Abstract
Let be a finite set of monomials of the same degree in a polynomial ring over an arbitrary field . We give some necessary and/or sufficient conditions for the birationality of the ring extension , where is the th Veronese subring of . One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygytheoretic birationality criterion in characteristic zero obtained in [1].
Linear syzygies and birational combinatorics
^{0}^{0}footnotetext: 2000 Mathematics Subject Classification. Primary 13H10; Secondary 14E05,14E07,13B22.^{1}^{1}footnotetext: Key words. Birational map, linear syzygies, monomial subring, Jacobian matrix, Cremona transformations.Aron Simis and Rafael H. Villarreal*
1 Introduction
By the expression “birational combinatorics” we mean the theory of characteristicfree rational maps defined by monomials, along with natural criteria for such maps to be birational onto their image varieties. Both the theory and the criteria are intended to be simple and typically reflect the monomial data, as otherwise one falls back in the general theory of birational maps in projective spaces (cf., e.g., [12], [15]).
A first incursion in this kind of theory was made in [16]. There one focused mainly on monomial rational maps whose base ideal (ideal theoretic base locus) was normal. Though the results were fairly complete and some of the techniques used there are repeated here, one felt that normality was a special case obscuring the general picture.
In the present paper we envisage a general theory focusing on the underlying combinatorial elements rather than on special algebraic properties of the base ideal. In this sense, what we accomplish goes in the opposite direction of recent work on birational maps, where the emphasis fell on special behavior of the base locus. On the other hand, we did draw upon [12] and [15] (also upon the ongoing [1]) by invoking the role played by the socalled linear syzygies of the coordinates of the rational map. The methods in the first two of these references are specially suited for the explicit computation of the inverse map of a birational map onto the image. To compromise between the two approaches, we show a bridge between them by means of comparing the respective linear algebra gadgets  from modules over the ground polynomial ring to modules over . The challenge remains as to how one computes the inverse map by a purely combinatorial method.
We now describe the content of the paper in more detail. It goes without saying that the language throughout is algebraic or combinatorial, although we do add frequent remarks as to the geometric meaning of the results.
Section 2 sets up the scenario for the basic pertinent integer combinatorics. We emphasize two criteria of birationality in this setup  the arithmetical principle of birationality and the determinantal principle of birationality. These criteria were used in [16] and seem to be part of the folklore in the scattered literature. Then, we introduce the various versions of matrices that will play a distinctive role in the theory and, in particular, replay in more generality the passage from the transposed Jacobian matrix to the logmatrix of a set of monomials, as devised in [14]. Since we wish to remain characteristicfree, we take the formal Jacobian matrix rather than the ordinary one, as is explained in the section. The soobtained numerical matrices allow for a first birationality criterion (Proposition 2.3). We then proceed to a full arithmetical characterization of birationality (Theorem 2.6).
Section 3 deals with the role of the Fitting ideals of monomial structures. We expand on the topic only enough in order to compare ranks between matrices over and matrices over . As a side bonus, we characterize totally unimodular logmatrices in terms of Fitting ideals of the formal Jacobian matrix. The main result of the section is Theorem 3.7, which extends one of the results of [1] to all characteristics for monomial rational maps.
Section 4 focuses on the case of monomials of degree . Here, we give complete results, covering all previously known results and establishing facts that do not extend to higher degrees. We introduce the notion of cohesiveness for rational maps of any degre inspired by the graph theoretic concept of connectedness. We show, preliminarily, that the lack of cohesiveness is an early obstruction for birationality and for the existence of “enough” linear syzygies. If, moreover, the degree is we show that cohesiveness is a necessary and sufficient condition for having a linear syzygy matrix of maximal rank (Proposition 4.6). We proceed to one of the main theorems of the section (Theorem 4.7) saying that a rational map of degree is birational onto its image if and only if it is cohesive and the corresponding logmatrix has maximal rank. This comes to us as a bit of a surprise as it says that any cohesive coordinate projection of the Veronesean that preserves dimension is birational onto the image; moreover, this holds in any characteristic. We have not met any explicit mention of this fact in the previous literature. We give examples to show how easily this fails for nonmonomial rational maps and for monomial ones in degrees . Finally we care to translate the results into the language of graphs with loops.
The last section has the purpose of describing sufficiently ample classes of monomial rational maps that are birational. It is further subdivided in two subsections, the first of which is entirely devoted to classes of Cremona maps. We characterize Cremona transformations of degree as those cohesive ones whose logdeterminant is nonzero. The corresponding graph theoretic characterization is suited to construct other Cremona transformations of higher degree via a certain duality principle. The second subsection is a pointer to a recently studied class of combinatorial objects called polymatroidal monomial sets. This class includes the toric algebras of Veronese type which, from the geometric angle, constitutes a vast class of dimension preserving projections of the ordinary Veronese embeddings.
2 Birationality of monomial subrings
Let be a polynomial ring over a field . As usual we set if . In the sequel we consider a finite set of distinct monomials of the same degree and having no nontrivial common factor. We also assume throughout that is not conic, i.e., that every divides at least one member of . By trivially contracting to less variables, any set of monomials can be brought to this form.
Two integer matrices naturally associated to are:
where the ’s are regarded as column vectors. We will often refer to as the logmatrix of .
If is an integer matrix with rows, we denote by (resp. ) the subgroup of (resp. subspace of ) generated by the columns of . will denote the greatest common divisor of all the nonzero minors of .
An extension of integral domains is said to be birational if it is an equality at the level of the respective fields of fractions. In the sequel let denote the set of all monomials of degree in . Then is the th Veronese subring of . Our main aim is the birationality of the ring extension .
For convenience of reference, we quote the following easy results stated in [16]:
Lemma 2.1
(Arithmetical Principle of Birationality (APB)) Let and be finite sets of monomials of such that , and let be their respective logmatrices. Then is a birational extension if and only if .
Proof. In this situation, the ring extension is birational if and only every monomial of can be written as a fraction whose terms are suitable power products of the monomials of . Clearing denominators of such a fraction and taking log of both members establishes the required equivalence.
Lemma 2.2
(Determinantal Principle of Birationality (DPB)) Let be a finite set of monomials of the same degree . Then is a birational extension if and only if .
Proof. See [16, Proposition 1.2].
By we denote the canonical basis vectors of the vector space (sometimes of the free module , respectively, the vector space ). Let, as before, be a set of monomials of the same degree .
Consider the following basic matrices:

