Products of polymatroids with the strong exchange property
aa r X i v : . [ m a t h . A C ] F e b PRODUCTS OF POLYMATROIDS WITH THE STRONGEXCHANGE PROPERTY
LISA NICKLASSON
Abstract.
It was conjectured by White in 1980 that the toric ring associatedto a matroid is defined by symmetric exchange relations. This conjecture wasextended to discrete polymatroids by Herzog and Hibi, and they prove that theconjecture holds for polymatroids with the so called strong exchange property .In this paper we generalize their result to polymatroids that are products ofpolymatroids with the strong exchange property. This also extends a resultby Conca on transversal polymatroids. Polymatroidal bases
Let K be a field, and let K [ X ] denote the polynomial ring over K on theset of variables X = { x , . . . , x n } . We equip K [ X ] with the standard grading,meaning that deg( x α · · · x α n n ) = α + · · · + α n . We will also use the notationdeg i ( x α · · · x α n n ) = α i . Definition 1.1.
We call a set B of monomials in K [ X ] a polymatroidal basis if allthe monomials in B are of the same degree and the following property holds. If f and g are monomials in B with deg i f > deg i g for some i , then there is a j suchthat deg j f < deg j g and x j x i f ∈ B .The name polymatroidal comes from the fact that a polymatroidal basis is the setof base elements of a discrete polymatroid, phrased in the language of monomials.An ideal generated by a polymatroidal basis is called a polymatroidal ideal. Thestructure of a polymatroidal basis is actually more symmetric than the definitionreveals.
Theorem 1.2 (Symmetric exchange property) . Let B be a polymatroidal base, andsuppose f, g ∈ B with deg i f > deg i g . Then there is a j with deg j f < deg j g suchthat x j x i f, x i x j g ∈ B . For a proof of this theorem, as well as a more extensive background on discretepolymatroids, see [5].To a polymatroidal basis B we can associate the toric ring K [ B ]. This is thesubring of K [ X ] generated by the monomials in B . In the language of discretepolymatroids K [ B ] is also known as the base ring of B , or in the language ofmonomial ideals as the fiber ring of the ideal generated by B . If B = { f , . . . , f m } ,let Y = { y , . . . , y m } and define a homomorphism ϕ : K [ Y ] → K [ X ] by ϕ ( y i ) = f i .Then J B := ker ϕ is called the toric ideal of B , and K [ B ] ∼ = K [ Y ] / J B . It isa well known fact that such an ideal is generated by binomials, more preciselyby differences of monomials. If we take two variables y r , y s ∈ Y , Theorem 1.2induces binomials of degree two in J B in the following way. Say ϕ ( y r ) = f and ϕ ( y s ) = g . Assuming f = g Theorem 1.2 tells us that there are i and j such that deg i f > deg i g , deg j f < deg j g , and x j x i f, x i x j g ∈ B . Then there are some y t , y u ∈ Y so that ϕ ( y t ) = x j x i f and ϕ ( y u ) = x i x j g and y r y s − y t y u ∈ J B . The elements in J B of this type are called symmetric exchange relations . To be precise, y r y s − y t y u isa symmetric exchange relation if there are i and j such thatdeg i ϕ ( y r ) > deg i ϕ ( y s ) , deg j ϕ ( y r ) < deg j ϕ ( y s ) ,ϕ ( y t ) = x j x i ϕ ( y r ) , and ϕ ( y u ) = x i x j ϕ ( y s ) . Conjecture 1.3 (White 1980, Herzog-Hibi 2002) . The toric ideal J B of a polyma-troidal basis B is generated by the symmetric exchange relations. This conjecture was originally posed in [9] for matroids, which correspond tosquarefree polymatroidal bases, and for discrete polymatroids in [5]. Since thesymmetric exchange relations are quadratic it is natural to ask whether the algebra K [ B ] is Koszul, and if the toric ideal has a quadratic Gr¨obner basis. Recall thata K -algebra R is Koszul if K has a linear free resolution over R . Every Koszulalgebra is defined by an ideal generated in degree two, and every algebra definedby an ideal with a quadratic Gr¨obner basis is Koszul. Example 1.4.
