Binomial fibers and indispensable binomials
aa r X i v : . [ m a t h . A C ] O c t BINOMIAL FIBERS AND INDISPENSABLE BINOMIALS
HARA CHARALAMBOUS, APOSTOLOS THOMA, AND MARIUS VLADOIU
Abstract.
Let I be an arbitrary ideal generated by binomials. We show that certainequivalence classes of fibers are associated to any minimal binomial generating set of I .We provide a simple and efficient algorithm to compute the indispensable binomials ofa binomial ideal from a given generating set of binomials and an algorithm to detectwhether a binomial ideal is generated by indispensable binomials. Introduction
Let R = K [ x , . . . , x n ] where K is a field. A binomial is a polynomial of the form x u − λx v where u , v ∈ N n and λ ∈ K \ { } , and a binomial ideal is an ideal generatedby binomials. We say that the ideal I of R is a pure binomial ideal if I is generated by pure difference binomials , i.e. binomials of the form x u − x v with u , v ∈ N n . Binomialideals were first studied systematically in [10] and this class of ideals also includes latticeideals. Recall that if L ⊂ Z n is a lattice, then the corresponding lattice ideal is definedas I L = ( x u − x v : u − v ∈ L ) and the lattice is saturated exactly when the lattice idealis toric, i.e. prime. The study of binomial ideals is a rich subject: the classical referenceis [22] and we also refer to [16] for recent developments. It has applications in variousareas in mathematics, such as algebraic statistics, integer programming, graph theory,computational biology, code theory, see [7, 9, 13, 18, 19, 23], etc.A particular problem that arises is the efficient generation of binomial ideals by a setof binomials. Up to now, it has mainly been addressed for toric and lattice ideals, see[3, 4, 5, 11, 12, 22] among others. In Section 1, we consider this problem in the case ofbinomial ideals. For this we study the fibers of binomial ideals: in [8, Proposition 2.4] anequivalence relation on N n was introduced for any binomial ideal I of R , namely u ∼ I v if x u − λx v ∈ I for some λ = 0. For each such equivalence class, we get a fiber on the set ofmonomials: the I -fiber of x u is the set { x v : u ∼ I v } . When I := I L is the lattice ideal of L , the equivalence class of u consists precisely of all v such that u − v ∈ L and the I -fibersare finite exactly when L ∩ N n = { } . In this case, for each I -fiber one can use a graphconstruction, see [7, 4], that determines the I -fibers that appear as invariants associatedto any minimal generating set of I . We also note that in [5] the fibers of I L were studiedeven when L ∩ N n = { } . In all cases finite or not, it is clear that divisibility of monomialsdoes not induce necessarily a meaningful partial order on the set of I -fibers. In this paperfor any binomial ideal I we define an equivalence relation on the set of I -fibers and thenorder the equivalence classes of I -fibers, see Definition 1.6. We note that this was firstdone in [5] for the case I = I L . This partial order allows us to prove that a certain set Mathematics Subject Classification.
Key words and phrases.
Binomial ideals, Markov basis, lattices.The third author was partially supported by project PN-II-RU-TE-2012-3-0161, granted by the Ro-manian National Authority for Scientific Research, CNCS - UEFISCDI. The paper was written when thethird author visited University of Ioannina, and he gratefully acknowledges for the warm environment thatfostered the collaboration of the authors. f equivalence classes of I -fibers is an invariant, associated to any generating set of I , seeTheorem 1.12. We note that lattice ideals have all fibers either finite or infinite and theequivalence classes of fibers for lattice ideals have the same cardinality, see [5, Propositions2.3 and 3.5]. However this might not be the case for a binomial ideal and this constitutesan added degree of difficulty, see Examples 1.2 c) and 1.7 c). Moreover we show thatbinomial ideals which contain monomials have a unique maximal fiber consisting of allmonomials of I , see Theorem 1.8.A related question that attracted a lot of interest in the recent years is whether there isa unique minimal binomial generating set for a binomial ideal. One of the first papers todeal with this question for lattice ideals from a purely theoretical point of view was [21].As it turns out, the positive answer has applications to Algebraic Statistics: [1, 2, 18].Thus in [19] and [2] the notions of indispensable monomials and binomials were defined.Let I be a binomial ideal. A binomial is called indispensable if (up to a nonzero constant)it