(a) the matrix of the socalled linear syzygies of , whose columns are the set of vectors of the form such that ;

(b) the numerical linear syzygy matrix obtained from by making the substitution for all ;

(c) the matrix whose columns are the set of difference vectors such that , for some pair of indices – in other words, ;

(d) the formal Jacobian matrix
A word in order to explain the last matrix. The notion of derivative of a polynomial usually requires the specification of a base field. However, if is an ordinary monomial its formal partial derivative with respect to is defined to be
regarded as a term in the polynomial ring (in particular it is always nonzero provided ). The formal Jacobian matrix of is accordingly defined. Of course, by applying the unique homomorphism from to we find the ordinary partial derivatives and the ordinary Jacobian matrix over this ring.
Notice that the matrices in the first row of the diagram:
specialize to the matrices in the second row by making for all . The matrices and have order , while the matrices and have order .
Here is a couple of uses of these matrices. The following notion will be used in the proof below: a matrix is called totally unimodular if each minor of is or for all .
Proposition 2.3
Let be a finite set of monomials of the same degree .

If , then is a birational extension.

If and , then . In particular is birational.
Proof. (i) Let such that . By APB (Lemma 2.1) it suffices to prove that . Let be the column vectors of the matrix . Each is of the form for a unique pair and suitable . Hence because . Therefore we can write
Taking inner product with yields
(1)  
Consider the digraph with vertex set such that the directed edges correspond bijectively to the column vectors of . The incidence matrix of is , thus is totally unimodular [13, p. 274] and is torsionfree. Hence from Eq. (1) we get and , as required.
(ii) Consider the linear maps
Letting denote the restriction of to , we have a linear map
By hypothesis, and . Hence
Therefore . On the other hand, since is generated by vectors of the form , certainly , hence .
Remark 2.4
To proceed with a full arithmetical characterization of birationality, we will need the following results on modules over .
Lemma 2.5
(i) Let be the canonical basis
vectors of the free module and let be the submodule
generated by the difference vectors . Then is
freely generated by and the quotient
is torsionfree of rank one.
(ii) Let be arbitrarily given.
Then the injective homomorphism , , induces an injective homomorphism of modules
which is an isomorphism at the level of the respective torsion submodules.
Proof. (i) This is simply the fact that is the kernel of the homomorphism , .
(ii) Clearly, there is an induced map as argued – because maps to – and the induced map is injective – because the two equations and easily imply that .
Next, clearly any homomorphism maps torsion to torsion, so it remains to check surjectivity at the torsion level. Let then be a torsion element of . This implies a relation
where , and . Then
Hence it follows that the class is a torsion element and maps to , as required.
Theorem 2.6
Let be a finite set of monomials of the same degree . The following conditions are equivalent

(a) is birational.

