Asymptotic behavior of Integer Programming and the stability of the Castelnuovo-Mumford regularity
AASYMPTOTIC BEHAVIOR OF INTEGER PROGRAMMING ANDTHE STABILITY OF THE CASTELNUOVO-MUMFORDREGULARITY
LE TUAN HOA
Dedicated to Professor Ngo Viet Trung on the occasion of his 65th birthday.
Abstract.
The paper provides a connection between Commutative Algebra and In-teger Programming and contains two parts. The first one is devoted to the asymptoticbehavior of integer programs with a fixed cost linear functional and the constraintsets consisting of a finite system of linear equations or inequalities with integer coeffi-cients depending linearly on n . An integer N ∗ is determined such that the optima ofthese integer programs are a quasi-linear function of n for all n ≥ N ∗ . Using resultsin the first part, one can bound in the second part the indices of stability of theCastelnuovo-Mumford regularities of integral closures of powers of a monomial idealand that of symbolic powers of a square-free monomial ideal. Introduction
This paper provides a case when a problem in Integer Programming is raised fromCommutative Algebra and its solution leads to solving the original problem in thelatter field.Let I be a proper homogeneous ideal of a polynomial ring R = K [ X , ..., X r ] over afield K . The Castelnuovo-Mumford regularity reg( R/I ) (see (3.2) for the definition) isone of the most important invariants of I . This notion was introduced by D. Mumfordfor sheaves in 1966, and then extended to graded modules by D. Eisenbud and S.Goto in 1984, see Chapter 4 of [11]. Let ¯ I denote the integral closure of I . Veryfew are known about reg( R/I n ) and reg( R/I n ). However, if n (cid:29) n . Unfortunately the proofs in both papers do not give any hintto when these invariants become linear. In order to address this problem, indices ofstability are introduced, see Definition 3.1. So, reg-stab( I ) (resp., reg-stab( I )) is thesmallest number such that reg( R/I n ) (resp., reg( R/I n )) is a linear function for all n ≥ reg-stab( I ) (resp., n ≥ reg-stab( I )).It is of great interest to bound reg-stab( I ) and reg-stab( I ). However these problemsseem to be very difficult and are currently solved for only few cases. If I is an m -primary ideal, where m = ( X , ..., X r ), then some bounds on reg-stab( I ) are establishedin [4, 8, 12] in terms of other invariants, which are not easy to compute. Under theadditional assumption that I is a monomial ideal, an explicit bound is given in [4,Theorem 3.1]. If I is not an m -primary ideal, then in a series of papers it is shown Mathematics Subject Classification.
Key words and phrases.
Linear Programming, Integer Programming, monomial ideal, integral clo-sure, Castelnuovo-Mumford regularity. a r X i v : . [ m a t h . A C ] N ov hat for some very special monomial ideals (often generated by square-free monomialsof degree two), reg-stab( I ) is a rather small number, see [1, 2, 3, 5, 17, 22, 23].Our purpose is to provide an explicit bound on reg-stab( I ) for all monomial ideals,in terms of the maximal generating degree of I and the number r of variables. In orderto do that, together with studying reg( R/I n ) we also study the so-called a i -invariants a i ( R/I n ), which can be regarded as partial Castelnuovo-Mumford regularities, see (3.1).Using a technique of computing local cohomology modules H i m ( R/I ) of a monomialideal given in [32] and developed further in a series of papers [15, 19, 21, 25, 33], onecan translate the problem of computing H i m ( R/I n ) into studying the sets of integerpoints in some rational polyhedra. Now, dealing with partial Castelnuovo-Mumfordregularities, for each n , the computation of a i ( R/I n ) can be formulated as a finite setof integer programs, see Theorem 3.8 and Corollary 3.5. This is a crucial point in ourapproach, which allows to connect two branches of Mathematics.Now we get a parametric family (depending on n ) of integer programs, and one wayto study the asymptotic behavior of the a i -invariants is to study the behavior of themaximum M n , as a function of n , of the integer program ( IQ n ):( IQ n ) max { d x + · · · + d r x r | x ∈ Q n ∩ N r } , where Q n = { x ∈ R r | (cid:80) a ij x j ≤ nb i + c i ∀ i ≤ s ;and x j ≥ ∀ j = 1 , ..., r } , and a ij , d i , b i , c i ∈ Z .This idea was applied in a recent paper [17], but the constraint set there is muchmore simple. In our case, we have to deal with a rather general situation. If all c i = 0,then for each n , the set of feasible solutions to the corresponding integer program isexactly the set of integer points in the convex polyhedron: P n = { x ∈ R r | (cid:88) a ij x j ≤ nb i ∀ i ≤ s ; and x j ≥ ∀ j = 1 , ..., r } . Of course, we can restrict to the case P := P is not empty. One can consider Q n as a relaxation of P n = n P . If all vertices of P are integral, then so is P n , and themaximum over integer points of P n is a linear function of n for all n , since the maximumof the corresponding linear program is attained at a vertex of P n = n P (this is thecase in [17]). However, if not all vertices of P are integral or if c i (cid:54) = 0 for some i , theneven the feasibility of the integer program ( IQ n ) is unclear; that means it is unclearif the polyhedron Q n is empty or not, and if it has an integer point. Note that thestudy of integer points in the family of the polyhedra P n was initiated by Ehrhart in[13, 14], which has lead to many applications and stimulated a lot of research untilthe present time. There are also interesting relationships between rational points inintegral polyhedra and Commutative Algebra, see e.g., Stanley’s book [30].The study of asymptotic integer programs was initiated in the work by Gomory [16],where a much more general situation is considered: the right hand side of constraintscan be any vector in some cone. He showed certain