The set B = { x x x , x x , x x , x x , x x x , x x } is a polyma-troidal basis. The toric ideal J B is the kernel of the homomorphism ϕ : K [ X ] → K [ Y ] defined by ϕ ( y ) = x x x , ϕ ( y ) = x x , ϕ ( y ) = x x ,ϕ ( y ) = x x , ϕ ( y ) = x x x , ϕ ( y ) = x x . A computation in Macaulay2 [4] gives that J B = ( y y − y y , y y − y y , y y − y y ) . This generating set is a Gr¨obner basis w. r. t. the Lex order. We also check thatthe generators are symmetric exchange relations. The fact that deg ( x x ) > deg ( x x ), deg ( x x ) < deg ( x x ), and x x x x = x x , x x x x = x x x both belongs to B gives rise to the symmetric exchange relation y y − y y . Moreover, deg ( x x x ) > deg ( x x ), deg ( x x x ) < deg ( x x ), and x x x x x = x x x , x x x x = x x both belongs to B gives the symmetric exchange relation y y − y y . Last, y y − y y is a sym-metric exchange relation which comes from the fact that deg ( x x ) > deg ( x x ),deg ( x x ) < deg ( x x ), and x x x x = x x , x x x x = x x x both belongs to B. △ We finish this section with a lemma regarding the symmetric exchange relations,which we will need later.
Lemma 1.5.
Let B be a polymatroidal basis and let E B be the ideal generated bythe symmetric exchange relations. Suppose we have y r , y s , y t , y u such that ϕ ( y t ) = x j x i ϕ ( y r ) and ϕ ( y u ) = x i x j ϕ ( y s ) for some i and j . Then y r y s − y t y u ∈ E B . RODUCTS OF POLYMATROIDS WITH THE STRONG EXCHANGE PROPERTY 3
Proof.
If deg i ϕ ( y r ) > deg i ϕ ( y s ) and deg j ϕ ( y r ) < deg j ϕ ( y s ) then y r y s − y t y u is asymmetric exchange relation.Suppose deg i ϕ ( y r ) > deg i ϕ ( y s ) and deg j ϕ ( y r ) ≥ deg j ϕ ( y s ). Since B is poly-matroidal there is some index k such that deg k ϕ ( y r ) < deg k ϕ ( y s ) and x k x i ϕ ( y r ), x i x k ϕ ( y s ) ∈ B . Then there are some y v , y w such that ϕ ( y v ) = x k x i ϕ ( y r ) and ϕ ( y w ) = x i x k ϕ ( y s ). It follows that y r y s − y v y w ∈ E B . We also have y t y u − y v y w ∈ E B becausedeg j ϕ ( y t ) = deg j ϕ ( y r ) + 1 > deg j ϕ ( y s ) − j ϕ ( y u ) , deg k ϕ ( y t ) = deg k ϕ ( y r ) < deg k ϕ ( y s ) = deg k ϕ ( y u ) ,ϕ ( y v ) = x k x i ϕ ( y r ) = x k x i x i x j ϕ ( y t ) = x k x j ϕ ( y t ) and similarily ϕ ( y w ) = x j x k ϕ ( y u )Then y r y s − y t y u = ( y r y s − y v y w ) − ( y t y u − y v y w ) ∈ E B .Suppose instead deg i ϕ ( y r ) ≤ deg i ϕ ( y s ) and deg j ϕ ( y r ) < deg j ϕ ( y s ). Since B is polymatroidal, deg j ϕ ( y s ) > deg j ϕ ( y r ) implies that there is some index k suchthat deg k ϕ ( y s ) < deg k ϕ ( y r ) and x k x j ϕ ( y s ) , x j x k ϕ ( y s ) ∈ B . Then there are y v and y w such that ϕ ( y v ) = x k x j ϕ ( y s ) and ϕ ( y w ) = x j x k ϕ ( y r ), and we have y s y r − y v y w ∈ E B .In a similar way as in the previous case one can verify that deg i ϕ ( y u ) > deg i ϕ ( y t ),deg k ϕ ( y u ) < deg k ϕ ( y t ), ϕ ( y v ) = x k x i ϕ ( y u ) and ϕ ( y w ) = x i x k ϕ ( y t ), so that y u y t − y v y w ∈ E B . Then y r y s − y t y u = ( y r y s − y v y w ) − ( y t y u − y v y w ) ∈ E B .Last we consider the case deg i ϕ ( y r ) ≤ deg i ϕ ( y s ) and deg j ϕ ( y r ) ≥ deg j ϕ ( y s ).In this case the binomial y t y u − y r y s is a symmetric exchange relation, as ϕ ( y r ) = x i x j ϕ ( y t ), ϕ ( y s ) = x j x i ϕ ( y u ), deg i ϕ ( y t ) < deg i ϕ ( y u ), and deg j ϕ ( y t ) > deg j ϕ ( y u ).We have now seen that y r y s − y t y u ∈ E B in all possible cases. (cid:3) The Strong Exchange Property
It was proved in [5] that Conjecture 1.3 holds for discrete polymatroids with theso called strong exchange property . Definition 2.1.