(b) is free of rank .

(c) The logmatrix of has maximal rank and .
Proof. First we observe that, quite generally, there is an exact sequence of finite abelian groups
(2) 
(here and , for – see [16, Proof of Theorem 1.1].
If, moreover, has full rank then is torsion, hence if and only if is torsionfree, and in this case the th Fiting ideal of is the same as that of , i.e., .
On the other hand, we have an exact sequence of modules
(3) 
Again, if has full rank then the leftmost module has rank and, since the rightmost module is torsion, the mid module has rank . Now apply Lemma 2.5(ii) with and to get
Therefore, the equivalence (a) (b) follows from DBP of Lemma 2.2.
It remains to show that (b) (c). First, (c) (b) is clear by Lemma 2.5(i). For the reverse implication, since the mid term of the sequence (3) is assumed to be torsionfree of rank one and , then must have full rank and, moreover, is torsionfree of rank one. In particular, there is a splitting which, after extending to , implies
(4) 
Hence we get the desired equality because of the torsion freeness hypothesis. Notice that Eq. (4) also follows directly. Indeed if is a basis for the column space of , then
is also a basis because for all . Hence each can be written as
Taking inner products with the vector yields . Therefore we have shown the containment “” in Eq. (4). A symmetric argument proves the equality.
3 When are the Fitting ideals monomial ideals?
In [14, Lemma 1.1] was shown that the minors of the Jacobian matrix of a set of monomials are always monomials (possibly zero). The following result extends and clarifies the above assertion.
Proposition 3.1
Let be a graded ring with grading given by an additive abelian monoid . Let be a finitely generated graded module over . Then the Fitting ideals of are homogenous ideals of .
Proof. By assumption, there is an exact sequence of graded modules over
A Fitting ideal of is an ideal generated by the minors of , for a suitable . This ideal is the image of the wellknown induced graded homomorphism
Therefore, is a homogeneous ideal of .
Corollary 3.2
Let be given the standard multigrading (i.e., the grading with of degree ). If is a finitely generated multigraded module, then the Fitting ideals of are monomial ideals. In particular, any minor of the Jacobian matrix, respectively, of the syzygy matrix of arbitrary order, of a finite set of monomials is a monomial.
Proof. Apply Proposition 3.1 while noticing that a homogeneous polynomial in the standard multigrading is necessarily a monomial.
We can also apply the previous result in the case of the standard multigraded ring , with in degree . The result is that, in particular, the formal Jacobian matrix of a finite set of monomials has monomial Fitting ideals. We wish to emphasize this in the following form:
Corollary 3.3
The formal Jacobian matrix and the logmatrix of a finite set of monomials have the same number of zero or nonzero minors. In particular, these matrices have the same rank. Also, there are at most finitely many field characteristics over which the Jacobian matrix of over these characteristics has rank strictly smaller than the rank of the corresponding logmatrix.
There is also a consequence tied up with the notion of a unimodular matrix.
Corollary 3.4
The following are equivalent for a finite set of monomials.

The logmatrix of is totally unimodular

Every nonzero minor of the formal Jacobian matrix of has unit leading coefficient