asymptotic periodicity of theoptimal solution. His study was then extended by Wolsey in [35]. From their result,Theorem 2.4, one can quickly derive that M n is an eventually quasi-linear function, seeProposition 2.6. Note that an explicit formulation (for a much more general situation)of this fact can be also found in recent papers [36, 29]. owever, for our application, the main concern here is to determine a number N ∗ in terms of data A = ( a ij ) , b = ( b i ) , c = ( c i ) , d = ( d i ), such that M n becomes aquasi-linear function of n for all n ≥ N ∗ . In order to do this, we have to find a newand independent proof of the fact that M n is an eventually quasi-linear function, seeTheorem 2.12. In the proof, we need some facts on the finite generation of a semigroupring (see Lemma 2.7) and Schur’s bound on the Frobenius number of a finite sequenceof positive integers.Back to the study of the a i -invariants, we further need an auxiliary result to guaranteethat we can apply Theorem 2.12 of the first part, see Lemma 3.9. From that we finallycan show that a i ( R/I n ) is a quasi-linear function of the same slope for all n ≥ N † ,where N † is explicitly defined in terms of the maximal generating degree d ( I ) of I and r , see Theorem 3.11. A bound on reg-stab( I ) can be now quickly derived, see Theorem3.13.The technique of bounding reg-stab( I ) can be applied to studying the asymptoticbehavior of the Castelnuovo-Mumford regularity reg( R/I ( n ) ) and a i ( R/I ( n ) ) of the so-called symbolic powers I ( n ) of I , where I is an arbitrary square-free monomial ideal. Inthis case we are able to show that a i ( R/I ( n ) ) and reg( R/I ( n ) ) are quasi-linear functionsof n for all n ≥ r r/ , see Theorem 3.17.Now we briefly describe the content of the paper. In Section 1 we give some auxiliaryresults. Here a condition for the feasibility of the set of solutions to the correspondinglinear programs is presented. The asymptotic behavior of Integer Programming isstudied in Section 2. Firstly, a condition on n is given to guarantee the existence ofinteger solutions, see Lemma 2.2. Secondly, we show how to use a result by Gomoryand Wolsey to derive the property of M n being a quasi-linear function for n (cid:29) m n and M n of the integer programs over n P and its relaxation Q n , see Lemma 2.8, Lemma 2.10 and Remark 2.11. Finally, wecan give a proof of the main result of this section, Theorem 2.12. The study of theasymptotic behavior of a i ( R/I n ) and reg( R/I n ) as well as of a i ( R/I ( n ) ) and reg( R/I ( n ) )is carried out in Section 3. After describing the Newton polyhedron of I (Lemma 3.3)and simplicial complexes associated to I n (Lemma 3.7) we give a proof of Theorem 3.8connecting the two branches in Mathematics. Using this theorem and Theorem 2.12we can formulate and prove the main results of the papers, Theorems 3.11, 3.13 and3.17. 1. Asymptotic behavior of Linear Programming
In this section we present some preliminary results which give the feasibility of theset of solutions to the corresponding linear programs as well as some properties of itsasymptotic behavior. These results will be used in the next sections.The symbols Z , N , Q , R and R + denote the sets of integers, non-negative integers,rationals, real numbers and non-negative real numbers, respectively. Vectors usually(but not always) are column vectors. If a = ( a , ..., a r ) and a (cid:48) = ( a , ..., a r ) are rowvectors, we write a ≤ a (cid:48) if a i ≤ a (cid:48) i for all i = 1 , ..., r . Similarly for column vectors. Fortwo subsets U, V ⊆ R r and α ∈ R , let U + V = { u + v | u ∈ U and v ∈ V } , and αV = { αv | v ∈ V } . e refer the reader to the book [28] for unexplained terminology and notions in Linearand Integer Programming. Given a ( s × r )-real matrix A = ( a ij ) ∈ M s,r ( R ), vectors b = ( b , ..., b s ) T , c = ( c , ..., c s ) T ∈ R s , and a positive integer n , let us consider thefollowing polyhedra: P = { x ∈ R r | A x ≤ b ; and x j ≥ ∀ j = 1 , ..., r } , P n = { x ∈ R r | A x ≤ n b ; and x j ≥ ∀ j = 1 , ..., r } , Q n = { x ∈ R r | A x ≤ n b + c ; and x j ≥ ∀ j = 1 , ..., r } . In [14], Ehrhart called the system A x ≤ n b (i.e., c = 0) homothetic. Otherwise itis called a bordered system. Note that P n = n P . So P (cid:54) = ∅ if and only if P n (cid:54) = ∅ . Weoften use the following property P n + Q m ⊆ Q n + m , for all n, m ∈ N . In particular, Q n + m (cid:54) = ∅ , provided that both P n and Q m are notempty. The following result gives a lower bound such that starting from this number,all int( Q n ) (cid:54) = ∅ , where int( ∗ ) denotes the interior of ∗ . Lemma 1.1.
Assume that P is full-dimensional. Let γ ∈ int( P ) . Set (1.1) ε γ = min { b i − A i γ | i = 1 , ..., s } , where A i denotes the i -th row of A , and (1.2) N γ = 1 + 1 ε γ max { , − c , ..., − c s } . Then n γ ∈ int( Q n ) for all n ≥ N γ .Proof. If γ ∈ int( P ), then n γ ∈ int( P n ). When n is small, n γ does not necessarilybelong to the relaxation Q n of P n . However, since the slack ε in γ := nb i − A i ( n γ ) > n γ in the i -th constraint increases linearly on n , the point n γ also satisfies the i -thconstraint of Q n for all n large enough. More precisely, for all n ≥ N γ > − c i /ε γ ≥− c i ε i γ , we have nb i − A i ( n γ ) = nε i γ > c i for all i , which yields n γ ∈ int( Q n ). (cid:3) As pointed out by a referee, one can obtain a smaller N γ using the number1 + max ≤ i ≤ s { , − c i /ε i γ } . However, we prefer to use (1.2), which in some cases simplifies our computation.In the above lemma, the bound N γ depends on the existence of the interior point γ .In the following lemma, using the decomposition of P into the sum of a polytope anda polyhedral cone, one can find an r -simplex contained in P . Taking the barycenter ofthis simplex as γ , we can give an explicit bound N for N γ in terms of P . Lemma 1.2.
