A polynomial ideal is called binomial if it is generated by polynomials with at most two terms. Binomial ideals were first studied by Eisendbud and Sturmfels , which were further studied in [1, 6, 19, 16, 23] and were applied in algebraic statistics , chemical reactions , and error-correcting codes .
In this paper, we initiate the study of binomial difference ideals and hope that they will play similar roles in difference algebraic geometry to their algebraic counterparts. Difference algebra and difference algebraic geometry were founded by Ritt  and Cohn , who aimed to study algebraic difference equations in the way polynomial equations were studied in commutative algebra and algebraic geometry [4, 13, 17, 26].
We now describe the main results of this paper. In Section 3, we prove basic properties of -lattices. By a -lattice, we mean a -module in . -lattices play the same role as -lattices do in the study of binomial ideals. Here, is used to denote the difference operator . For instance, is denoted as . Since is not a PID, the Hermite normal form for a matrix with entries in does not exist. In this section, we introduce the concept of generalized Hermite normal form and show that a matrix is a generalized Hermite normal form if and only if its columns form a reduced Gröbner basis for a -lattice.
In Section 4, we give three canonical representations for Laurent binomial difference ideals in terms of reduced Gröbner bases of -lattices, difference characteristic sets, and partial characters. Gröbner bases play an important role in the study of binomial ideals . In general, a binomial difference ideal is not finitely generated and does not have a finite Gröbner basis. Instead, the theory of characteristic set for difference polynomial systems  is used for similar purposes. It is shown that any Laurent binomial difference ideal can be written as , where is a regular and coherent difference ascending chain consisting of binomial difference polynomials.
Let be a proper Laurent binomial difference ideal and the support lattice of , which is a -lattice. In Section 5, we give criteria for a Laurent binomial difference ideal to be prime, reflexive, well-mixed, and perfect in terms of its support lattice. The criterion for prime ideals is similar to the algebraic case, but the criteria for reflexive, well-mixed, and perfect difference ideals are unique to difference algebra and are first proposed in this paper. Furthermore, it is shown that the reflexive, well-mixed, and perfect closures of a Laurent binomial difference ideal with support lattice are still binomial, whose support lattices are the -, -, and the -saturation of , respectively. It is further shown that any perfect Laurent binomial difference ideal can be written as the intersection of Laurent reflexive prime binomial difference ideals whose support lattices are the --saturation of the support lattice of .
In Section 6, binomial difference ideals are studied. It is shown that a large portion of the properties for binomial ideals proved in  can be easily extended to the difference case. We also identify a class of normal binomial difference ideals which are in a one to one correspondence with Laurent binomial difference ideals. With the help of this correspondence, most properties proved for Laurent binomial difference ideals are extended to the non-Laurent case.
In Section 7, algorithms are given to check whether a -lattice is -, -, M-, or P-saturated, or equivalently, whether a Laurent binomial difference ideal is prime, reflexive, well-mixed, or perfect. If the answer is negative, we can also compute the -, -, M-, or P-saturation of and the reflexive, well-mixed, or perfect closures of . Based on these algorithms, we give an algorithm to decompose a finitely generated perfect binomial difference ideal as the intersection of reflexive prime binomial difference ideals. This algorithm is stronger than the general decomposition algorithm in that for general difference polynomials, it is still open on how to decompose a finitely generated perfect difference ideal as the intersection of reflexive prime difference ideals .
A distinctive feature of the algorithms presented in this paper is that problems about difference binomial polynomial ideals are reduced to problems about -lattices which are pure algebraic and have simpler structures.
2 Preliminaries about difference algebra
2.1 Difference polynomial and Laurent difference polynomial
An ordinary difference field, or simply a -field, is a field with a third unitary operation satisfying: for any , , , and if and only if . We call the transforming operator of . If , is called the transform of and is denoted by . For , is called the -th transform of and denoted by , with the usual assumption . If is defined for each , is called inversive. Every -field has an inversive closure . A typical example of inversive -field is with .