A set B of monomials in K [ X ] is called a basis with the strongexchange property (SEP) if all the monomials in B are of the same degree and if f, g ∈ B with deg i f > deg i g and deg j f < deg j g , for some i and j , then x j x i f ∈ B .Notice that we will also have x i x j g ∈ B in the above definition. Theorem 2.2 (Herzog-Hibi 2002) . If B is a basis with the SEP then the toric ideal J B is generated by the symmetric exchange relations, and has a quadratic Gr¨obnerbasis. We say that a monomial ideal has the SEP if it is generated by a basis withthe SEP. Every basis with the SEP is polymatroidal, but the converse does nothold. It was proved in [6] that bases with the SEP are isomorphic to bases ofVeronese type. In Theorem 2.3 we make this characterization a bit more explicit.For a basis B with the SEP, let max B ( i ) = max { deg i ( f ) | f ∈ B } and min B ( i ) =min { deg i ( f ) | f ∈ B } . Theorem 2.3.
Let B be a basis with the SEP of monomials of degree d . Then B = ( x α · · · x α n n (cid:12)(cid:12)(cid:12) n X i =1 α i = d, min B ( i ) ≤ α i ≤ max B ( i ) ) LISA NICKLASSON
Proof.
We begin by proving the following claim: For any i and j there is a monomial f ∈ B with deg i f = min B ( i ) and deg j f = max B ( j ). To prove this, let f be amonomial in B such that deg i f = min B ( i ) and deg j f is as high as possible fora monomial f with deg i f = min B ( i ). If deg j f < max B ( j ), take g ∈ B withdeg j g = max B ( j ). Then deg j g > deg j f and there is a k with deg k g < deg k f and x j x k f ∈ B . But this contradicts our choice of f , and we can conclude thatdeg j f = max B ( j ).A consequence of the claim is that if f ∈ B and deg i f > min B ( i ) and deg j f < max B ( j ) we have x j x i f ∈ B . Now let x α · · · x α n n be a monomial of degree d suchthat min B ( i ) ≤ α i ≤ max B ( i ) for each i . Take any monomial x β · · · x β n n ∈ B . If x β · · · x β n n = x α · · · x α n n then β i > α i and β j < α j for some i and j ,say i = 1 and j = 2. Then β > min B (1) and β < max B (2), and we have x β − x β +12 x β · · · x β n n ∈ B . If x β − x β +12 x β · · · x β n n = x α · · · x α n n we repeat thesame argument. Eventually we will get that x α · · · x α n n ∈ B , which is what wewanted to prove. (cid:3) We define the product of two polymatroidal bases B and B as B B = { b b | b ∈ B , b ∈ B } . If we consider B and B as generating sets of ideals I and I , then clearly B B is a generating set for the ideal I I . A nice feature of polymatroidal bases,proved in [2], is that their products are again polymatroidal. Products of baseswith the SEP does not necessarily have the SEP. For example { x , x }{ x , x } = { x x , x x , x x , x x } does not have the SEP. However, powers of bases with theSEP do have the SEP. Proposition 2.4. If B has the SEP, so does B k for any positive integer k .Proof. By Theorem 2.3 B = n x α · · · x α n n | min B ( i ) ≤ α i ≤ max B ( i ) , n X i =1 α i = d o . Let M = n x α · · · x α n n | k min B ( i ) ≤ α i ≤ k max B ( i ) , n X i =1 α i = dk o We can see that M is a basis with the SEP, and B k ⊆ M . We shall prove B k ⊇ M .Let x α · · · x α n n ∈ M , and write α i = kβ i + r i with 0 ≤ r i < k . Since P ni =1 α i = dk ,the number P ni =1 r i must also be divisible by k . Since r i < k we can factorize x r · · · x r n n = M · · · M k where M , . . . M k are squarefree monomials all of the samedegree. Then we have a factorization x α · · · x α n n = k Y i =1 ( x β · · · x β n n M i )and x β · · · x β n n M i ∈ B sincemin B ( j ) ≤ j α j k k ≤ deg j ( x β · · · x β n n M i ) ≤ l α j k m ≤ max B ( j ) . We have now proved that x α · · · a α n n ∈ B k , and hence B k has the SEP. (cid:3) RODUCTS OF POLYMATROIDS WITH THE STRONG EXCHANGE PROPERTY 5
The main results of this paper is Theorem 2.5 which describes the structure ofthe toric ring K [ B ] when B is a product of bases with the SEP. In particular, weshow that Conjecture 1.3 holds in this case.For two graded K -algebras R = L i ≥ R i and S = L i ≥ S i we let R ◦ S denotetheir Segre product , i. e. R ◦ S = L i ≥ R i ⊗ K S i . Theorem 2.5.