Assume that P is full-dimensional and P admits the following decompo-sition: P = conv( α , ..., α p ) + cone( β , ..., β q ) , for some points α i ∈ R r and direction vectors β j . Set β = , (1.3) ε = min i,j,k { b i − A i ( α j + β k ) | A i ( α j + β k ) < b i } , nd (1.4) N = 1 + r + 1 ε max { , − c , ..., − c s } . Then, for all n ≥ N , int( Q n ) (cid:54) = ∅ .Proof. Under the assumption one can choose r + 1 points γ , ..., γ r +1 from the set { α i + β j | i = 0 , ..., p and j = 0 , ..., q } such that γ , ..., γ r +1 are affinely independent.Set(1.5) γ = 1 r + 1 ( γ + · · · + γ r +1 ) . Fix i ≤ s . Then there is a point γ j which does not lie on the hyperplane defined by A i x = b i . This means A i γ j < b i , whence A i γ j ≤ b i − ε . Then A i γ = 1 r + 1 ( A i γ j + (cid:88) l (cid:54) = j A i γ l ) ≤ r + 1 ( b i − ε + rb i ) = b i − ε r + 1 . (1.6)Hence b i − A i γ ≥ ε / ( r + 1), or by (1.1), ε γ ≥ ε / ( r + 1). Using (1.2) this implies N γ ≤ r + 1 ε max { , − c , ..., − c s } = N . By Lemma 1.1, we have n γ ∈ int( Q n ) for all n ≥ N . (cid:3) Assume now that P is full-dimensional and the optimum of the following linearprogram ( LP ) max { d T x | A x ≤ b , x ≥ } , is finite, where d = ( d , ..., d r ) T ∈ R r . Let n ≥ N . By Lemma 1.2, Q n (cid:54) = ∅ . Hence theoptimum ϕ n of the following linear program( LQ n ) max { d T x | A x ≤ n b + c , x ≥ } , is either finite or equal to ∞ . Assume that ϕ n = ∞ . Then Q n contains a half-line suchthat the value d T x along this half-line tends to ∞ , i.e., there is y ∈ Q n and v suchthat d T v > y + α v ∈ Q n for all α ≥
0. Then P also contains a half-line with thesame direction v . Indeed, since A i ( y + α v ) ≤ nb i + c i , A i v ≤ lim α →∞ nb i + c i − A i y α = 0.Let z be an arbitrary point of P . Then A i ( z + α v ) = A i ( z ) + αA i v ≤ b i + 0 = b i , for all i ≤ s and α ≥
0. This means z + α v ∈ P . Now we have d T ( z + α v ) = d T z + α d T v −→ ∞ , when α −→ ∞ , a contradiction. Therefore we always have ϕ n < ∞ .The fact that ϕ n is a linear function of n , where n (cid:29)
0, is perhaps well-known.However we cannot find a reference with an explicit formulation of this result, so thatwe include a proof for the sake of completeness. It easily follows from the duality, cf.the proof of [34, Lemma on p. 468]. roposition 1.3. Assume that P is full-dimensional and the optimum ϕ of the linearprogram ( LP ) is finite. Then the optimum ϕ n of the linear program ( LQ n ) is a linearfunction of n of slope ϕ , i.e., ϕ n = ϕn + ϕ , for some ϕ and for all n (cid:29) .Proof. Let n ≥ N . By the duality theorem for linear programming (see, e.g., [28,Corollary 7.1g and (25) on page 92]), we have ϕ n = max { d T x | A x ≤ n b + c , x ≥ } = min { y T ( n b + c ) | y ≥ , y T A ≥ d T } , where y ∈ R s . The dual programs have the same feasible region for all n . Since ϕ n is bounded, it is attained by a vertex of the polyhedron { y ≥ , y T A ≥ d T } . Let y ∗ , ..., y ∗ m be all vertices of this polyhedron. Then ϕ n = min ≤ i ≤ m { (( y ∗ i ) T b ) n + ( y ∗ i ) T c } . If v ∗ is the largest coordinate of intersection points of different lines among u =(( y ∗ i ) T b ) v + ( y ∗ i ) T c , i = 1 , ..., m , then for all v ≥ v ∗ , there is one line, say u =(( y ∗ ) T b ) v +( y ∗ ) T c , which lies below all other lines. Clearly ( y ∗ ) T b is the smallest slopewhich is equal to ϕ . This means ϕ n = ϕn + ( y ∗ ) T c for all n ≥ N := max { N , v ∗ } . (cid:3) Remark 1.4.
1. Assume that
P (cid:54) = ∅ . The property Q n (cid:54) = ∅ always holds if all c i = 0,since in this case Q n = P n = n P . However, if c i (cid:54) = 0 for some i , then the assumptionthat P is full-dimensional cannot be omitted. Example . Let
P ⊂ R be a segment defined by P : x + x ≤ , − x − x ≤ − ,x , x ≥ . Let c = − , c = 0. Then Q n : x + x ≤ n − , − x − x ≤ − n,x , x ≥ . We have
P (cid:54) = ∅ , dim P = 1, but Q n = ∅ for all n ≥ P is full-dimensional and c , ..., c s are fixed. Then Q n could beempty for some small n . However, for n (cid:29) Q n (cid:54) = ∅ . Example . Let P : − x − x ≤ − ,x ≤ ,x ≤ ,x , x ≥ . If c = 2 m, m ≥
2, and c = c = 0, then Q n = ∅ if and only if n < m . . From the proof of Proposition 1.3 one can give an estimation on the value N from which ϕ n becomes a linear function.2. Asymptotic behavior of Integer Programming
In this section we always assume that a ij , b i , c i , d j ∈ Z . As usual, let d = ( d , ..., d r ) T , x =( x , ..., x r ) T and n ∈ N . Consider the following integer program( IQ n ) max d T x (cid:80) rj =1 a ij x j ≤ nb i + c i ( i = 1 , ..., s ); x j ∈ N ( j = 1 , ..., r ) . The corresponding polyhedron of this problem is Q n . In particular, when c = · · · = c s = 0 we get the following integer program( IP n ) max d T x (cid:80) rj =1 a ij x j ≤ nb i ( i = 1 , ..., s ); x j ∈ N ( j = 1 , ..., r ) , and the corresponding polyhedron is P n = n P .From Proposition 1.3 we immediately get the following sufficient criterion for M n tobe an eventually linear function. Another criterion is given in Corollary 2.13. Corollary 2.1.
Assume that P is full-dimensional and the optimum of the linearprogram ( LP ) is finite. Assume further that all vertices of Q n are integral for all n ≥ . Then the maximum M n of ( IQ n ) is a linear function of n for all n (cid:29) . If the matrix A is totally unimodular, i.e., if all its subdeterminants are either 0 or ±
1, then the second assumption in the above corollary holds. On the other hand, ingeneral even the maximum m n of ( IP n ) is not an asymptotically linear function of n (see Example 2.14).If α is a real number, then (cid:98) α (cid:99) , (cid:100) α (cid:101) and { α } = α − (cid:98) α (cid:99) denote the lower integer part, the upper integer part and the fractional part, respec-tively, of α .Denote by A ( i , ..., i k ; j , ..., j k ) the submatrix of A with elements in the rows i , ..., i k and the columns j , ..., j k , k ≤ min { r, s } , and D ( i , ..., i k ; j , ..., j k ) = det( A ( i , ..., i k ; j , ..., j k )) . In the sequel, set(2.1) I = ( i , ..., i k ; j , ..., j k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ≤ min { r, s } , ≤ i < · · · < i k ≤ s, ≤ j < · · · < j k ≤ r,D ( i , ..., i k ; j , ..., j k ) (cid:54) = 0 . f I = ( i , ..., i k ; j , ..., j k ) ∈ I and u ∈ R r , we denote by z I , u the solution of thesystem(2.2) (cid:40) A ( i , ..., i k ; j , ..., j k )( x j , ..., x j k ) T = ( u i , ..., u i k ) T ,x l = 0 ∀ l (cid:54)∈ { j , ..., j k } . In the following result we give a sufficient condition to guarantee that the integerprogram ( IQ n ) has a feasible solution. It is a kind of improvement of Lemma 1.2. Theintuition for this result is the following: since the data are integers, the slack ε definedin (1.3) is not too small. Therefore, one can give an estimation of n such that theslacks of the point n γ (defined by (1.5)) in Q n are so big, that one of its approximateinteger points still belongs to Q n . This idea was used in the proof of [21, Lemma 3.2]. Lemma 2.2.