In this paper, is assumed to be inversive and of characteristic zero. Furthermore, we use - as the abbreviation for difference or transformally.
We introduce the following useful notation. Let be an algebraic indeterminate and . For in any -over field of , denote
For instance, . It is easy to check that for , we have
By we mean the set . If is a set of elements, we denote .
Let be a subset of a -field which contains . We will denote respectively by , , , and the smallest subring, the smallest subfield, the smallest -subring, and the smallest -subfield of containing and . If we denote , then we have and .
Now suppose is a set of -indeterminates over . The elements of are called -polynomials over in , and itself is called the -polynomial ring over in . A -polynomial ideal, or simply a -ideal, in is an ordinary algebraic ideal which is closed under transforming, i.e. . If also has the property that implies that , it is called a reflexive -ideal. A prime -ideal is a -ideal which is prime as an ordinary algebraic polynomial ideal. For convenience, a prime -ideal is assumed not to be the unit ideal in this paper. A -ideal is called well-mixed if implies for . A -ideal is called perfect if for any and , implies . If is a subset of , we use , , , and to denote the algebraic ideal, the -ideal, the well-mixed -ideal, and the perfect -ideal generated by .
An -tuple over is an -tuple of the form where the are selected from a -overfield of . For a -polynomial , is called a -zero of if when substituting by in , the result is .
For , we define . is called a Laurent -monomial in and is called its support
. A nonzero vectoris said to be normal if the leading coefficient of is positive, where is the largest subscript such that .
A Laurent -polynomial over in is an -linear combination of Laurent -monomials in . Clearly, the set of all Laurent -polynomials form a commutative -ring under the obvious sum, product, and the usual transforming operator , where all Laurent -monomials are invertible. We denote the -ring of Laurent -polynomials with coefficients in by . Let be a Laurent -polynomial in . An -tuple over with each is called a nonzero -solution of if
2.2 Characteristic set for a difference polynomial system
Let be a -polynomial in . The order of w.r.t. is defined to be the greatest number such that appears effectively in , denoted by . If does not appear in , then we set . The order of is defined to be , that is, .
The elimination ranking on is used in this paper: if and only if or and , which is a total order over . By convention, for all .
Let be a -polynomial in . The greatest w.r.t. which appears effectively in is called the leader of , denoted by and correspondingly is called the leading variable of , denoted by . The leading coefficient of as a univariate polynomial in is called the initial of and is denoted by .
Let and be two -polynomials in . is said to be of higher rank than if or and .
Suppose . is said to be reduced w.r.t. if for all .
A finite sequence of nonzero -polynomials is said to be a difference ascending chain, or simply a -chain, if and or , and is reduced w.r.t. for .
A -chain can be written as the following form
where for and for . The following are two -chains
Let be a -chain with as the initial of , and any -polynomial. Then there exists an algorithm, which reduces w.r.t. to a polynomial that is reduced w.r.t. and satisfies the relation
where the and is called the -remainder of w.r.t. .
A -chain contained in a -polynomial set is said to be a characteristic set of , if does not contain any nonzero element reduced w.r.t. . Any -polynomial set has a characteristic set. A characteristic set of a -ideal reduces to zero all elements of .
Let be a -chain, , . is called regular if for any , is invertible w.r.t  in the sense that contains a nonzero -polynomial involving no . To introduce the concept of coherent -chain, we need to define the -polynomial first. If and have distinct leading variables, we define . If and () have the same leading variable , then . Define
Then is called coherent if for all .
Let be a -chain. Denote to be the minimal multiplicative set containing the initials of elements of and their transforms. The saturation ideal of is defined to be
The following result is needed in this paper.
[9, Theorem 3.3] A -chain is a characteristic set of if and only if is regular and coherent.
In this section, we prove basic properties of -lattices, which will play the role of lattices in the study of binomial ideals.