The formal Jacobian matrix of has characteristicfree Fitting ideals (i.e., the Fitting ideals of over any field are generated by the same set of nonzero monomials).
As for the syzygies of , we observe that, in particular, any minor of the first Taylor syzygy matrix of (see [4] for an explanation of the Taylor complex) is a monomial with coefficient . We next include an alternative elementary proof of this fact alone, as the method of the proof might be useful in some other context.
Lemma 3.5
Let denote the Taylor syzygy matrix of . Then any nonzero minor of is a monomial with coefficient .
Proof. We proceed by induction on the size of the minor. The case being obvious, we assume that . We may clearly assume that the given minor is formed by the submatrix with the first rows and columns of . Let denote the submatrix of with the first columns. By definition of the Taylor syzygy matrix of , any column of the latter has exactly two nonzero entries. It follows that the complementary rows in to the rows of cannot all be zero as otherwise would be a matrix of syzygies of the initial monomials of , which is impossible since while the entire syzygy matrix of these monomials has rank .
Thus, there must be a nonzero entry in some complementary row to in , say, the th column, with . By the Taylor construction, there is exactly one further nonzero entry on the th column. This entry must belong to as otherwise . Also, this entry is again monomial with coefficient . Expanding by the th column yields the product of this monomial by the minor of a suitable submatrix of . By induction, this minor has the required form, hence so does .
Corollary 3.6
Let be any submatrix of and let denote the specialized matrix over obtained by sending . Then .
The next result complements one of the results of [1], where a criterion is given for a rational map to be birational in characteristic zero. The present proposition extends the latter result in all characteristics for monomial rational maps.
Theorem 3.7
Let be a finite set of monomials of the same degree . If and , then is birational.
Proof. By Corollary 3.6 (or by Corollary 3.2) the matrix obtained from by making for all has also rank . By Proposition 2.3(ii), the extension is birational.
Corollary 3.8
If the logmatrix has maximal rank and the ideal has a linear presentation, then is birational.
Proof. It follows at once from Theorem 3.7 because in this case the rank of is .
4 Monomials of degree two
The birational theory of monomials of degree two can be completely established using elementary graph theory as we show in the sequel.
We start with a general auxiliary result which holds, more generally, for any rational map between projective spaces.
Lemma 4.1
Let be forms of fixed degree . Suppose one has a partition of the variables such that , where the forms in the set (respectively, ) involve only the variables (respectively, variables). If neither nor is empty then:

The extension is not birational

The linear syzygy matrix of does not have maximal rank.
Proof. (i) Suppose to the contrary, i.e., that . Since clearly , it follows that . Say and . Then one has and, similarly, . But this is a contradicition as, e.g., (for instance, by APB (Lemma 2.1)).
(ii) The linear syzygy matrix of is a blockdiagonal matrix
where and are the linear syzygy matrices of and , respectively. Since and , then .
Definition 4.2
A set of forms of fixed degree will said to be cohesive if the forms have no nontrivial common factor and cannot be disconnected as in the hypothesis of the previous lemma.
Remark 4.3
The reason to assume that the forms have no nontrivial common factor is technical: multiplying a set of forms of the same degree by a given form yields the same rational map. To make the rational map correspond uniquely to a set of forms, one usually assumes that their gcd is one, i.e., that the ideal generated by these forms in the polynomial ring has codimension at least two (for further details on this and similar matters see [15]).
Yet another concept that fits the scene is a convenient extension of the notion of an ideal of linear type.
Definition 4.4
Let be forms of fixed degree , with . Consider a presentation of the Rees algebra where and is a bihomogeneous ideal. We will say that is of residual linear type if is generated in bidegrees and , where denotes an arbitrary integer .
Ideals of residual linear type are called ideals of fiber type in [11], it is shown there that polymatroidal ideals (see Section 5) are of fiber type. Thus, is of residual linear type if its relations are generated by the relations that define the symmetric algebra and the polynomial relations of with coefficients in the base field . A conjecture – perhaps only a question – regarding these ideals can be phrased as follows.
Conjecture 4.5
Let be a finite set of monomials of the same degree such that the ideal is of residual linear type. Then the following conditions are equivalent:

Both the logmatrix and the linear syzygy matrix of have maximal rank.