belongs to every minimal generating set of I consisting of binomials. This implies ofcourse that (up to a nonzero constant) it belongs to every binomial generating set of I . Amonomial is called indispensable if it is a monomial term of at least one binomial in everysystem of binomial generators of I . How does one compute these elements?When I := I L is a lattice ideal and L ∩ N n = { } , there are several works in the literaturethat deal with this problem. In particular, in [19] it was shown that to compute theindispensable binomials of I L , one computes all lexicographic reduced Gr¨obner bases andthen their intersection: there are n ! such bases; a corresponding result for indispensablemonomials was shown in [2]. In [20], it was shown that to compute the indispensablebinomials of I L , it is enough to compute certain degree-reverse lexicographic reducedGr¨obner bases of I L ( n of them), and then compute their intersection. In [4, Proposition3.1], it was shown that to find the indispensable monomials of I L , it is enough to considerany one of the binomial generating sets of I L . Moreover in [4, Theorem 2.12] it wasshown that in order to find the indispensable binomials of I L , it is enough to consider anyminimal binomial generating set of I L , assign Z n /L -degrees to the binomials of this setand to compute their minimal Z n /L -degrees. More recently in [14, Theorem 1.1, Corollary1.3], it was shown that if I is a pure binomial ideal then there is a d ∈ N such that any I is A -graded for some A ⊂ Z d : when N A ∩ ( − N A ) = { } and all fibers are finite a sufficientcondition was given in [14] for the indispensable binomials and a characterization for theindispensable monomials, involving the I -fibers of a minimal generating set of I .In this paper, we significantly improve all previously known results regarding indispens-able binomials. Moreover our results apply to the general case of all binomial ideals. InSection 2 we show that as in [14], the indispensable monomials are the elements of theminimal generating set of the monomial ideal of I , see Remark 2.3. Then we go on andare able to express this condition into three necessary and equivalent conditions involvinga graph whose vertices are the (possibly infinitely many) elements of the fiber, see The-orem 2.5. This result is then applied to provide sufficient and necessary conditions for abinomial in I to be indispensable, see Theorem 2.6.In Section 3, we prove that an arbitrary system of binomial generators of I gives allnecessary information to decide whether a given binomial is indispensable, see Theorem3.3. As an immediate application of Theorem 3.3 we obtain an algorithm which computesthe indispensable elements of a binomial ideal I , given any system of binomial generatorsof I , see Algorithm 1. This algorithm bypasses the computation of a reduced Gr¨obnerbasis unlike the previous methods. As a result we show that Algorithm 1 is a polynomialtime algorithm, see Remark 3.5, in contrast to the other methods, see [15] for details egarding the complexity of reduced Gr¨obner basis computation for binomial ideals. Wealso show, that to decide whether a minimal system of binomial generators of a binomialideal is in fact a system of indispensable binomials it is enough to compute the cardinalityof the minimal generating set of an associated monomial ideal, see Corollary 3.6 and theresulting Algorithm 2.In Section 4, we generalize the notion of primitive elements to pure binomial ideals.The set of all primitive elements is the Graver basis of I . This set is extremely importantin theory and all computations involving lattice ideals, see [22]. We prove that the Graverbasis of any pure binomial ideal is finite, see Proposition 4.3. We show that the Graverbasis includes as a subset the universal Groebner basis of I, see Theorem 4.2. Finally,we show that a Lawrence lifting construction gives a pure binomial ideal generated byindispensable binomials, see Theorem 4.4.We thank the two anonymous referees for their careful reading and suggestions whichgreatly improved our paper, in particular for the questions on the complexity of the algo-rithms and on the generalization to binomial ideals, which we could answer affirmatively.1. Fibers of a Binomial Ideal
Let R = K [ x , . . . , x n ] where K is a field. We denote by T n the set of monomials of R including 1 = x , where x u = x u · · · x u n n . If J is a monomial ideal of R we denote by G ( J )the unique set of minimal monomial generators of J . For B = x u − λx v with λ = 0 we letsupp( B ) := { x u , x v } . Definition 1.1.