For brevity, a -module in is called a -lattice. Since is a Noetherian ring, any -lattice has a finite set of generators :
A matrix representation of or is
with to be the -th column of . We also denote . The rank of a -lattice is defined to be the rank of any matrix representation of , which is clearly well defined.
We list some basic concepts and properties of Gröbner bases of modules. For details, please refer to .
Denote to be the -th standard basis vector , where lies in the -th row of . A monomial in is an element of the form , where and . The following monomial order of will be used in this paper: if , or and , or , , and .
With the above order, any can be written in a unique way as a linear combination of monomials, where and . The leading term of is defined to be For any , we denote by the set of leading terms of .
The order can be extended to elements of as follows: for , if and only if .
Let and . We say that is G-reduced with respect to if any monomial of is not a multiple of by an element in for any .
A finite set is called a Gröbner basis for the -lattice generated by if for any , there exists an , such that . A Gröbner basis is called reduced if for any , is G-reduced with respect to . In this paper, it is always assumed that .
Let be a Gröbner basis. Then any can be reduced to a unique normal form by , denoted by , which is G-reduced with respect to .
Let , . Then the S-polynomial of and is defined as follows: if then ; otherwise
The following basic property for Gröbner basis is obviously true for -lattices and a polynomial-time algorithm to compute Göbner bases for -lattices is given in .
Theorem 3.3 (Buchberger’s Criterion)
The following statements are equivalent.
is a Gröbner basis.
for all .
if and only if .
We will study the structure of a Gröbner basis for a -lattice by introducing the concept of generalized Hermite normal form. First, we consider the case of .
Let be a reduced Gröbner basis of a -module in , , and . Then
2) and for .
3) for . If is the primitive part of , then for .
4) The S-polynomial can be reduced to zero by for any .
Proof: 1) and 4) are consequences of Theorem 3.3. To prove 2), assume that there exists an such that but . Let , where . Then and . Since , we have . Let . Then which is reduced w.r.t. and , contradicting to the definition of Gröbner bases.
We prove 3) by induction on . When , let and . Then, and . Let , we need to show . Since the S-polynomial can be reduced to zero by , we have , where and . Then, , and follows since is a primitive polynomial in . The claim is true. Assume that for , the claim is true, then for . We will prove the claim for . Since can be reduced to zero by . We have with and . Then, . By induction, is a factor of the right hand side of the above equation. Thus . Let for , we have where and . Since , we have and . For any , assume . We will show that . Since , we have is a factor of the right hand side of the above equation, for . Then, and . The claim is proved.
Here are three Gröbner bases in : , , .
To give the structure of a reduced Gröbner basis similar to that in Example 3.5, we introduce the concept of generalized Hermite normal form. Let
whose elements are in . It is clear that and . We denote by to be the column of the matrix whose last nonzero element is
Then the leading monomial of is . It is clear that .
The matrix in (6) is called a generalized Hermite normal form if it satisfies the following conditions:
for any .
can be reduced to zero by the column vectors of the matrix for any .
is G-reduced w.r.t. the column vectors of the matrix other than , for any .
is a reduced Gröbner basis such that if and only if is a generalized Hermite normal form.
The following matrices are generalized Hermite normal forms
whose columns constitute the reduced Gröbner bases of the -lattices.
Let be a reduced Gröbner basis. Let be the S-polynomial of and be the normal representation of in terms of the Gröbner basis . Then the syzygy polynomial
is an element in , where is the -th standard basis vector of . Define an order in as follows: if in . By Schreyer’s Theorem [5, p. 212], we have
Let be a generalized Hermite normal form. Then the syzygy polynomials form a Gröbner basis of the -lattice under the newly defined order .
Let be defined in (6) and . Introduce the following notations:
We need the following properties about . By saying the infinite set is linear independent over , we mean any finite subset of is linear independent over . Otherwise, is said to be linear dependent.
The columns of in (8) are linear independent over .
Proof: Suppose is given in (6). The leading term of is for and . Furthermore, for two different and in such that and