Assume that the polyhedron P is full-dimensional. Let Q I,n = Q n ∩ N r .Set a ∗ := max i,j | a ij | and c ∗ = min ≤ i ≤ s c i . Then Q I,n (cid:54) = ∅ for all n ≥ κ := max { , ( r + 1) D ( r a ∗ − c ∗ ) } , where D is the maximum absolute value of the subdeterminants of the matrix A .Proof. Keep the notation in Lemma 1.2 and its proof.First we give a lower bound for the number ε defined in (1.3). Note that each α j in Lemma 1.2 is a solution of a system of linear equation of the type (2.2) with( i , ..., i k ; j , ..., j k ) ∈ I (see the notation (2.1)). Hence(2.3) D α j ∈ Z r , where D := D ( i , ..., i k ; j , ..., j k ) . One can also assume that each vector β j (cid:48) of the cone cone( β , ..., β q ) in Lemma 1.2 isa a solution of a system of linear equation of the type A ( i (cid:48) , ..., i (cid:48) l − ; j (cid:48) , ..., j (cid:48) l − )( x j (cid:48) , ..., x j (cid:48) l − ) T = ,x j (cid:48) l = 1 ,x m = 0 ∀ m (cid:54)∈ { j (cid:48) , ..., j (cid:48) l } , where ( i (cid:48) , ..., i (cid:48) l − ; j (cid:48) , ..., j (cid:48) l − ) ∈ I and j (cid:48) l (cid:54)∈ { j (cid:48) , ..., j (cid:48) l − } . Then D β j (cid:48) ∈ Z , where D := D ( i (cid:48) , ..., i (cid:48) l − ; j (cid:48) , ..., j (cid:48) l − ) . Hence 0 (cid:54) = D D ( b i − A i ( α j + β j (cid:48) )) ∈ Z for all i such that A i ( α j + β j (cid:48) ) < b i . Thisimplies | D D | ( b i − A i ( α j + β j (cid:48) )) ≥
1, whence(2.4) ε ≥ /D , Now, let γ be the same point defined in (1.5). Define w ∈ N r as follows w l = (cid:40) (cid:98) nγ l (cid:99) if { nγ l } ≤ / , (cid:100) nγ l (cid:101) if { nγ l } > / . his means w l − nγ l ≤ / l . Fix n ≥ κ . Using (1.6) and (2.4), for all i ≤ s , weget A i w = A i ( n γ ) + A i ( w − n γ ) ≤ n ( b i − ε r +1 ) + r a ∗ ≤ nb i − n ( r +1) D + r a ∗ ≤ nb i + c ∗ (since n ≥ κ ) ≤ nb i + c i . Thus w ∈ Q n . (cid:3) Remark 2.3.
Even in the case c = · · · = c r = 0 the assumption P being full-dimensional in Lemma 2.2 cannot be omitted. For an example, P : (cid:40) x + · · · + x r ) = 1 ,x , ..., x r ≥ , has dimension r − P n = n P has an integer point if and only if n is divisible by 3.Gomory [16] considered a family of integer programs P ( b (cid:48) ) : (cid:40) M ( b (cid:48) ) = max d (cid:48) T x (cid:48) , s. t. A (cid:48) x (cid:48) = b (cid:48) , x (cid:48) j ∈ N ( j = 1 , .., s (cid:48) + r (cid:48) ) , where A (cid:48) ∈ M s (cid:48) ,s (cid:48) + r (cid:48) ( Z ), b (cid:48) ∈ Z s (cid:48) and rank( A (cid:48) ) = s (cid:48) . Here A (cid:48) , d (cid:48) are fixed and b (cid:48) is considered as a vector parameter. Of course, our case b (cid:48) = n b + c is only a veryspecial case. Let B be an optimal basis for the linear programming relaxation of P ( b (cid:48) ).Without loss of generality we can assume that B consists of the first s (cid:48) columns, so A (cid:48) = ( B, N ) , where N ∈ M s (cid:48) ,r (cid:48) ( Z ). For a vector x (cid:48) ∈ R s (cid:48) + r (cid:48) , we write x (cid:48) = ( x B , x N ), where x B ∈ R s (cid:48) and x N ∈ R r (cid:48) . If b (cid:48) is sufficiently deep in the cone { z ∈ R s (cid:48) | B z ≥ } , Gomory gave aformula for an optimal solution to P ( b (cid:48) ) and showed that this solution is periodic inthe columns B j of B , see [16, Theorem 5]. This result was extended by Wolsey to any b (cid:48) as following: Theorem 2.4. ([16, Theorem 5] and [35, Theorem 1])
Given fixed A (cid:48) and d (cid:48) for which { y (cid:48) T A ≥ d (cid:48) T } (cid:54) = ∅ , there exists a finite dictionary containing a list of nonnegativeinteger vectors { x qN } q ∈ Q B for each dual feasible basis B with the following property: Let Q b (cid:48) = { q ∈ Q B | x qB ( b (cid:48) ) = B − b (cid:48) − B − N x qN ≥ and integer } . Either Q b (cid:48) (cid:54) = ∅ implying that P ( b (cid:48) ) is infeasible, or Q b (cid:48) (cid:54) = ∅ implying that if ¯ d TN x q ( b (cid:48) ) N =min Q b (cid:48) { ¯ d TN x qN } , where ¯ d = d N B − N − d N ≥ , the vector ( x q ( b (cid:48) ) B ( b (cid:48) ) , x q ( b (cid:48) ) N ) is anoptimal solution to P ( b (cid:48) ) . To proceed further, we need the following notion:
Definition 2.5.