Let B = B · · · B s where B , . . . , B s are bases with the SEP. Then (1) J B is generated by symmetric exchange relations, and (2) K [ B ] ∼ = K [ B ] ◦ · · · ◦ K [ B s ] /L where L is generated by binomials of degreeone. The theorem is proved in Section 3. Here we should point out that Conjecture1.3 is still open, since not every polymatroidal basis is the product of bases withthe SEP. Consider for example the polymatroidal basis B = { x x , x x , x x , x x , x x } . It is not of the form in Theorem 2.3, so it does not have the SEP. If B would bea product of two bases, it would be the product of two sets of variables, but oneeasily checks that this is not the case.In the proof of Theorem 2.5 we will need the following lemma. Lemma 2.6.
Let B be a basis with the SEP. Suppose f, g, x i x j f, x k x ℓ g ∈ B . Thenthere is an index m such that x i x m f, x ℓ x m f, x m x ℓ g ∈ B .Proof. If x i x ℓ f ∈ B we are done, as we can choose m = ℓ . Suppose x i x ℓ f / ∈ B . Since x i x j f ∈ B we have deg i f < max B ( i ). Then x i x ℓ f / ∈ B implies deg ℓ f = min B ( ℓ ). As x k x ℓ g ∈ B we have deg ℓ g > min( ℓ ) = deg ℓ f . Then there is an m such that deg m g < deg m f and x m x ℓ g, x ℓ x m f ∈ B . Also, deg m f > min B ( m ) and deg i f < max B ( i ) gives x i x m f ∈ B . (cid:3) Proof of Theorem 2.5
Let B , . . . , B s be bases in K [ X ] with the SEP, and let B = B · · · B s . We let Y be the set of Q sj =1 | B j | variables y v indexed by vectors v = ( f , . . . , f s ) where f j ∈ B j . We define our homomorphism ϕ : K [ Y ] → K [ X ] so that y v maps tothe product of the entries in the vector v . For a monomial y v · · · y v d ∈ K [ Y ] weassociate a d × s matrix with row vectors v , . . . , v d . The matrix associated to amonomial is unique up to permutations of the rows, and ϕ ( y v · · · y v d ) is the productof all the entries in this matrix.Notice that the toric ideal J B = ker ϕ , as we defined it here, may containbinomials of degree one, as it can happen that ϕ ( y v ) = ϕ ( y w ) even though v = w .Let L B denote the ideal generated by the binomials of degree one in J B . Let E B be the ideal generated by the symmetric exchange relations in J B . Also, let P B be the ideal generated by the binomials F − G ∈ J B such that the matrices of F and G differ in at most one column (up to a permutation of the rows). Wewill see that J B = E B + L B = P B + L B . This proves that J B is generatedby the symmetric exchange relations, as the given generating set of E B will stillcorrespond to symmetric exchange relations after reducing modulo L B . The factthat J B = P B + L B proves the second part of Theorem 2.5, as K [ Y ] P B ∼ = K [ Y ] J B ◦ · · · ◦ K [ Y s ] J B s . LISA NICKLASSON
Now it remains to prove J B = E B + L B = P B + L B , or equivalently J B ⊆ ( E B + L B ) ∩ ( P B + L B ). Lemma 3.1.