The extension is birational.
A comment on the reasonableness of the conjecture. The implication (i) (ii) is just Theorem 3.7.
The reverse implication (ii) (i) follows from the principle of linear obstruction [15, Proposition 3.5] (see also [1]) in the case of an ideal of linear type (necessarily, ). In order to suitably extend to ideals of residual linear type, one could in principle use the main criterion of [15] and the terminology thereof. Let denote the linear syzygy matrix of . Thus, the weak Jacobian matrix of ([15, Definition 2.2]) can be thought of as the Jacobian matrix of the quadrics in obtained by replacing every product in by , if , and by if (thus, we need char). In the case , an easy strong duality works here to yield that and the Jacobian dual of define the same cokernel, hence have the same rank. But the Jacobian dual of is , hence . For ideals of residual linear type, one needs an analogue that says , where , with a (prime) toric ideal. Since is monomial, a sufficiently elaborated application of Corollary 3.2 shows that the minors of are monomials. Since is toric, then . Therefore, we are reduced to show that . It is this the missing argument, for which one may have to bring in the other underlying facts of birationality – e.g., the logmatrix of has maximal rank and, moreover, coker is torsion free as module (the latter issues from the criterion in [15]).
Henceforth we assume that for all . It is convenient to interpret a set of monomials of degree two in terms of graphs, possibly with loops. Thus, consider the graph on the vertex set whose set of edges and loops correspond bijectively to the pairs such that (possibly ). Denote by the underlying simple graph obtained by omitting all loops. Notice that, in our situation, the logmatrix of is the incidence matrix of and the monomial subring is the edge subring of the graph .
One basic result for cohesive sets of monomials in degree reads as follows.
Proposition 4.6
If is a set of forms of degree with no nontrivial common factor. Then if and only if is cohesive.
Proof. One implication follows immediately from Lemma 4.1. For the reverse implication, assume that is cohesive. Then the corresponding graph as above is connected, hence the underlying simple graph has a spanning tree . Being a tree, has edges. The required result is easily verified in this case by induction on the number of vertices: consider the subtree obtained by removing a vertex of degree one and the corresponding edge, say, . By the inductive assumption, , so let denote an submatrix thereof of rank . If is any edge of then, by restoring the removed vertex and edge, yields a linear syzygy of involving edges and and a submatrix of formed by bordering with the corresponding column syzygy and a last rows of zeros. It is clear that this has rank .
This takes care of the spanning tree . Next, one successively restores edges and loops on to in order to recover the whole , this time with no new vertices. By a similar token, adding one such edge or loop at a time to the connected subgraph , will increase by one the rank of the new submatrix of formed by bordering as before the previous one with the column corresponding to the added edge or loop.
Before we set ourselves to state the main result of this section, the following observation seems pertinent. Quite generally, as used in the proof of Lemma 4.1 and easily shown, the field of fractions of the Veronese algebra is generated by the fractions and the pure power . Thus, a simple condition in order that be birational is that be expressed as a fraction whose terms are products of the monomials in . Now, in particular, if all these monomials are squarefree then a reasonable tour de force may be needed in order to accomplish it. Thus, e.g., for it is not difficult to guess that the corresponding simple graph must have a cycle of odd length. At the other end of the spectrum it is possible, by such elementary considerations, to guess conditions under which one has enough fractions out of the monomials in .
We chose to follow a more conceptual thread.
The next result generalizes [16, Corollary 3.2] and gives a complete answer for monomial birationality in degree two.
Theorem 4.7
Let be a finite set of monomials of degree two having no nontrivial common factor and let denote the corresponding graphs as above. Let denote the incidence matrix of . The following conditions are equivalent:

is cohesive and has maximal rank.

The extension is birational.