Let I be a binomial ideal of R . We say that F ⊂ T n is an I - fiber if thereexists a x u ∈ T n such that F = { x v : v ∼ I u } . If x u ∈ T n , and F is an I -fiber containing x u we write F u or F x u for F . If B ∈ I and B = x u − λx v with λ = 0 we write F B for F u .It is trivial that | F u | = 1, that is F u is a singleton, if and only if there is no binomial0 = B ∈ I such that x u ∈ supp( B ). If J ⊂ I gives the containment between two binomialideals and F is a J -fiber, then clearly F is contained in an I -fiber. Example 1.2. a) Let a ∈ N , r ∈ Z ≥ , L = r Z , λ ∈ K \ { } and I = ( x a − λx a + r ). It is immediate that x a + j − λ nr x a + nr + j ∈ I for all j, n ∈ N . The I -fibers are either singletons or infinite. Thereare exactly a singletons and r distinct infinite I -fibers of the form F a + j = { x a + j + nr : n ∈ N } , for 0 ≤ j ≤ r −
1. Moreover since x a − λx a + r = (1 − λx r )( x a − λ r x a +3 r ) + λ r x r − ( x a +1 − λx a + r +1 ) ,I = ( x a − λ r x a +3 r , x a − λx a + r +1 ). Thus I has no indispensable binomials. It is clearthat the the only indispensable monomial of I is x a .b) Let I = ( y − x y, y − xy , y − y , y − y ). The I -fibers are as follows: • F x i = { x i } for all i ∈ N • F y = { yx n : n ∈ N }• F yx = { yx n +1 : n ∈ N }• F y = { y x n : n ∈ N }• F y x = { y x n +1 : n ∈ N }• F y = { y x n : n ∈ N }• F y = { y m x n : m, n ∈ N } , nd they are independent of the characteristic of K . Note that when char( K ) = 3 then I = ( y − x y, y − xy , y ) and is the binomial ideal ( y − x y, y − xy , y − y , y + y ).The I -fibers are depicted in the left of Figure 1. • • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • • xy • • • • • • • • • ⊲ ◦ ⋄ ∗ ◦ x ••••••••• ⊳ ⋄◦∗⋄ y • • • • • • • ⊲ ◦ ⋄ ∗ ◦ ⋄ ∗• • • • • ⋆ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄• • • ⋆ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦• • • ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗• • ⋆ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄• • ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦• ⊳ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗• ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ⊳ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ ◦ ⋄ ∗ I I Figure 1.
Fibers of binomial idealsc) Consider the pure binomial ideal I = ( y − xy , x y − x y , x y − x y , x y − x ). Itsfibers are depicted in the right part of Figure 1. There are 29 singleton I -fibers, depictedby dots. The other fibers are: • F xy = { xy , y }• F x y = { x y , x y , x y }• F x y = { x y, x }• F xy = { xy , y }• F x y = { x y, x }• F x y = { x a y b | ( a, b ) ∈ N and ( a, b ) ∈ (4 ,
3) + N ( − ,
2) + N (2 , − }• F x y = { x a y b | ( a, b ) ∈ N and ( a, b ) ∈ (3 ,
4) + N ( − ,
2) + N (2 , − }• F x y = { x a y b | ( a, b ) ∈ N and ( a, b ) ∈ (4 ,
4) + N ( − ,
2) + N (2 , − } .d) Consider the ideal I = ( y − xy , x y − x y , x y − x y , x y − x ) ⊂ K [ x, y ]. If K is a field of characteristic different from 5, then all of the fibers are exactly the same as inthe previous example (c) except of the three fibers F x y , F x y , F x y of I which becomeone fiber of I . On the other hand, if char( K ) = 5 then I = ( y − xy , x y − x y , x y − x y , x y ) is the binomial ideal ( y − xy , x y − x y , x y − x y , x y − x y , x y + x y )and there are infinitely many singleton I -fibers: all of the singleton fibers of example(c) and all of the singleton fibers corresponding to monomials of the form x n . The other I -fibers are: • F xy = { xy , y }• F x y = { x y , x y , x y }• F xy = { xy , y }• F x y which contains all the monomials belonging to the monomial ideal M =( x y, x y , x y , x y , xy , y ).Note that this example reveals that fibers are dependent on the characteristic of thefield. f G ⊂ T n and t ∈ N n we let x t G := { x t x u : x u ∈ G } . Theorem 1.3.
