On Syzygy Modules over Laurent Polynomial Rings

Abstract

In this paper, we present a dynamical method for computing the syzygy module of multivariate Laurent polynomials with coefficients in a Dedekind ring (with zero divisors) by reducing the computation over Laurent polynomial rings to calculations over a polynomial ring via an appropriate isomorphism.

1. Introduction

Our goal is to give a dynamical method for computing a finite basis for the syzygy module of finitely many multivariate Laurent polynomials with coefficients in a Dedekind ring . More precisely, given nonzero polynomials , we will compute generating the syzygy module . The technique consists in reducing the computation over the Laurent polynomial ring to a problem over a polynomial ring via an appropriate isomorphism. One advantage of this indirect approach is that we can use techniques over polynomial rings more efficient than the corresponding methods in Laurent polynomial rings. Such kind of algorithm can be used in signal processing for the computation of the inverse FIR filter of a given multidimensional FIR filter. Our approach is inspired by the theory developed in the papers [1–13] and outlined in the next section.

2. Computing Dynamically a Basis for Syzygies of Polynomials over Dedekind Rings

Let be a multiplicative subset of a ring ; then, the localization of at is the ring and the elements of are forced to be invertible. If , the localization of at the multiplicative subset generated by will be denoted by . Moreover, by induction, for each , we define .

Now, let be a Dedekind ring and consider . First of all, we need to present a dynamical process [6] for computing a basis for . This method works like the case where the basic ring is a Noetherian valuation ring [11]. The Noetherian hypothesis is added so that a dynamical Gröbner basis for the ideal of can be computed. The only difference is when one has to handle two incomparable (under division) elements in . In this situation, one should first compute such that

Henceforth, one opens two branches: the computations are pursued in and . Note that contrary to [11], the localization instead of is used in order to avoid redundancies. The Dedekind ring is forced to behave like a valuation ring and this situation will produce a binary tree in which leaves correspond to localizations , , of at comaximal multiplicative subsets . The fact that a basis for can be computed at each leaf together with Lemma 1 will yield the desired one.

Let denote a basis for over , . There exists a such that , for each , and is a generator for over . As explained in Theorem II.3.6 [10], we have the following concrete local-global principle for coherent modules.

Lemma 1. (syzygy, coherent modules). Let be a ring, be comaximal monoids, be a module, and .(1)The syzygy module of the vector whose elements are seen as vectors in is finitely generated if and only if each syzygy module of the ’s vector ( are seen as vector in ) is finitely generated.(2) is coherent if and only if each is coherent.(3)The ring is coherent if and only if each is coherent.

Proof. (1)Let be a monoid in and be the syzygy module of the vector whose elements are considered in . We will prove that . It is clear that . Conversely, if in , let us denote and , such that in and in for . We have and in .(2)Let and be the module of relations for . For all monoids , is the module of relations for in .(3)Is a particular case of 2.

Theorem 1. Let and , where is a polynomial in such that . The syzygy module of over is the same than the syzygy module of over .

Proof. Let us denote by the syzygy module of over and that of in .
If over , then , so we have .
Conversely if over , we can suppose that (the set of generators of over can be supposed with only elements of ) and we can write . Finally, .

3. Syzygies of Laurent Polynomials over Dedekind Ring

Now, we can give a method for computing a set of generators for the syzygy module, , over the Laurent polynomial ring . For this, we need the following.

Lemma 2 (see [14]). Let be the ideal of generated by ; then,

Proof. Let be an -algebra homomorphism defined from to byand extended to as follows: for and ,Considering such isomorphism, our problem is reduced to a computation over the polynomial ring . More precisely, we are going to rely on such isomorphism between and to obtain an algorithm computing syzygy basis in .
The previous isomorphism is well defined, as it does not depend on the way elements of and are represented. First, every Laurent polynomial can be written as a polynomial in the variables , and to get an element of from a Laurent polynomial by this isomorphism, we have to consider the relation on .
Note that if with being a term of and , we haveConversely, each element of is the image of a polynomial in through the canonical (surjective) homomorphism of -algebras:which converts into
Given , we obtain by replacing by and by , bearing in mind that . Each can be expressed as algebraic combination of , with coefficients in . By taking this expression of without the bars over variables, we get a polynomial such that . Also, each with as image can be written as where .
By the isomorphism between and , we get the image of replacing by and by .

Theorem 2. Let be a set of generators for ; then, is a set of generators for .

Proof. Since is equivalent to with , the -homomorphismis surjective. We conclude that if is a set of generators for , then is a set of generators for .

4. Illustrative Examples

Example 1. Let be the ideal of generated byTo compute a set of generators for , we rely on the isomorphism between and . It is equivalent to compute a set of generators for withFirst of all, let us compute in a set of generators for withLet us use the lexicographic order with as monomial order to compute a dynamical Gröbner basis for in . As the leading coefficients of and are not comparable under division in and , we can open two following leaves to proceed:In , . Since the leading coefficients of and are not comparable, we need to open two new leaves asIn , is a special Gröbner basis for at the leaf . And over , we have asIn , we find as a special Gröbner basis for at the leaf , and we have the following over :In , we proceed as above, and we will open two leaves: is a special Gröbner basis for at the leaf , and we get asNote that .
Finally, over isNote that . Hence, a set of generators for isTherefore, a set of generators for isFinally, a set of generators of over isNow, let us see an example where the basic ring is not principal.

Example 2. Let us consider the ring , where is its ideal generated byTo compute a set of generators for , we rely on the isomorphism between and . It is equivalent to compute a set of generators for withFirst of all, let us compute in a set of generators for withUsing the lexicographic order with as monomial order, we proceed as above by computing a dynamical Gröbner basis for in .
Let us denote by and the leading coefficients of and , respectively. Since and are not comparable, we have to find such thatWith the solution of this system given by ,, and , we can open two leaves:In :Thus, is a special Gröbner basis for at the leaf .
And we have on , asIn , as 2 and 3 are not comparable under division in , in order to pursue the computations, we need to open two new leaves:The final evaluation tree is given byIn , we get as a special Gröbner basis forat the leaf . And over , isIn , we get as a special Gröbner basis forat the leaf . And on , isHence, over , a generating set of isNote that we have , and over , the generation is reduced to asIn fact the trick in Lemma 2 can be used for computing syzygies of a finite system in any module over when is a coherent module. Here, and .
Note that a generator set for the syzygy module of over the Laurent polynomial ring can be directly computed in by multiplying by a polynomial such that the new vector obtained is in , by virtue of Theorem1. The syzygy module does not change and the problem is reduced to a computation of a generator set for the syzygy module in .

Example 3. Let in . The problem of computing a generator for the syzygy module of can be reduced to find a generator for the syzygy module of in .
is obtained by multiplying by .
Let us consider the lexicographic order with .
As and we have to open two leaves: and .
In , . The leading coefficients of and are comparable under division. As , we open in two new leaves and .
In :So, is a special Gröbner basis for over .
We obtainIn , we obtain as special Gröbner basis for over .
Also,We find as special Gröbner basis for over .
Also,Finally, we obtainThe two approaches used give similar results. In the approach used in the last example, there is one less polynomial than using the isomorphism approach, but the calculations remain similar and relation allows simplifications with the isomorphism approach.

Data Availability

The data used to support the findings are included within the article and are cited as references.

Conflicts of Interest

The authors declare that they have no conflicts of interest.