The theory of toric varieties has been extensively studied since its foundation in the early 1970s by Demazure , Miyake-Oda , Mumford et al. , and Satake , due to its deep connections with polytopes, combinatorics, symplectic geometry, topology, and its applications in physics, coding theory, algebraic statistics, and hypergeometric functions [4, 9, 19].
In this paper, we initiate the study of toric difference varieties and expect that they will play similar roles in difference algebraic geometry to their algebraic counterparts in algebraic geometry. Difference algebra and difference algebraic geometry [2, 10, 12, 23] were founded by Ritt  and Cohn , who aimed to study algebraic difference equations as algebraic geometry to polynomial equations.
Similar to the algebraic case, a difference variety is called toric if it is the Cohn closure of the values of a set of Laurent difference monomials. To be more precise, we introduce the notion of symbolic exponent. For and in a difference field with the difference operator , denote Then a Laurent difference monomial in the difference indeterminates has the form , where . For
define the following map
where is any difference extension field of and . Then, the toric difference variety defined by is the Cohn closure of the image of .
A -lattice is a -submodule of , which plays the similar role as lattice does in the study of toric algebraic varieties. A -lattice is called toric if for any and . We show that a difference variety is toric if and only if the defining difference ideal for is where is a toric -lattice and is a set of difference indeterminants. An algorithm is given to decide whether a -lattice is toric, and consequently, to decide whether defines a toric difference variety.
Similar to the algebraic case, a difference variety is toric if and only if contains a difference torus as a Cohn open subset and with a difference algebraic group action of on extending the natural group action of on itself. Distinct from the algebraic case, a difference torus is not necessarily isomorphic to , and this makes the definition of the difference torus more complicated.
Many properties of toric difference varieties can be described using affine -semimodules. An affine -semimodule generated by in (1) is . It is shown that a difference variety is toric if and only if , where is an affine -semimodule in and . Furthermore, there is a one-to-one correspondence between irreducible invariant subvarieties of a toric difference variety and faces of the corresponding affine -semimodule. A one-to-one correspondence between orbits of a toric difference variety and faces of the corresponding affine -semimodule is also established.
Toric difference varieties connect difference Chow forms  and sparse difference resultants . Precisely, it is shown that the difference Chow form of is the difference sparse resultant of generic difference polynomials with monomials . As a consequence, a Jacobi style order bound for a toric difference variety is given.
The rest of this paper is organized as follows. In section 2, preliminaries for difference algebra are introduced. In section 3, the concept of difference toric variety is defined and its coordinate ring is given in terms of affine -semimodules. In section 4, the one-to-one correspondence between toric difference varieties and toric difference ideals is given. In section 5, a description of toric difference varieties in terms of group action is given. In section 6, deeper connections between toric difference varieties and affine -semimodules are given. In section 7, an order bound for a toric difference variety is given. In section 8, an algorithm is given to decide whether a given -lattice is -saturated. Conclusions are given in Section 9.
A difference ring, or -ring for short, is a ring together with a ring endomorphism . If is a field, then we call it a difference field, or a -field for short. A morphism between -rings and is a ring homomorphism which preserves the difference operators. In this paper, all -fields have characteristic and is a base -field.
A -algebra is called a --algebra if the algebra structure map is a morphism of -rings. A morphism of --algebras is a morphism of -algebras which is also a morphism of -rings. A -subalgebra of a --algebra is called a --subalgebra if it is stable under . If a --algebra is a -field, then it is called a -field extension of . Let and be two --algebras. Then is naturally a --algebra by defining for and .
Let be a -field and a --algebra. For a subset of , the smallest --subalgebra of containing is denoted by . If there exists a finite subset of such that , we say that is finitely -generated over . If moreover is a -field, the smallest --subfield of containing is denoted by .
Now we introduce the following useful notation. Let be an algebraic indeterminate and . For in a -field, denote with and . It is easy to check that .