Let F be a partition of T n . There exists a binomial ideal I such that F isthe set of I -fibers if and only if for any u ∈ N n and any F ∈ F there exists a G ∈ F suchthat x u F ⊂ G .Proof. Let I be a binomial ideal. Let F := F t be an I -fiber and let u ∈ N n . It is clear x u F ⊂ F t + u . For the converse we let F = ( F i : i ∈ Λ) and I be the ideal generated by theset { x u − x v : u , v ∈ F i , i ∈ Λ } . Consider the set G of I -fibers. We will show that G = F .Indeed let G be an I -fiber and x u ∈ G . Since F is a partition of T n , there is an F ∈ F suchthat x u ∈ F . From the definition of I , it is clear that F ⊂ G . For the converse inclusionsuppose that x v ∈ G and thus x v − x u ∈ I . Hence x v − x u = P lk =1 c k x w k ( x v k − x u k )where x u k , x v k ∈ F i k , i k ∈ Λ, c k ∈ K , and l ≥
1. Therefore for some k , x w k x u k = x u ∈ F .By the hypothesis on the elements of F we obtain x w k x v k ∈ F . An easy induction on l finishes the proof. (cid:3) In the proof of Theorem 1.3 we showed even more, namely that for a given set of fibersthere exists a pure difference binomial ideal whose set of I -fibers is the given one.A vector u ∈ Z n is called pure if u ∈ N n or − u ∈ N n . If B = x u − λx v ∈ I with λ = 0 welet v ( B ) = u − v . If F is a fiber of a binomial ideal I , we let L F := h v ( B ) : F B = F i ⊂ Z n .We also consider L pure,F , the sublattice of L F generated by the set { w ∈ N n : ∃ x u , x v ∈ F such that w = u − v } , and denote by L + pure,F the semigroup generated by the same set. Finally we let M F bethe monomial ideal of R generated by the elements of F . In some cases it might be thatthe monomials in M F are precisely the elements of F , for example if I = (1 − x ) in K [ x ],but usually this is far from being the case. If I is a lattice ideal and I = I L then L F ⊂ L and by [5, Proposition 2.3] or [17, Theorem 8.6] it follows that F is an infinite fiber if andonly if L F contains a nonzero pure element. In the case of an arbitrary binomial ideal I ,one can extend Proposition 2.6 of [5]. Proposition 1.4.
Let I be a binomial ideal and F be an I -fiber. If { x a , . . . , x a s } is theminimal monomial generating set of M F then F = s [ i =1 { x a i x w : w ∈ L + pure,F } . In particular, F is infinite if and only if L pure,F = 0 .Proof. It follows immediately from definitions the inclusion ⊆ . For the other inclusionlet w ∈ L + pure,F and fix an i . Then there exist x u ∈ F and λ ∈ K \ { } such that x u − λx u + w ∈ I . Since x a i ∈ G ( M F ) then x a i ∈ F and there exists µ ∈ K \ { } such that x a i − µx u ∈ I . We have x a i − λx a i + w = ( x a i − µx u ) + µ ( x u − λx u + w ) − λx w ( x a i − µx u ) , hence x a i − λx a i + w ∈ I and consequently x a i + w ∈ F . Therefore we obtain the desiredequality. (cid:3) The next example comments on certain subtleties of the above proposition. xample 1.5. Let I be the ideal of Example 1.2 c). The fiber F = F x y is finiteeven though L F = h (2 , − , ( − , i contains (1 , , / ∈ L pure,F and L pure,F = ( ). For the fiber F = F x y , one can see that M F = ( x , x y, x y , x y , x y , x y , xy , y ) , thus L pure,F = h (1 , , (0 , i and L + pure,F = (0 , N + (1 , N + (3 , N . Therefore we have { x x b +3 c y a + b : a, b, c ∈ N } ⊂ F .Let I be a binomial ideal. We define an equivalence relation ” ≡ ” on the set of I -fibersand a partial order ” < ” of the equivalence classes which generalize those from [5]: Definition 1.6. If F , G are I -fibers, we let F ≡ G if there exist u , v ∈ N n such that x u F ⊂ G and x v G ⊂ F and denote by F , the equivalence class of F . We set F ≤ G ifthere exists u ∈ N n such that x u F ⊂ G . We write F < G if F ≤ G and F = G .In [5, Proposition 3.5] it was shown for lattice ideals that any two equivalence classesof fibers have the same cardinality. This is no longer necessarily true for an arbitrarybinomial ideal I as you can see in the Example 1.7 b), but the cardinality of F for an I -fiber F can be computed similarly by replacing L F with L pure,F . Example 1.7. a) Let I be the ideal of Example 1.2 a). The infinite I -fibers are equivalent.b) Let I be the ideal of Example 1.2 b). We note that • F y = { F y , F xy } , • F y = { F y , F xy } , • F y = { F y } , • F y = { F y } .The set of equivalence classes of I -fibers is totally ordered, the maximal element being F y and the minimal one F , where F = { } .c) Let I be the ideal of Example 1.2 c). The three infinite fibers F x y , F x y , F x y areequivalent. The equivalence classes F x y , F xy and F x y are incomparable and minimalwhen restricting to equivalence classes of fibers of cardinality greater than one.Note that it is possible for a binomial ideal to contain monomials. If this is the case, thenit obviously contains infinitely many, but the surprising fact is that all of these monomialsform an I -fiber, which becomes the maximal I -fiber. Theorem 1.8.