We say that a function f : N → R is a quasi-linear function of period t if there are finitely many linear functions f , ..., f t − , t ≥
1, such that f ( n ) = f i ( n ) if n ≡ i mod t for all 0 ≤ i ≤ t −
1. For short, we denote f by ( f i ), i.e., f = ( f i ).A function g : N → R is an eventually quasi-linear function , if it agrees with aquasi-linear function for sufficiently large n . rom now on, assume that P is full-dimensional and that the following linear program( LP ) : max { d T x | A x ≤ b , x j ≥ j = 1 , ..., r ) } , has a finite optimum ϕ . If the integer program ( IP n ) (resp., ( IQ n )) has a feasiblesolution, then by Proposition 1.3, it also has a finite optimum. In this case, we denotethese optima by m n and M n , respectively. Otherwise we set m n = −∞ (respectively, M n = −∞ ) or we also say that m n (respectively, M n ) is not (well) defined. If b (cid:48) issufficiently deep in the cone { z ∈ R s (cid:48) | B z ≥ } , [16, Theorem 5] also claims thatthe r (cid:48) -vector x q ( b (cid:48) ) N is periodic in the columns B j of B , i.e., x q ( b (cid:48) + B j ) N = x q ( b (cid:48) ) N . In thegeneral case, it is unclear if this property still holds, but from (2.5) below it is easyto show that this solution has the asymptotic periodicity in columns, in the sensethat x q ( b (cid:48) +( n +1) B j ) N = x q ( b (cid:48) + nB j ) N for all n (cid:29)
0. This periodicity signifies that M n isan eventual quasi-linear function of n . This property is confirmed in the followingconsequence of Theorem 2.4. Proposition 2.6. M n is an eventually quasi-linear function of n of a period at most D - the maximum absolute value of the subdeterminants of the matrix A .Proof. By the standard technique, we can reformulate the integer program ( IQ n ) inform of P ( n b + c ) with A (cid:48) = ( A, I s ), where I s is the unit matrix of size s , x (cid:48) =( x , ..., x r , z , ...., z s ) T , d (cid:48) = ( d , ..., d r , , ..., T and b (cid:48) = n b + c . We may assume thatrank( A (cid:48) ) = s .Let n ≥ κ be an arbitrary integer, where κ is defined in Lemma 2.2. By Lemma2.2, P ( n b + c ) is feasible. By Theorem 2.4, for each n there is a dual feasible basis B such that Q n b + c (cid:54) = ∅ . Note that one can choose κ (cid:48) such that B ( n b + c ) ≥ for all n ≥ κ (cid:48) if and only if B ( κ (cid:48) b + c ) ≥ , i.e., all vectors n b + c , n ≥ κ (cid:48) lie in the samecone. Therefore one can choose the same B for all n ≥ max { κ, κ (cid:48) } . Fix B . So, thevector ( x q ( b (cid:48) ) B ( b (cid:48) ) , x q ( b (cid:48) ) N ) is well defined. Note that M n = M ( b (cid:48) ). In order to show that M n is an eventually quasi-linear function, it suffices to show the periodicity of q ( b (cid:48) ),or equivalently, to show the periodicity of the sets Q n b + c , provided n (cid:29) B j of B we have(2.5) x qB ( b (cid:48) + B j ) = B − ( b (cid:48) + B j ) − B − N x qN = x qB ( b (cid:48) ) + e j , where e j is the j -th basis vector of R s . We now show that this property implies theperiodicity of Q n b + c for n (cid:29) κ ” such that for any q ∈ Q B and n ≥ κ ”, we have x qB ( n b + c ) ≥ if and only if x qB ( κ ” b + c ) ≥ . Let δ = | det( B ) | . Then δ b = a B + · · · + a s B s , for some integers a , ..., a s .For any n ≥ max { κ, κ (cid:48) , κ ” } + δ and q ∈ Q B we have(i) x qB ( n b + c ) ≥ if and only if x qB (( n − δ ) b + c ) ≥ ,(ii) x qB ( n b + c ) = x qB (( n − δ ) b + c + δ b ) = x qB (( n − δ ) b + c ) + a e + · · · + a s e s .This means x qB ( n b + c ) is an integer vector if and only if so is x qB (( n − δ ) b + c ).These two properties imply that Q n b + c = Q ( n − δ ) b + c , as required. (cid:3) ote that the property M n being an eventually quasi-linear function directly followsfrom a much more general result in the recent work [36, Theorem 3.5(a), Property3a] (also see [29, Theorem 1.4]), where the parameters of integer programs can bepolynomials of one variable. It is also interesting to note that Gomory’s formula of anoptimal solution to P ( b (cid:48) ) in Theorem 2.4 is still used until now, see, e.g., [27, Theorem2.9]. On the other hand, in [31], a relationship between Integer Programming andGr¨obner bases was studied, when the cost function varies.For our application in Section 3, we need to find a number N ∗ , such that M n becomesa quasi-linear function for all n ≥ N ∗ . Unfortunately, neither results nor proofs in thepapers [29, 36] provide an estimation for N ∗ . Analyzing the proof of Proposition 2.6, wewould get such an estimation if we could understand better the complexity of vectors { x qN } q ∈ B . This is not a trivial task.In the rest of this section we give a totally different proof of Proposition 2.6, whichgives a way to find such a number N ∗ . Our proof is mainly combinatorial and basedon the module structure of the family of integer points {Q I,n } n ≥ . For that purpose,consider the following polyhedra in R r +1 ˜ P = (cid:26)(cid:18) x y (cid:19) | A x − y b ≤ ; x ≥ , y ≥ (cid:27) , and ˜ Q = (cid:26)(cid:18) x y (cid:19) | A x − y b ≤ c ; x ≥ , y ≥ (cid:27) . Note that ˜ P is a pointed polyhedral cone, (cid:18) v n (cid:19) ∈ ˜ P if and only if v ∈ P n and (cid:18) v n (cid:19) ∈ ˜ Q if and only if v ∈ ˜ Q n .By [28, Theorem 6.4], ˜ P I := ˜ P ∩ N r +1 forms a finitely generated semigroup, so thatthe K -vector space K [ ˜ P I ] is a Noetherian ring, where K is a field. The set K [ ˜ Q I ],where ˜ Q I := ˜ Q ∩ N r +1 , is a finitely generated module over K [ ˜ P I ]. The following resultis given in the proof of [28, Theorem 17.1]. For convenience of the reader we give asketch of the proof here. Lemma 2.7.