Let F be a monomial in K [ Y ] , and let ( f ij ) be the associated matrix.Let F ′ be the monomial in K [ Y ] with the associated matrix ( f ′ ij ) obtained from ( f ij ) by permuting the elements in one column. Then F − F ′ ∈ E B ∩ P B .Proof. It is clear from the definition of P B that F − F ′ ∈ P B , so we need toprove that F − F ′ ∈ E B . As any permutation can by performed as a sequenceof transpositions, it is enough to consider a transposition of elements in a column.Without loss of generality, we assume that f ′ = f , f ′ = f , and otherwise f ′ ij = f ij . We define u i = ( f i , . . . , f is ) and v i = ( f ′ i , . . . , f ′ is ), so that F = y u · · · y u d and F ′ = y v · · · y v d . Then F − F ′ = y u · · · y u d − y v · · · y v d = ( y u y u − y v y v ) y u · · · y u d , and we want to prove that y u y u − y v y v ∈ E B . Recall that f , f ∈ B , whichis a basis with the SEP. We have f = x a x a x b x b · · · x a k x b k f for some a , . . . , a k , b , . . . b k , and we may assume { a , . . . , a k } ∩ { b , . . . b k } = ∅ , sothat no variable occurs both in the numerator and in the denominator. By Theorem2.3 f ( m )11 := x a x a x b x b · · · x a m x b m f ∈ B , m = 1 , . . . , k, since with this construction deg f ( m )11 = deg f andmin B ( i ) ≤ min(deg i f , deg i f ) ≤ deg i f ( m )11 ≤ max(deg i f , deg i f ) ≤ max B ( i ) . In the same way f ( m )21 := x b x b x a x a · · · x b m x a m f ∈ B , m = 1 , . . . , k. Let u ( m )1 = ( f ( m )11 , f , . . . , f s ) and u ( m )2 = ( f ( m )21 , f , . . . , f s ) , for m = 1 , . . . , k and notice that u ( k )1 = v and u ( k )2 = v . By Lemma 1.5 y u y u = y u (1)1 y u (1)2 = y u (2)1 y u (2)2 = . . . = y u ( k )1 y u ( k )2 = y v y v in K [ Y ] / E B , and it follows that F = F ′ in K [ Y ] / E B . (cid:3) Lemma 3.2.
Let F and G be monomials in K [ Y ] such that their matrices areidentical in the first m < s columns, and F − G ∈ J B . If J B ′ = E B ′ + L B ′ = P B ′ + L B ′ , where B ′ = B m +1 · · · B s , then F − G ∈ ( E B + L B ) ∩ ( P B + L B ) . The analogous statement is also true if the matrices of F and G are identical inthe last m columns, or more generally in any subset of the columns. The proof inthis situation follows the same recipe. Proof.
As the first m columns of the matrices ( f ij ) and ( g ij ) representing F and G are identical, and F − G ∈ J B , we have Y ≤ i ≤ dm Let F be a monomial in K [ Y ] such that x x ϕ ( F ) ∈ B d . Then thereare monomials F ′ and H such that F − F ′ ∈ E B ∩ P B , ϕ ( H ) = x x ϕ ( F ′ ) , and thematrices associated to F ′ and H differ in at most one entry in each column.Proof. Let ( f ij ) be the matrix associated to F . Since x x ϕ ( F ) ∈ B d there are h ij ’s,such that x x ϕ ( F ) = Y ≤ i ≤ d ≤ j ≤ s h ij , h ij ∈ B j . For each pair i, j there is a multiset Λ ij ⊂ { , . . . , n } such that h ij = (cid:16) Y ( a,b ) ∈ Λ ij x a x b (cid:17) f ij . The choice of the h ij ’s is not unique, and the goal is the choose them so that foreach j there is only one i for which Λ ij in nonempty. First note that we can chooseΛ ij so that for each ( a, b ) ∈ Λ ij it holds thatdeg a f ij < deg a h ij and deg b f ij > deg b , h ij . It follows that x a x b f ij ∈ B j , since B j has the SEP. As (cid:16) Y ≤ i ≤ d ≤ j ≤ s Y ( a,b ) ∈ Λ ij x a x b (cid:17) ϕ ( F ) = Y ≤ i ≤ d ≤ j ≤ s Y ( a,b ) ∈ Λ ij x a x b f ij = Y ≤ i ≤ d ≤ j ≤ s h ij = x x ϕ ( F )we have (cid:16) Y ≤ i ≤ d ≤ j ≤ s Y ( a,b ) ∈ Λ ij x a x b (cid:17) = x x and we may order the factors x a x b in the left hand side so that(1) (cid:16) Y ≤ i ≤ d ≤ j ≤ s Y ( a,b ) ∈ Λ ij x a x b (cid:17) = x x a x a x a x a x a · · · x a k − x a k x a k x Each factor in the right hand side still corresponds to an index pair in some Λ ij .Now, if there are two different values of i such that Λ i is nonempty we do thefollowing. In the right hand side of (1) consider the leftmost and rightmost factorscorresponding to index pairs from any Λ i . Say for instance that it is ( a , a ) ∈ Λ and ( a , a ) ∈ Λ . Then f , f , x a x a f , and x a x a f all belong to B . By Lemma 2.6 there is an m such that x a x m f , x a x m f , and x m x a f ∈ B . Now we define F ′ as the matrix ( f ′ ij ) with f ′ ij = f ij except f ′ = x a x m f and f ′ = x m x a f . Notice that F ′ − F ∈ E B ∩ P B . In Λ we replace ( a , a ) by ( a , a ), LISA NICKLASSON and we remove ( a , a ) , ( a , a ) , ( a , a ) , and ( a , a ) from their corresponding Λ ij ’s.This means that Λ = { ( a , a ) } and Λ i = ∅ for i > 1. We also redefine the h ij ’sas h ij = (cid:16) Y ( a,b ) ∈ Λ ij x a x b (cid:17) f ′ ij . Notice that we still have h ij ∈ B j , in particular h = x a x a f ′ = x a x a x a x m f = x a x m f ∈ B and h i = f ′ i ∈ B for i > . Now we do the same procedure for j = 2, but with F replaced by F ′ and the righthand side of (1) replaced by x x a x a x a x a x a · · · x a k − x a k x a k x . We continue with this for all values of j . Finally we end up with h ij ’s that for each j there is at most one i for which h ij = f ′ ij . Then H given by the matrix ( h ij )together with F ′ has the desired properties. (cid:3) Now we are ready to prove that J B ⊆ ( E B + L B ) ∩ ( P B + L B ). Suppose we havemonomials F and G of degree d ≥ K [ Y ] such that their difference is in J B .We shall prove that F = G in K [ Y ] / (( E B + L B ) ∩ ( P B + L B )) by induction over s .The base case s = 1 is clear, as then B = B has the SEP, and the matrices haveonly one column. Now suppose the statement is true for any product of fewer than s bases with the SEP. The idea of the proof is the following. Let ( f ij ) and ( g ij ) bethe matrices corresponding to F and G . Say we are in the special situation where Q di =1 f i = Q di =1 g i . Then we also have Q ≤ i ≤ d ≤ j ≤ s f ij = Q ≤ i ≤ d ≤ j ≤ s g ij . Let H be themonomial in K [ Y ] associated to the matrix g f · · · f s g f · · · f s ... ... ... g d f d · · · f ds We have J B ′ = ( E B ′ + L B ′ ) ∩ ( P B ′ + L B ′ ) by the induction hypothesis. By Lemma3.2 we have H = G in K [ Y ] / (( E B + L B ) ∩ ( P B + L B )). If f i = g i for each i weare done. If not we can apply the induction hypothesis and Lemma 3.2 one moretime to F and H , since they are identical in the last column. It then follows that F = H = G in K [ Y ] / (( E B + L B ) ∩ ( P B + L B )).Now, what if we are not in the special situation with Q di =1 f i = Q di =1 g i ?If Q di =1 f i = Q di =1 g i then also Q ≤ i ≤ d ≤ j ≤ s f ij = Q ≤ i ≤ d ≤ j ≤ s g ij . Since B d · · · B ds ispolymatroidal there are say x and x so that(2) deg Y ≤ i ≤ d ≤ j ≤ s f ij < deg Y ≤ i ≤ d ≤ j ≤ s g ij and deg Y ≤ i ≤ d ≤ j ≤ s f ij > deg Y ≤ i ≤ d ≤ j ≤ s g ij RODUCTS OF POLYMATROIDS WITH THE STRONG EXCHANGE PROPERTY 9 and x x Y ≤ i ≤ d ≤ j ≤ s f ij ∈ B d · · · B ds . By Lemma 3.3 we can, after replacing F modulo E B ∩ P B , find h ij ’s 1 ≤ i ≤ d ,2 ≤ j ≤ s so that x x Y ≤ i ≤ d ≤ j ≤ s f ij = Y ≤ i ≤ d ≤ j ≤ s h ij , h ij ∈ B j such that for each j there is at most one i ’s for which h ij = f ij .Because of (2) we also havedeg d Y i =1 f i > deg d Y i =1 g i and deg , d Y i =1 f i < deg , d Y i =1 g i . By Proposition 2.4 B d has the SEP, and it follows that x x d Y i =1 f i ∈ B d . Here we can apply Lemma 3.3 once more, to get that x x d Y i =1 f i = d Y i =1 h i , h i ∈ B where h i = f i for at most one i . Now, let H be the monomial with the associatedmatrix ( h ij ), 1 ≤ i ≤ d , 1 ≤ j ≤ s . For each j there is at most one i for which h ij = f ij . We can use Lemma 3.1 to, for one j at a time, rearrange the h ij ’s so thatthe i where possibly h ij = f ij is i = 1. Since now F and H differ only in the first rowthey are equal modulo L B . If Q di =1 h i = Q di =1 g i we can repeat this process, nowwith H and G . After finitely many repetitions we will get Q di =1 h i = Q di =1 g i .Then we have arrived in the special situation described in the beginning of thisproof, and the result follows by induction.4. Rees algebras An object closely related to the toric ring K [ B ] is the Rees algebra of the mono-mial ideal generated by B . In general, the Rees algebra of an ideal I ⊆ K [ X ] is de-fined as R ( I ) = L j ≥ I j t j . If I = ( f , . . . , f m ) we have R ( I ) = K [ X ][ f t, . . . , f m t ].Let Y = { y , . . . , y m } and let K [ X, Y ] be the polynomial ring on the variables X ∪ Y with the bigrading deg x i = (1 , 0) and deg y i = (0 , k, ∗ ) if it has degree ( k, s ) for any s , and similarly for ( ∗ , k ). We can present R ( I ) as K [ X, Y ] / ker φ where φ : K [ X, Y ] → K [ X ][ t ] is the homomorphism definedby φ ( x i ) = x i and φ ( y i ) = f i t . The defining ideal ker φ is called the Rees ideal of I .An ideal I is said to be of fiber type if its Rees ideal is generated in degrees (0 , ∗ )and ( ∗ , Lemma 4.1 ([8, Lemma 3.2]) . If I has linear free resolution and is of fiber typethen its Rees ideal is generated in degrees (0 , ∗ ) and (1 , . Polymatroidal ideals are of fiber type and have linear free resolution, see [6] and[7], so Lemma 4.1 applies to polymatroidal ideals. If I has the SEP then its Reesideal is generated in degrees (0 , 2) and (1 , Theorem 4.2. The Rees ideal of a product of ideals with the SEP is generated indegrees (0 , and (1 , .Proof. Let B be a product of bases with the SEP, and let I be the Rees ideal ofthe ideal generated by B . By Lemma 4.1 I is generated in degrees (0 , ∗ ) and (1 , , ∗ ) in I is precisely the toric ideal J B . Then theresult follows by Theorem 2.5. (cid:3) As the generators of the Rees ideals in Theorem 4.2 has total degree 2 we concludethis section with the following question. Question 4.3. Is the Rees algebra of a product of ideals with the SEP Koszul? Gr¨obner bases and transversal polymatroidal bases Polymatroids that are given as products of subsets of X are called transversal .This is a special case of the class of polymatroids considered in this paper, as subsetsof X indeed have the SEP. Theorem 2.5 (2) was proved for transversal polymatroidsin [1, Theorem 3.5], which in this case says that K [ B ] is isomorphic to the Segreproduct of polynomial rings modulo an ideal of linear binomials. Moreover [1,Theorem 3.5] states that K [ B ] is Koszul when B is transversal. By Theorem 2.5(1) we now also know that White’s conjecture holds for transversal polymatroids.However, the following questions are still open. Question 5.1. Does the toric ideal of a transversal polymatroid, or more generallya product of bases with the SEP, have a quadratic Gr¨obner basis? Question 5.2. If B is a product of bases with the SEP, is K [ B ] Koszul? It is not obvious which monomial order to use when searching for a quadraticGr¨obner basis of J B . Let B = X · · · X s , where X , . . . , X s are subsets of X .Then J B is generated by the so called Hibi relations together with binomials