Let I be a binomial ideal and denote by F ( I ) the set { x w | x w ∈ I } . Then F ( I ) = ∅ if and only if there exist λ = µ ∈ K and monomials x u , x v such that x u − λx v and x u − µx v belong to I . Furthermore, if F ( I ) = ∅ then F ( I ) is an I -fiber, F ( I ) = { F ( I ) } and F ≤ I F ( I ) for every I -fiber F .Proof. Assume that F ( I ) = ∅ and let x w ∈ I . If x v is a monomial such that v ∼ I w thenthere exists λ ∈ K \ { } such that x u − λx v ∈ I . Since x w ∈ I then also x v ∈ I , andthus x v ∈ F ( I ). Hence F w ⊆ F ( I ). For the converse inclusion, note that if x v ∈ F ( I )then x v ∈ I and implicitly x w − x v ∈ I . Therefore v ∼ I w and x v ∈ F w , which implies F ( I ) ⊆ F w .It is easy to see from the definition of F ( I ) that x w F ( I ) ⊆ F ( I ) and x w F ⊆ F ( I ) forany I -fiber F . This implies in particular F ≤ I F ( I ) for every I -fiber F . On the otherhand, if x v F ( I ) ⊆ F for some I -fiber F with F = F ( I ) then we obtain a contradiction rom Theorem 1.3, since x v F ( I ) ⊆ F ( I ). Therefore it follows that F ( I ) = { F ( I ) } , andwe are done. (cid:3) Example 1.9.
Let I , . . . , I be the ideals of Example 1.2. Then:(a) F ( I ) = ∅ ;(b) F ( I ) = ∅ if char( K ) = 3, and F ( I ) = F y when char( K ) = 3;(c) F ( I ) = ∅ ;(d) F ( I ) = F x y ∪ F x y ∪ F x y if char( K ) = 5, where F x y , F x y , F x y are theunderlying sets of the corresponding fibers of I ; and F ( I ) = F x y if char( K ) = 5,where F x y is the corresponding fiber of I . Remark 1.10.
Whether a binomial ideal I contains or not monomials is computationallydetectable through a single Gr¨obner basis computation. Indeed, if we compute a reducedGr¨obner basis G with respect to any monomial order then G contains a monomial ifand only if I contains monomials. The proof of this remark follows immediately fromBuchberger’s algorithm for computing a reduced Gr¨obner basis.The proof of [5, Theorem 3.8] applies ad litteram for an arbitrary binomial ideal I andis based on the noetherian property of a chain of monomial ideals associated to the fibers. Theorem 1.11.