Assume that ˜ P I (cid:54) = ∅ and ˜ Q I (cid:54) = ∅ . Then(i) The semigroup ring K [ ˜ P I ] is generated by (integer) vectors with all componentsless than ( r + 1) D (cid:48) in absolute value, where D (cid:48) is the maximum absolute value of thesubdeterminants of the matrix [ A b ] .(ii) The module K [ ˜ Q I ] is generated over K [ ˜ P I ] by (integer) vectors with all compo-nents less than ( r + 2)∆ in absolute value, where ∆ is the maximum absolute value ofthe subdeterminants of the matrix [ A b c ] .Proof. (Sketch): By Cramer’s rule, the polyhedral cone ˜ P admits the following repre-sentation ˜ P = cone( β , ..., β q ) , where β , ..., β q are integer vectors with each component being a subdeterminant of[ A b ] - in particular each component is at most D (cid:48) in absolute value. Let x , ..., x t be he integer vectors contained in E := { µ β + · · · + µ q β q | ≤ µ j < j = 1 , ..., q );at most r + 1 of the µ j are nonzero } . Then each component of each x j is less than ( r + 1) D (cid:48) and the set { β , ..., β q , x , ..., x t } forms a basis of the semigroup ˜ P I .Similarly, ˜ Q admits the following decomposition,˜ Q = conv( α , ..., α p ) + cone( β , ..., β q ) , where β , ..., β q are defined as above, and α , ..., α p are vectors with each componentbeing a quotient of subdeterminants of [ A b c ]. In particular, each component of α i isat most ∆ in absolute value. Let x , ..., x u , u > t , be the integer vectors contained inconv( α , ..., α p ) + E . Then each component of each x j , j ≤ u , is less than ∆ + ( r + 1) D (cid:48) ≤ ( r + 2)∆, andthe set { x t +1 , ..., x u } forms a basis of the module K [ ˜ Q I ] over K [ ˜ P I ]. (cid:3) For short, we also set P I,n := P n ∩ N r . Note that(2.6) P I,n + P I,m ⊆ P
I,n + m and P I,n + Q I,m ⊆ Q
I,n + m , for all numbers n, m ∈ N . This implies that the ring K [ ˜ P I ] admits the so-called N -graded structure, namely K [ ˜ P I ] = ⊕ n ≥ K [ P I,n ] , where deg( u ) = n if u ∈ P I,n . Similarly, K [ ˜ Q I ] is a graded module over K [ ˜ P I ]: K [ ˜ Q I ] = ⊕ n ≥ K [ Q I,n ] , where deg( v ) = n if v ∈ Q I,n . This interpretation allows us to relate elements of P I,n and Q I,n to elements of other sets P I,m and Q I,m with smaller indices m . Below aresome elementary properties of m n and M m .From (2.6) we get the following inequalities for all i, j > m i + m j ≤ m i + j and m i + M j ≤ M i + j . Lemma 2.8.
Assume that P I,n (cid:54) = ∅ . Then m n ≤ ϕn . Moreover, there is j ∗ ≤ D suchthat m j ∗ = ϕj ∗ .Proof. The inequality m n ≤ ϕn is trivial. Since ( LP ) has a finite optimum, there isa vertex α of P such that d T α = ϕ . Note that all components of α are rationalnumbers of a common denominator which is a subdeterminant of A - in particular,there is j ∗ ≤ D such that j ∗ α ∈ N r ∩ P j ∗ = P I,j ∗ . Hence m j ∗ = d T ( j ∗ α ) = j ∗ ϕ . (cid:3) As we see from Lemmas 2.2, 2.7 and 2.8, the set of indices: J max := { n < ( r + 1) D (cid:48) | P I,n (cid:54) = ∅ and m n n = ϕ } (2.8) := { j < · · · < j p < ( r + 1) D (cid:48) } , contains j ∗ , where j ∗ ≤ D is defined in Lemma 2.8. In particular J max (cid:54) = ∅ and j ≤ D .Let δ := gcd( j , ..., j p ) . his number is in general much less than D and will be proved to be an upper boundfor the period of M n . By Schur’s bound on the Frobenius number, see [7], we have Lemma 2.9.
All multiples of δ bigger or equal to max { ( j − j p − , j } belong tothe numerical semigroup S := N j + · · · + N j p ⊆ N δ . The following simple result shows that the upper bound of m n in Lemma 2.8 isattained in many places. Lemma 2.10. If n ∈ S , then P I,n (cid:54) = ∅ and m n = ϕn .Proof. Assume that n = (cid:80) n i , n i ∈ J max . Let u i ∈ P I,n i such that d T u n i = m n i = ϕn i (see (2.8)). Then (cid:80) u i ∈ P I,n and d T ( (cid:80) u i ) = ϕn , whence m n = ϕn (by Lemma2.8). (cid:3) Remark 2.11.
We list here some further properties of m n , M n , P I,n and Q I,n , whichimmediately follow from Lemma 2.7, Lemma 2.10 and (2.6).(i) If u i ∈ P I,n i ( i = 1 ,
2) and v ∈ Q I,n , then deg( u + u ) = deg( u ) + deg( u )and deg( u + v ) = deg( u ) + deg( v ).(ii) If M n is well defined and j ∈ S , then M n + j is well defined. This is applied to j = mj for any m ≥ m n = d T u for some u = u + u ∈ P I,n with u i ∈ P I,n i , then m n i = d T u i for i = 1 , m n = d T u = d T u + d T u ≤ m n + m n . The reverse inequalityfollows from (2.7).Similarly, if M n = d T v for some v = u + v (cid:48) ∈ Q I,n with u ∈ P I,n and v (cid:48) ∈ Q I,n , then m n = d T u , M n = d T v (cid:48) .(iv) Assume that M n = d T ( u + v ), where u ∈ P I,n and v ∈ Q I,n . Assume furtherthat n = jn (cid:48) for some j ∈ S (defined in Lemma 2.10). Then we may assumethat u = n (cid:48) u (cid:48) , where u (cid:48) ∈ P I,j with d T u (cid:48) = ϕj . In particular this is applied to j .By Lemma 2.10, m n = ϕn , m j = ϕj = d T u (cid:48) for some u (cid:48) ∈ P I,j . By (iii), m n = d T u = n (cid:48) d T u (cid:48) . So, we can replace u + v by n (cid:48) u (cid:48) + v .(v) Let J be the set of generating degrees of the semigroup ring K [ ˜ P I ]. By Lemma2.8, j ∈ J ; and by Lemma 2.7(i), all elements in J are at most ( r + 1) D (cid:48) − u ∈ P I,n for some n , then u can be expressed as a sum of someelements u i ∈ P I, deg( u i ) with deg( u i ) ∈ J .We can now formulate and prove the main result of this section. Theorem 2.12.