Let a set of -indeterminates over . Then the -polynomial ring over in is the polynomial ring in the variables for and . It is denoted by and has a natural --algebra structure. A -polynomial ideal, or simply a -ideal, in is an algebraic ideal which is closed under , i.e. . If also has the property that implies that , it is called a reflexive -ideal. A -prime ideal is a reflexive -ideal which is prime as an algebraic ideal. A -ideal is called perfect if for any and , implies . It is easy to prove that every -prime ideal is perfect. If is a finite set of -polynomials in , we use , , and to denote the algebraic ideal, the -ideal, and the perfect -ideal generated by respectively.
For , is called a Laurent -monomial and is called its support. A Laurent -polynomial in is a linear combination of Laurent -monomials and denotes the set of all Laurent -polynomials, which is a --algebra.
Let be a -field. We denote the category of -field extensions of by and the category of by where . Let be a set of -polynomials. For any , define the solutions of in to be
Note that is naturally a functor from the category of -field extension of to the category of sets. Denote this functor by .
Let be a -field. An (affine) difference variety or -variety over is a functor from the category of -field extension of to the category of sets which is of the form for some subset of . In this situation, we say that is the (affine) -variety defined by . If there is no confusion, we will omit the word “affine” for short.
The functor given by for is called the -affine (-)space over . If the base field is specified, we often omit the subscript .
Let be a subset of . Then
is called the vanishing ideal of . It is well known that -subvarieties of are in a one-to-one correspondence with perfect -ideals of and we have for .
Let be a -subvariety of . Then the --algebra
is called the -coordinate ring of .
A --algebra isomorphic to some is called an affine --algebra. By definition, is an affine --algebra. Similar to affine algebraic varieties, the category of affine --varieties is antiequivalent to the category of affine --algebras . The following lemma is from [23, p.27].
Let be a --variety. Then for any , there is a natural bijection between and the set of --algebra homomorphisms from to . Indeed,
Suppose that is an affine --algebra. Let be the set of all -prime ideals of . Let . Set
It can be checked that is a topological space with closed sets of the form . Then the topological space of is equipped with the above Cohn topology.
Let be a -field and . Let . Two solutions and are called equivalent if there exists a --isomorphism between and which maps to . Obviously this defines an equivalence relation. The following theorem gives a relationship between equivalence classes of solutions of and -prime ideals containing . See [23, p.31].
Let be a --variety. There is a natural bijection between the set of equivalence classes of solutions of and .
We shall not strictly distinguish between a -variety and its topological space. In other words, we use to mean the -variety or its topological space.
3 Affine toric -varieties
In this section, we will define affine toric -varieties and give a description of their coordinate rings in terms of affine -semimodules.
Let be a -field. Let be the functor from to satisfying where and . In the rest of this section, always assume
a set of -indeterminates. We define the following map
Define the functor from to with for each which is called the quasi -torus defined by .
An affine -variety over the -field is called toric if it is the Cohn closure of a quasi -torus in . Precisely, let
Then the (affine) toric -variety defined by is . The matrix with as the -th column is called the matrix representation for .
defined above is an irreducible -variety of -dimension , where is the matrix representation of .
Let . Let be the -ideal generated by in . Then it is easy to check
Alternatively, let be a new -indeterminate and be a -ideal in , where are the postive and negative parts of , respectively, . Then
The following example shows that some might not appear effectively in .
Let . By (7), and does not appear in .
Next, we will give a description for the coordinate ring of a toric -variety in terms of affine -semimodules. is called an -semimodule if it satisfies (i) ; (ii) . Moreover, if there exists a finite subset such that , is called an affine -semimodule. A map between two -semimodules is an -semimodule morphism if for all and .
Let be a -field. For every affine -semimodule , we associate it with the following -semimodule algebra which is the vector space over with as a basis and multiplication induced by the addition of . More concretely,
with multiplication induced by . Make to be a --algebra by defining
If , then . Therefore, is a finitely -generated --algebra. When an embedding is given, it induces an embedding . So is a --subalgebra of generated by finitely many Laurent -monomials and it follows that is a -domain. We will see that is actually the -coordinate ring of a toric -variety.
Let be an affine -variety. Then is a toric -variety if and only if there exists an affine -semimodule such that . Equivalently, the -coordinate ring of is .
The map is surjective by the definition of . If , then , which is equivalent to . Then, and . Therefore . Conversely, if , where is an affine -semimodule, and for . Let be the toric -variety defined by . Then as we just proved, the coordinate ring of is isomorphic to . Then .