Let I be a binomial ideal. Any descending chain of equivalence classesof fibers F > · · · > F k > F k +1 > · · · is finite. It is easy to see that if F is finite then F = { F } . In the next example we compute theequivalence classes of the infinite fibers for the binomial ideals of Example 1.2.Let F be a fiber of a binomial ideal I . We define two binomial ideals contained in II Let I be a binomial ideal, S a binomial generating set of I and F an I -fiber. Then I Let I be the ideal of Example 1.2 b), which is minimally generatedby the four binomials. The set of equivalence classes of Markov fibers of I is the set { F y , F y , F y } . It is not hard to see that I has also a minimal generating set of cardinalitythree: I = ( y − x y, y − xy , y − y ).2. Indispensable Monomials and Binomials Let I be a binomial ideal and F an I -fiber. We consider the monomial ideals M F generated by the monomials of all fibers G with G ≡ F , and M I generated by all monomials x u ∈ supp( B ), where B ∈ I is a nonzero binomial. It is clear that M F ⊆ M F . We notethe following: Lemma 2.1. If I = ( x u − λ x v , . . . , x u s − λ s x v s ) with λ , . . . , λ s ∈ K \ { } then M I =( x u , x v , . . . , x u s , x v s ) .Proof. One inclusion is immediate. For ” ⊆ ” suppose that x u − λx v ∈ I for some λ ∈ K \{ } .Hence there are polynomials f j ∈ R such that x u − λx v = s X j =1 f j ( x u j − λ j x v j ) . Thus for some i, k ∈ [ s ], x u i or x v i divides x u , and x u k or x v k divides x v . Therefore x u , x v belong to ( x u , x v , . . . , x u s , x v s ). (cid:3) We also note that if x u is a minimal monomial generator of M I then x u is also a minimalmonomial generator of M F u and of M F u . Example 2.2. Let I be the ideal of Example 1.7 b). Then M I = ( y ). Also, if F = F xy then M F = ( xy ) and M F = ( y ). If G = F y then M G = M G = M I .In [14, Proposition 1.5] it was shown that if I is an A -homogeneous binomial ideal forsome A ⊂ Z d such that N A ∩ ( − N A ) = { } , then the indispensable monomials of I areexactly the minimal monomial generators of M I . The same proof applies to any binomialideal I . We isolate this remark: Remark 2.3. Let I be a binomial ideal and S a system of binomial generators of I . Theindispensable monomials of I are precisely the elements of G ( M I ). Moreover G ( M I ) comesfrom S B ∈ S supp( B ) by keeping the minimal elements according to divisibility.Next, for F an I -fiber, we define the graph Γ F (on possibly infinitely many vertices): Definition 2.4. Γ F is the graph with vertices the elements of F and edges { x u , x v } whenever x u − λx v ∈ I Let I be a binomial ideal. The monomial x u is a minimal monomial gen-erator of M I (and thus an indispensable monomial) if and only if the following conditionshold simultaneously(a) | F u | ≥ ,(b) x u is an isolated vertex of Γ F u ,(c) x u is a minimal generator of M F u .Proof. Assume first that x u is a minimal monomial generator of M I . Conditions (a) and(c) follow immediately. For condition (b) we suppose that x u is not an isolated vertex ofΓ F u : there exists x v ∈ F u so that { x u , x v } is an edge of Γ F u and thus x u − λx v ∈ I 2, we conclude that there exists amonomial x v such that x u − λx v ∈ I for some λ ∈ K \ { } . Since x u / ∈ G ( M I ) we concludethat there exists a minimal binomial system of generators S of I and B ∈ S so that x u = x w x u ′ with x u ′ ∈ supp( B ) and x w = 1. In particular we get that F u ′ ≤ F u . We firstexamine the case when B ∈ I Theorem 2.6. Let I be a binomial ideal. The binomial B ∈ I is indispensable if and onlyif the graph Γ F B consists of two isolated vertices.Proof. Let B = x u − λx v for some λ ∈ K \ { } and F = F B . Assume first that the graphΓ F consists of two isolated vertices, that is F = supp( B ) and B / ∈ I Let I be the ideal of Example 1.2 c). Since G ( M I ) = { x , x y, x y , x y , x y , xy , y } then I has seven indispensable monomials by Remark 2.3. This implies that a minimalgenerating set of I can have cardinality no less than four. The indispensable binomialsof I are x y − x and xy − y as follows from Theorem 2.6 and the study of the fibersof the indispensable monomials. See Examples 1.2 c), 1.7 c).The following result generalizes [14, Corollary 1.11]. Corollary 2.8. Let J ⊂ I be two binomial ideals. If B is an indispensable binomial of I and B ∈ J then B is indispensable in J .Proof. By Theorem 2.6 we have that the I -fiber F B is equal to supp( B ). Since the fiberof B in J is a subset of F B and contains supp( B ), then it equals F B . Moreover since J Let F ( S ) be the graph whose vertices are the monomials in S B ∈ S supp( B )and edges { x u , x v } whenever, up to a nonzero scalar multiplication, x u − λx v ∈ S for some λ ∈ K \ { } .Note that the graph F ( S ) may not be simple, since it may have multiple edges as thefollowing example shows. xample 3.2. Let I ⊂ C [ x, y ] be the binomial ideal generated by S = { y − xy , x y − x y , x y − x y , x y − x , x y − x } . The graph F ( S ) is depicted below: • • y xy • •• x y x y x y F ( S ) • • x x y Figure 2. The graph associated to a generating set S of a binomial idealMoreover, if S is a minimal binomial generating set for a binomial ideal I with F ( I ) = ∅ then F ( S ) is a forest. Theorem 3.3. Let S be a binomial generating set of I . The binomial B ∈ S is indispens-able if and only if supp( B ) ⊂ G ( M I ) and the induced graph on the vertices of supp( B ) isa connected component of F ( S ) consisting of a simple edge.Proof. If B ∈ S is indispensable then the first assertion follows from Remark 2.3. Moreoverby Theorem 2.6, | F B | = 2. Thus the induced graph on the vertices of supp( B ) is necessarilya connected component of F ( S ). Furthermore, if this induced graph was not a simple edgethen there would exist λ = λ ∈ K \ { } such that x u − λ x v , x u − λ x v ∈ S . This impliesthat x u , x v ∈ I , thus F B = F ( I ) is an infinite fiber, a contradiction since | F B | = 2.For the converse, assume that B = x u − λx v for some λ ∈ K \ { } , supp( B ) ⊂ G ( M I )and the induced graph on the vertices of supp( B ) is a connected component of F ( S )consisting of a simple edge. The last condition implies that supp( B ) ∩ supp( B ′ ) = ∅ forall B ′ ∈ S , B ′ = B . We can assume that S = { B , B , . . . , B s } (where B = B ) and that B j = x u j − λ j x v j . Suppose now that there exists a monomial x w ∈ F B \ supp( B ). Thisimplies that there exists µ ∈ K \ { } such that x u − µx w ∈ I . Thus x u − µx w = X j,t c α j,t x α j,t B j , (1)where c α j,t ∈ K and the monomials x α j,t are such that x α j,t = x α j,t for any j and t = t .Since x u ∈ G ( M I ) and x u / ∈ supp( B j ) for j = 1 we can assume that c α , = 1 and x α , = 1. By our assumption x w = x v = x v . Thus x v must appear at least twice in theRHS of Equation 1. It follows that x v is divisible by x u j or x v j for some j = 1. Since x v ∈ G ( M I ) this would imply that x v = x u j or x v = x v j . However this is impossiblesince the induced graph on the vertices of supp( B ) is a connected component of F ( S ).Therefore our assumption is wrong and F B = supp( B ). Assume now that B ∈ I Note that for the ideal I of Example 3.2 there is only one connectedcomponent consisting of a simple edge, namely the one corresponding to supp( y − xy ).Thus I has just one indispensable binomial. he following algorithm is an immediate application of Theorem 3.3. Algorithm 1 Computing the indispensable binomials of a binomial ideal I Input: F = { B , . . . , B s } ⊆ K [ X ], with B i = x u i − λ i x v i for i ∈ [ s ], where λ , . . . , λ s ∈ K \ { } . Output: F ′ ⊂ F , the set of indispensable binomials of I = ( B , . . . , B s ). Compute G ( M I ), a subset of { x u , x v , . . . , x u s , x v s } , and set T = { i : { x u i , x v i } ⊂ G ( M I ) } . If T = ∅ then F ′ = ∅ . Otherwise, for every i ∈ T check whether x u i ∈ Supp( B j ) or x v i ∈ Supp( B j ) for some j = i . F ′ = { x u i − λ i x v i : i ∈ T, x u i / ∈ Supp( B j ) , x v i / ∈ Supp( B j ) , for all j = i } . Remark 3.5. We note that step 1 of Algorithm 1 involves checking the divisibility rela-tions of the elements of { x u , x v , . . . , x u s , x v s } . Checking one such divisibility is equivalentto computing the difference vector of the two exponent vectors and see if it belongs to N n . Thus the running time of checking such a divisibility is O ( n ). Since there are s (2 s − such divisibility relations to check, the total running time of having the output of step 1is O ( s n ). Therefore Algorithm 1 is a polynomial-time algorithm. Corollary 3.6. Let I be a binomial ideal minimally generated by s binomials. I is gen-erated by indispensable binomials if and only if | G ( M I ) | = 2 s .Proof. Assume that I is minimally generated by the binomials x u − λ x v , . . . , x u s − λ s x v s , where λ , . . . , λ s ∈ K \ { } . If the binomials are indispensable then | G ( M I ) | = 2 