ofdegree one, see [1, Proposition 3.7]. The Hibi relations are defined in the followingway. We fix an order of the variables in each X i . The orderings do not have tobe compatible with each other. Then we let Y be the set of variables indexedby integer vectors α = ( α , . . . , α s ) so that 1 ≤ α j ≤ | X j | . The toric ideal J B is the kernel of ϕ : K [ Y ] → K [ X ] where ϕ ( y α ) is defined as the product of the α -th element of X , the α -th element of X , and so on. The Hibi relations areall relations y α y β − y α ∧ β y α ∨ β where α ∧ β = (max( α , β ) , . . . , max( α s , β s )) and α ∨ β = (min( α , β ) , . . . , min( α s , β s )) . We have a partial ordering on the variables Y by y α ≤ y β if α i ≤ β i for all i . Notice that the Hibi relation y α y β − y α ∧ β y α ∨ β is nonzero only if y α and y β are incomparable. The Hibi relations are a Gr¨obnerbasis w. r. t. the DegRevLex order with any order on Y extending the partial order.The leading term of y α y β − y α ∧ β y α ∨ β is y α y β . We make the following observation. Proposition 5.3. Let B = X · · · X s be a transversal polymatroidal basis suchthat J B is generated by the Hibi relations together with one binomial of degree one.Then J B has a quadratic Gr¨obner basis. Moreover, if | X | = · · · = | X s | then K [ B ] is Gorenstein. RODUCTS OF POLYMATROIDS WITH THE STRONG EXCHANGE PROPERTY 11 Proof. We may order the variables in each X i so that the one binomial of degree onein J B is y α − y β with α = ( | X | , . . . , | X s | ) and β = (1 , , . . . , y α by y β . First notice that y α does not divide the leading term ofany Hibi relation. Indeed, y α ≥ y γ ≥ y β for any y γ ∈ Y . This shows that replacing y α by y β does not change the leading term of any binomial in J B .Now assume | X | = · · · = | X s | = m . Then K [ B ] ∼ = S/L there S is the Segreproduct of s copies of the polynomial ring in m variables, and L is generated by abinomial of degree one. The Segre product of several of copies of a Gorenstein ringis again Gorenstein, by a result in [3]. The polynomial ring is Gorenstein, so T isGorenstein. T is also a domain, and hence T /L is Gorenstein. (cid:3) Example 5.4. Let B = { x , x }{ x , x }{ x , x }{ x , x }{ x , x } . Then Y is theset of variables indexed by integer vectors α = ( α , . . . , α ) where 1 ≤ α i ≤ 2. Wehave ϕ ( y ) = x x x x x and ϕ ( y ) = x x x x x , and y − y isthe only linear relation in J B . Hence the Hibi relations, with every occurrence of y replaced by y , is a quadratic Gr¨obner basis defining K [ B ]. One exampleof a Hibi relation is y y − y y where ϕ ( y ) = x x x x x , ϕ ( y ) = x x x x x ,ϕ ( y ) = x x x x x , and ϕ ( y ) = x x x x x . The Segre product of five polynomial rings in two variables is a Gorenstein algebrawith Hilbert series 1 + 26 t + 66 t + 26 t + t (1 − t ) . Hence K [ B ] is Gorenstein with Hilbert series1 + 26 t + 66 t + 26 t + t (1 − t ) . △ We finish with a question inspired by Proposition 5.3. Question 5.5. Which transversal polymatroidal bases define Gorenstein algebras? Acknowledgement I would like to thank Aldo Conca for our discussion about Gr¨obner bases andtransversal polymatroids which led to the content of Section 5. References [1] A. Conca. Linear spaces, transversal polymatroids and ASL domains. J. Algebr. Comb. , 25:25–41, 2007.[2] A. Conca and J. Herzog. Castelnuovo-Mumford regularity of products of ideals. Collect. Math. ,54(2):137–152, 2003.[3] S. Goto and K. Watanabe. On graded rings I. J. Math. Soc. Japan , 30(2):179–213, 1978.[4] D. R. 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