We further have
Let be an affine -semimodule and let be the toric -variety associated with . Then there is a one-to-one correspondence between and , . Equivalently, .
Proof: By Lemma 2.3, an element of is given by a --algebra homomorphism , where . This corresponds to such a morphism satisfying such that .
In the rest of this paper, we will identity elements of with morphisms from to and use to denote these elements.
4 Toric -ideal
In this section, we will show that -toric varieties are defined exactly by toric -ideals. We first define the concept of -lattice which is introduced in .
A -module which can be embedded into for some is called a -lattice. Since is Noetherian as a -module, we see that any -lattice is finitely generated. Let be generated by , which is denoted as . Then the matrix with as the -th column is called a matrix representation of . Define the rank of to be the rank of its representing matrix. Note may not be a free -module, thus the number of minimal generators of could be larger than its rank.
A -lattice is called toric if it is -saturated, that is for any nonzero and , implies .
Associated with a -lattice , we defined a binomial -ideal
where are the positive part and the negative part of , respectively. If is toric, then the corresponding -lattice ideal is called a toric -ideal.
has the following properties.
A -variety is toric if and only if is a toric -ideal.
Let be the toric -variety defined in (5). Then is a toric -ideal whose support lattice is , where is the matrix with columns .
Proof: is clearly a toric -lattice. Then it suffices to show that , where is defined in (5). For , we have . As a consequence, and . Since is generated by for , we have .
To prove the other direction, consider a total order for the -monomials , which extends to a total order over by comparing the largest -monomial in a -polynomial. We will prove . Assume the contrary, and let be a minimal element in under the above order. Let be the biggest -monomial in . From , we have . Since is a -monomial about and , there exists another -monomial in such that . As a consequence, , from which we deduce and hence . Then , which contradicts to the minimal property of , since .
Let be a -lattice. Define the orthogonal complement of to be
where is the dot product of and . It is easy to show that
Let be a matrix representation for . Then and hence . Furthermore, if is a toric -lattice, then .
The following lemma shows that the inverse of Lemma 4.3 is also valid.
If is a toric -ideal in , then is a toric -variety.
Proof: Since is a toric -ideal, then the -lattice corresponding to , denoted by , is toric. Suppose is a set of generators of . Regard as a matrix with columns and let be the set of rows of . Consider the toric -variety defined by the . To prove the lemma, it suffices to show or . Since toric -ideals and toric -lattices are in a one-to-one correspondence, we only need to show . This is clear since .
Use notations introduced in Example 3.3. Let . Then . By Lemma 4.3, we have Conversely, let be the support lattice of . Then is the defining matrix for . By Lemma 4.5, is the defining matrix for the toric -variety . In Example 3.3, we need to use the difference characteristic set method to compute . Here, the only operation used to compute is Gröbner basis methods for -lattices .
Finally, we have the following effective version of Theorem 4.2.
A toric variety has the parametric representation and the implicit representation , where is given in (3) and for . Then, there is a polynomial-time algorithm to compute from and vise versa.
Proof: The proofs of Lemma 4.3 and Lemma 4.4 give algorithms to compute from , and vice versa, provided we know how to compute a set of generators of for a matrix with entries in . In , a polynomial-time algorithm to compute the Gröbner basis for -lattices is given. Combining this with Schreyer’s Theorem on page 224 of , we have an algorithm to compute a Gröbner basis for as a -module. Note that, when a Gröbner basis of the -lattice generated by the columns of is given, the complexity to compute a Gröbner basis of using Schreyer’s Theorem is clearly polynomial.
In other words, toric -varieties are unirational -varieties, and we have efficient implicitization and parametrization algorithms for them.
5 -torus and toric -variety in terms of group action
In this section, we will define the -torus and give another description of toric -varieties in terms of group actions by -tori.
Let be the quasi -torus and the toric -variety defined by in (4). In the algebraic case, is a variety, that is, , where is the field of complex numbers and . The following example shows that this is not valid in the -case.
In Example 3.3, .
Let . Then . On the other hand, assume which means
, or the -equations have a solution in . In what below, we will show that this is impossible. That is, .
Let . We have . Then, . Since . Then