s by Theorem 3.3. Conversely, if | G ( M I ) | = 2 s we obtain the desired conclusion fromAlgorithm 1. (cid:3) Algorithm 2 Testing whether a binomial ideal is generated by indispensable binomials Input: F = { B , . . . , B s } ⊆ K [ X ], a set of binomials generating I . Output: Is I generated by indispensable binomials? YES or NO Compute S ⊂ F , a set of minimal generators for I . Compute G ( M I ) from S . If | G ( M I ) | = 2 | S | then I is generated by indispensable binomials, otherwise not.As an example we recover immediately that the toric ideal of A , the 3 × × I A :it has cardinality 114. Then we minimize this set to get a generating set of cardinality 81and finally we compute | G ( M I A ) | , which is 162. The criterion of Corollary 3.6 finishesthe proof. 4. Graver bases of binomial ideals The notion of primitive binomials plays an important role in the theory of lattice idealsand all applications in terms of computations. We generalize this notion to arbitrary purebinomial ideals, see also [9]. efinition 4.1. Let I be a pure binomial ideal. A binomial 0 = x u − x v ∈ I is called a primitive binomial of I if there exists no other binomial 0 = x u ′ − x v ′ ∈ I such that x u ′ divides x u and x v ′ divides x v . The set of all primitive binomials of I is called the Graverbasis of I , and denoted by Gr( I ).As in the case of lattice ideals, one can generalize [22, Lemma 4.6] to show that allelements of the universal Gr¨obner basis of I are primitive. Below we include the proof forcompleteness. Proposition 4.2. Let I be a pure binomial ideal. Every binomial in the universal Gr¨obnerbasis of I is contained in Gr( I ) . In particular, Gr( I ) is a generating set for the ideal I .Proof. We argue by contradiction. Assume that f = x u − x v / ∈ Gr( I ) and f ∈ G < , thereduced Gr¨obner basis of I according to the monomial order ” < ”. Moreover supposethat in < ( f ) = x u > x v , i.e. x u ∈ G (in < ( I )). Since f is not primitive, there exists g = x u ′ − x v ′ ∈ I such that f = g and x u ′ | x u , x v ′ | x v . If in < ( g ) = x v ′ then x v ′ ∈ in < ( I )and thus x v is divisible by an element of G (in < ( I )). This leads to a contradiction since f belongs to the reduced Gr¨obner basis G < of I . If in < ( g ) = x u ′ then x u ′ = x u . Thus f − g = x v − x v ′ ∈ I and in < ( x v − x v ′ ) = x v . This, as before, leads to a contradiction. (cid:3) To prove that Gr ( I ) is finite, first we remark that if S is an infinite set of monomialsthen an arbitrary element of S is divisible by some element of G ( h S i ). Thus if whenever m, m ′ ∈ S we have that m does not divide m ′ neither m ′ divides m then S is necessarilyfinite. Proposition 4.3. Let I be a pure binomial ideal. The Graver basis of I , Gr( I ) , is a finiteset.Proof. We consider the set S = { x u y v , x v y u : x u − x v ∈ Gr( I ) } . It is immediate thatthere are no divisibility relations among distinct elements of S . Thus S is finite. It followsthat Gr( I ) is finite. (cid:3) One can think of the pairs that form S in the above proof, as the support of binomialsthat generate an ideal closely resembling the binomial ideal of the Lawrence lifting of I ,see [22, Theorem 7.1]. In the next theorem we study this ideal. Theorem 4.4. Let I = (0) be a pure binomial ideal and let Λ( I ) := ( x u y v − x v y u : x u − x v ∈ Gr( I )) ⊂ K [ x , . . . , x n , y , . . . , y n ] . The set { x u y v − x v y u : x u − x v ∈ Gr( I ) } is a minimal system of generators of Λ( I ) consisting of indispensable binomials.Proof. The conclusion follows immediately by Algorithm 2 and Proposition 4.3. (cid:3) Remark 4.5. Note that the conclusion of Theorem 4.4 holds for any binomial ideal J ,where J = ( x u y v − x v y u : x u − x v ∈ A, ∅ 6 = A ⊂ Gr( I )).We remark that when I = I A , a toric ideal, the ideal Λ( I ) is equal to the toric ideal ofthe second Lawrence lifting of A , see [22, Theorem 7.1]. 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Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, 54124,Greece E-mail address : [email protected] Department of Mathematics, University of Ioannina, Ioannina 45110, Greece E-mail address : [email protected] Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei14, Bucharest, RO-010014, Romania, and imion Stoilow Institute of Mathematics of Romanian Academy, Research group of theproject PN-II-RU-TE-2012-3-0161, P.O.Box 1–764, Bucharest 014700, Romania E-mail address : [email protected]@gta.math.unibuc.ro