is connected and, moreover, either it is non bipartite or else it is bipartite and .
Proof. We first show that (i) and (ii) are equivalent. Clearly, (ii) implies that and cohesiveness follows from Proposition 4.6. The converse is a consequence of Theorem 3.7 and Proposition 4.6.
We next show that (ii) and (iii) are equivalent.
First, (iii) (ii).
Since is connected, there is a spanning tree of containing all the vertices of , see [9].
If is a bipartite graph and is a loop of . We may then regard as a tree with a loop at . Notice that has exactly simple edges plus a loop. The incidence matrix of has order , is non singular, and we may assume that the last column of is the transpose of . Consider the matrix obtained from by removing the last column. The matrix is totally unimodular because it is the incidence matrix of a simple bipartite graph [13, p. 273]. Therefore and . From Lemma 2.2 we obtain that is birational, hence is birational as well.
Now, let be a non bipartite graph. Then . Since has a spanning tree and has at least one odd cycle ([9, pp. 3739 and p. 42]), then admits a connected simple subgraph with vertices and edges with a unique cycle of odd length. By [16, Corollary 3.2] the extension is birational, hence so is .
Finally, we show the implication (ii) (iii).
By Proposition 4.6, must be cohesive, i.e., is connected. We have already seen that . Suppose that is bipartite. Then the logmatrix of has rank , hence has at least one loop.
Example 4.8
A geometer would summarize the result of Theorem 4.7 by saying that any cohesive coordinate projection of the Veronesean that preserves dimension is birational onto the image. This is clearly false if the projection is a non coordinate cohesive projection, e.g., if is a set of forms forming a cohesive regular sequence (the simplest example with would be ). At the other end, for , a cohesive coordinate projection of the Veronesean preserving dimension can fail to be birational for the simple reason that it may be composed with a noncohesive set. The simplest example of this phenomenon is . Here, is not birational, but its “reparametrization” is the Veronesean. For , one of the simplest examples is , which is cohesive of maximal rank, nonreparametrizable and nonbirational: the ideal is of linear type, but the linear syzygy matrix is of rank , hence falls below the needed value (of course, this apparatus in such a simple example is worthless since one immediately sees that does not belong to the field of fractions of ).
Corollary 4.9
Let be a connected simple bipartite graph. Assume that is an edge of an even cycle of . Then is a birational extension, where is the graph on the vertices obtained by contracting the edge to a loop around the vertex .
Proof. By the contractinglooping transformation, the resulting graph acquires an odd cycle. Therefore, the simple subgraph induced by is nonbipartite and the assertion follows from Theorem 4.7.
Remark 4.10
(a) The fact that a connected graph on vertices having exactly edges and a unique cycle of odd length induces a birational (Cremona) map had been guessed in [12, Conjecture 2.8] and proved in [16, Corollary 3.3] in a characteristicfree way. In characteristic zero, the more general context envisaged in [1] includes this result.
(b) If , Corollary 4.9 has a pretty geometric interpretation. The given ring extension ( bipartite) translates into a rational map
whose image is Proj, after normalizing the grading of . The induced ring extension corresponds to the restriction of to the hyperplane defined by and its image can be identified with the image of (actually, the algebras and are isomorphic as graded algebras by the contracting isomorphism sending for and ). Thus, restricts to a birational map of onto im.
5 Hall of examples
5.1 Monomial Cremona transformations
Among monomial birational maps, the Cremona ones form a wellknown distinguished class. A Cremona map is a birational map of onto itself. A recent surprising result ([8]) showed that the monomial Cremona transformations of is generated by the ones of degree and by the projective linear group, thus partially extending the classical result of M. Noether to higher dimension. The question as to which are the “standard ones” in dimension , if any at all, remains open as far as we know.
5.1.1 Monomial Cremona transformations of degree
We add a tiny contribution towards further understanding the structure of such maps. The next result extends a bit [16, Corollary 3.3] and likewise clarifies the algebraic/combinatorial background of the involved Cremona maps.
Proposition 5.1
Let be a cohesive finite set of monomials of degree two having no nontrivial common factor and let denote the corresponding graphs as above. Let denote the incidence matrix of . The following conditions are equivalent:


defines a Cremona transformation of

Either

(i.e., no loops), has a unique cycle and this cycle has odd length;
or else

is a tree with exactly one loop.


The ideal is of linear type.
Proof. The equivalence of (i) through (iii) follows immediately from Theorem 4.7, by noticing that if the underlying simple graph is bipartite and has exactly edges and loops, then the latter has to be a tree with exactly one loop. To see that the first three conditions are also equivalent to (iv), notice that (iv) implies (i) since the generators of an ideal of linear type are analytically independent, hence algebraically independent as they are forms of the same degree. Now, when , the implication (iii)(a) (iv) is part of [16, Corollary 3.3] but has really been noticed way before in [17, Corollary 3.2] (see also [18, Corollary 8.2.4]). Thus, it remains to see that (iii)(b) (iv) in the case where effectively has loops. This follows from Lemma 5.2 below using induction and noticing that an edge with a loop is clearly of linear type.
Lemma 5.2
Let be a set of monomials of degree two and let be a monomial in , where is a new variable and . If is of linear type, then is of linear type.
Proof. Let be the Rees algebra of over the extended polynomial ring . Let denote the presentation ideal of , i.e., the kernel of the graded epimorphism:
We may assume that extends the presentation ideal of the Rees algebra over via the natural inclusion . We know that is a graded ideal in the standard grading of with . To show that is of linear type we have to show that for all . We proceed by induction on , the result being vacuous for . Thus, assume . Since is a toric ideal, it is generated by binomials. Therefore, by the inductive hypothesis, it suffices to show that any binomial in belongs to . Let
be a binomial in , where are distinct integers between and , for all and . We may assume that , otherwise because is of linear type. From the equality
follows that divides , since no on the right side of this equality involves the variable . Thus there is a relation
(5) 
where one of the ’s may be zero and . We may assume that because not all ’s are zero. Consider the equality
(6) 
where and