), there are some h 1 , . . . , h s ∈ A so that where h is si− reduced any further with respect to F.
If we follow the above process inductively, then f ⟶ G A , * 0. where Note that and deg s j�1 Hence, (11) and (15) give the HSG representation. □ e second major example of gradings leads to the SAGBI-Gröbner basis concept. Here, Γ � N n with component-wise addition equipped with a total ordering satisfying (11). In addition, c ≥ 0, ∀c ∈ Γ. For arbitrary c � (c 1 , . . . , c n ) ∈ Γ, the space P (Γ) c is a vector space of dimension 1, namely, e relation ⟶ G A , * is constructed as above.
A SAGBI-Gröbner basis G (with respect to a given monomial ordering and a given ideal I A in a subalgebra A) is a set of polynomials generating I A and satisfying one of the following equivalent conditions: (i) Every f ∈ I A has a representation: where h i ∈ A and g i ∈ G.
(iii) Every f ∈ I A si-reduces to 0 with respect to G. e 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, Discrete Dynamics in Nature and Society 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. e converse is false, as the following example shows.
ese polynomials belong to the subalgebra A � Q[x 2 , y 2 ]. en, we can see that f 1 , f 2 , and f 3 already constitute an HSG-basis for ideal If we order the monomials by degree lexicographical ordering, then . Every SAGBI-Gröbner basis G with respect to this ordering contains at least four elements, for instance, G � g 1 , g 2 , g 2 , g 4 with Obviously, this SAGBI-Gröbner basis is an HSG-basis as well.

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.
. , x n ] and a subset G � g 1 , . . . , g s ⊆ A, We call an element of syz A (G) an A− syzygy of G.
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. e analogue for constructing HSG-bases by means of syzygies is connected to the following result [7].  1 g 1 , . . . , a m 0 g m 0 are contributing to p 0 , i.e., M (H) (a i g i ) � p 0 for all 1 ≤ i ≤ m 0 . If we set a → � (a 1 , . . . , a m 0 , 0, . . . , 0), we can see that

Theorem 2 (HSG-basis criterion). Let
where m i�1 p j,i g i is an HSG representation for m i�1 q j,i g i since m i�1 q j,i g i ⟶ G 0. If we define H j � max (M (H) (p j,i g i )), then