Let
D, D (cid:48) and ∆ be the maximum absolute value of the subdetermi-nants of the matrices A , [ A b ] and [ A b c ] , respectively. Set N ∗ := ( r + 1) D (cid:48) ( D −
1) + ( r + 2)∆ − D. Assume that P is full-dimensional and the linear program ( LP ) has a finite maximum ϕ . Then the maximum M n of ( IQ n ) is a quasi-linear function of n with the slope ϕ and period δ (defined before Lemma 2.9) for all n ≥ N ∗ . roof. The idea of the proof is the following: Assume that M n = d T v for some v ∈Q I,n . By Lemma 2.7 and Remark 2.11(v), v can be expressed as a sum of one elementfrom Q I,h , h < ( r +2)∆, and some other elements from P I,j , j ∈ J . Since these degrees j are bounded, most of the elements in such an expression can be grouped into oneelement, say u , in P I,mj for some m ∈ N . This element can be replaced by a multiple m u of u ∈ P I,j (by Remark 2.11(iv)). If we take m as large as possible, then thesum v of the rest of the elements in an expression of v gives an element of a boundeddegree. Hence this degree, say t , must be repeated when n (cid:29)
0. This implies that M n is “nearly” quasi-linear. In order to show that it is really quasi-linear, we should makethis degree t periodically unchanged. We can achieve this by adding to v a multipleof u from m u (this explains why we need to go down to elements of small degrees).We now give the detail of the proof. Claim 1 . Assume that M n is defined (i.e., Q I,n (cid:54) = ∅ ) and M n = d T v for some v ∈ Q I,n .Then one can assume that v = m u + v , for some m ∈ N , u ∈ P I,j , v ∈ Q I, deg( v ) and deg( v ) ≤ N ∗ . Proof of Claim 1 : By Lemma 2.7(ii) and Remark 2.11(v), there are vectors u i ∈P I, deg( u i ) , deg( u i ) ∈ J ( i = 1 , ..., l ) and v h ∈ Q I,h such that v = l (cid:88) i =1 u i + v h , (2.9) where h < ( r + 2)∆ and n = l (cid:88) i =1 deg( u i ) + h. Here l ≥ l = 0 means that there is no element u i in the sub-sum of (2.9). Let U = (cid:80) li =1 u i . Using the containment P I,i + P I,j ⊆ P
I,i + j we can write U in the form U = u + u i + · · · + u i l (cid:48) , where u ∈ P I,mj for some m ≥ u = and 1 ≤ i < · · · < i l (cid:48) ≤ l ( l (cid:48) ≥ l (cid:48) .If l (cid:48) ≥ j , then consider the sums of degrees(2.10) deg( u i ) , deg( u i ) + deg( u i ) , ..., deg( u i ) + · · · + deg( u i j ) . If one of these numbers, say deg( u i ) + · · · + deg( u i j ) = j m (cid:48) for some m (cid:48) >
0, where1 ≤ j ≤ j , then taking u (cid:48) := u + u i + · · · + u i j ∈ P I,j ( m + m (cid:48) ) , as a new u , we get U = u (cid:48) + u i j +1 + · · · + u i l (cid:48) , a contradiction to the minimality of l (cid:48) .So, all numbers in (2.10) are not divisible by j . But then two of these sums, saydeg( u i ) + · · · + deg( u i j ) and deg( u i ) + · · · + deg( u i j (cid:48) ), where 1 ≤ j < j (cid:48) ≤ j , are ongruent modulo j . Hence, deg( u i j +1 ) + · · · + deg( u i j (cid:48) ) = j m ” for some m ” > u ” := u + u i j +1 + · · · + u i j (cid:48) ∈ P I,j ( m + m ”) , as a new u , we get U = u ” + u i + · · · + u i j + u i j (cid:48) +1 + · · · + u i l (cid:48) , again a contradiction to the minimality of l (cid:48) .Summing up, we always have l (cid:48) ≤ j −
1. Let v = v h + u i + · · · + u i l (cid:48) . Then, by (2.9), we havedeg( v ) = deg( v h ) + deg( u i ) + · · · + deg( u i l (cid:48) ) ≤ ( r + 2)∆ − j − r + 1) D (cid:48) − ≤ ( r + 2)∆ − D − r + 1) D (cid:48) − r + 1) D (cid:48) ( D −
1) + ( r + 2)∆ − D = N ∗ . Since v = u + v and deg( u ) = mj , by Remark 2.11(iv), we can assume that u = m u for some u ∈ P I,j , which yields v = m u + v , as required. Claim 2 . Let k be a number such that N ∗ ≤ k < N ∗ + j . Let n = n j + k , where n ≥
1. Assume that M n is well defined. Then M k is well defined, and M n = ϕn + M k − ϕk. Proof of Claim 2 : By Claim 1, one can write M n = d T ( m u + v ) for some u ∈P I,j , v ∈ P I,h and h ≤ N ∗ . Since n = mj + h = n j + k and h ≤ N ∗ ≤ k ,we have k − h = ( m − n ) j ≥
0. By Remark 2.11(ii), M k is well defined. Notethat deg(( m − n ) u + v ) = ( m − n ) j + h = k . Since M n = d T ( m u + v ) and m u + v = n u + [( m − n ) u + v ], by Remark 2.11(iii), M n = m n j + M k = ϕn j + M k = ϕn − ϕk + M k . Claim 2 already says that M n is a quasi-linear function of period j for n (cid:29)
0. Thenext claim shows that M n is a quasi-linear function of period j for all n ≥ N ∗ . Claim 3 . If n ≥ N ∗ , then M n is well defined. Moreover, for any n, m ≥ N ∗ such that n ≡ m (mod j ), we have M n = M m + ϕ ( n − m ) . Proof of Claim 3 : For the first statement: If n ≥ κ , then M n is well defined byLemma 2.2. So, we may assume that n < κ . Assume that n = n j + k for some N ∗ ≤ k < N ∗ + j . Let n (cid:48) = n (cid:48) j + n = ( n (cid:48) + n ) j + k with n (cid:48) large enough, suchthat n (cid:48) > κ . Since M n (cid:48) is well defined, by Claim 2, we get that M k is well defined, i.e., P I,k (cid:54) = ∅ . By Remark 2.11(ii), M n is well defined. or the second statement, assume that n ≡ m ≡ k (mod j ), for some N ∗ ≤ k Assume that P is full-dimensional and the linear program ( LP ) hasa finite maximum ϕ . Then (i) m n is an eventually linear function if and only if δ = 1 . In this case m n = ϕn for n (cid:29) . (ii) If δ = 1 , then M n is an eventually linear function. The converse does not hold.Proof. If δ = 1, then by Theorem 2.12, both m n and M n are eventually linear functions(quasi-linear functions of period 1).Assume that m n is an eventually linear function. By Lemma 2.10, m kj = ϕkj forall k ≥ 1. Hence we must have m n = ϕn for all n (cid:29) 0. Assume that δ > 1. Fix n (cid:29) n is not divisible by δ and ϕn = m n = d T u for some u ∈ P I,n . By Remark2.11(v), u = (cid:80) u i , where l i := deg( u i ) ∈ J . Since n is not divisible by δ , there isone degree, say l , is not divisible by δ . Then l (cid:54)∈ J max . That means m l < ϕl .By Lemma 2.8, d T u i ≤ ϕl i . All these imply that d T u = (cid:80) d T u i < ϕ (cid:80) l i = ϕn , acontradiction. Hence δ = 1. inally, let us consider the following integer programs M n = max( x + x )4 x + 2 x ≤ n + 1 , − x ≤ − n. In this case c = 1 , c = 0. The polytope P has three vertices v = (1 / , , v =(3 / , , v = (1 / , / ϕ = 1 is attained at only the vertex v .Hence, we have δ = 2. However, M n = n for all n ≥ (cid:3) We know by Lemma 2.2 that Q I,n (cid:54) = ∅ for all n (cid:29) 0, provided that P is full-dimensional. However, the following example shows that P I,n (cid:54) = ∅ does not imply that P I,n +1 (cid:54) = ∅ . Proposition 2.6 as well as Theorem 2.12 say that the period of M n isbounded by D . The following example also shows that this bound is sharp even for m n . Example 2.14. Let p < · · · < p r be relatively prime positive integers ( r ≥ max( x + · · · + x r ) p i x i ≤ i = 1 , ..., r ) , p x + · · · + p r x r ≥ ,x i ≥ i = 1 , ..., r ) , and(2.12) max( x + · · · + x r ) p i x i ≤ n ( i = 1 , ..., r ) , p x + · · · + p r x r ≥ n,x i ∈ N ( i = 1 , ..., r ) . Then the polyhedron P of (2.11) is full-dimensional and (2.11) has the maximum ϕ = (cid:80) ri =1 /p i . The polyhedron of the linear relaxation of (2.12) is P n . Let x i := (cid:98) n/p i (cid:99) > n/p i − 1. Then r (cid:88) i =1 p i x i > rn − ( p + · · · + p n ) ≥ n, for all n > ( p + · · · + p r ) / ( r − n > ( p + · · · + p r ) / ( r − n < p .We also have (1 , , ..., ∈ P I,p , (1 , , , ..., ∈ P I,p + p , ... However, if p ≥ p + a and p ≥ a , where a ≥ P I,n = ∅ for all p + 1 ≤ n ≤ p + a − P I,n (cid:54) = ∅ does not imply that P I,n +1 (cid:54) = ∅ .Note that if P I,n (cid:54) = ∅ , then the maximum of (2.12) reaches at the point ( x , ..., x r )and the maximum is equal to m n = r (cid:88) i =1 (cid:98) n/p i (cid:99) = ( r (cid:88) i =1 /p i ) n − r (cid:88) i =1 { n/p i } . This function is a quasi-linear function of period p · · · p r for n > ( p + · · · + p r ) / ( r − D = p · · · p r . he next example shows that Q I,n (cid:54) = ∅ only if n is quite big. Hence the lowestnumber from which M n becomes a quasi-linear function of n could be also quite big.In this example it is D . Example 2.15. Let a be a positive integer. Consider the following linear and integerprograms:(2.13) max( x + · · · + x r ) ax ≤ , ax − x ≤ , ..., ax r − x r − ≤ , − x r ≤ ,x i ≥ i = 1 , ..., r ) , (2.14) max( x + · · · + x r ) ax ≤ n, ax − x ≤ , ..., ax r − x r − ≤ , − x r ≤ ,x i ∈ N ( i = 1 , ..., r ) , and(2.15) max( x + · · · + x r ) ax ≤ n, ax − x ≤ , ..., ax r − x r − ≤ , − x r ≤ − ,x i ∈ N ( i = 1 , ..., r ) . That means we consider the case c = · · · = c r − = 0 and c r = − 1. The polyhedra of(2.13) and of the linear relaxations of (2.14) and (2.15) are P , P n and Q n , respectively.Then P is full-dimensional, all P I,n (cid:54) = ∅ as they contain the point . However it is easyto check that Q I,n (cid:54) = ∅ if and only if n ≥ a r . The maximum of (2.15) is equal to M n = (cid:98) n/a (cid:99) + (cid:98) n/a (cid:99) + · · · + (cid:98) n/a r (cid:99) , which is a quasi-linear function of slope (cid:80) ri =1 /a i and period a r .The last example shows that M n can be an eventually linear function, even if D isa big number. Example 2.16. Consider the following integer programs M n = max( x + x ) x + x ≤ n + c , − x ≤ − n + c , − x ≤ − n + c ,x , x ∈ N . The polytope P in this example has three vertices: v = (1 / , / , v = (1 / , / , v = (2 / , / ϕ = 1 is attained at v and v . m is not defined (thatmeans, equal −∞ ), m is attained at the point (1 , 1) which is not a vertex of P . Thesemigroup S contains all numbers 2 , , 4. In particular δ = 1 and M n is an eventuallylinear function for all c , c , c . In this example D = 12. . Stability of the Castelnuovo-Mumford regularity In this section we study a problem in Commutative Algebra. First let us recall thenotion of the Castelnuovo-Mumford regularity. Let R := K [ X , ..., X r ] be a polynomialring over a field K . Consider the standard grading in R , that is deg( X i ) = 1 for all i = 1 , ..., r. Let m := ( X , ..., X r ). For a finitely generated graded R -module E , set(3.1) a i ( E ) := sup { t | H i m ( E ) t (cid:54) = 0 } , where H i m ( E ) is the local cohomology module with the support m . The Castelnuovo-Mumford regularity of E is defined by(3.2) reg( E ) = max { a i ( E ) + i | ≤ i ≤ dim E } . Another way to define the Castelunovo-Mumford regularity is to use the gradedminimal free resolution of E . Since this invariant bounds the maximal degrees of allgenerators of syzygy modules, one can use it as a measure of the cost of extractinginformation about a module E . Beyond its interest to algebraic geometers and commu-tative algebraists, the Castelnuovo-Mumford regularity “has emerged as a measure ofthe complexity of computing Gr¨obner bases”, see the itroduction of [6]. The interestedreaders are referred to [6] for a survey and to the book [11] for some basic propertiesand results on the Castelunovo-Mumford regularity.Let I be a proper homogeneous ideal of a polynomial ring R . Then, reg( I ) andreg( R/I ) are well defined, and reg( I ) = reg( R/I ) + 1.The integral closure of I is the set of elements x in R that satisfy an integral relation x n + α x n − + · · · + α n − x + α n = 0 , where α i ∈ I i for i = 1 , . . . , n . This is again a homogeneous ideal and is denoted by I .The Castelnuovo-Mumford regularities of R/I n and R/I n could be huge numbers fora fixed number n . However, when n (cid:29) 0, a striking result of Cutkosky, Herzog andTrung, and independently of Kodiyalam, says that both reg( R/I n ) and reg( R/I n ) arelinear functions of n (see [9, Theorem 1.1] or [24, Theorem 5]). In order to understandwhen these functions become linear functions, let us introduce indices of stability ofCastelnuovo-Mumford regularities. Definition 3.1. i) Assume that reg( R/I n ) = an + b for all n (cid:29) 0. Setreg-stab( I ) = min { t ≥ | reg( R/I n ) = an + b ∀ n ≥ t } . ii) Similarly, assume that reg( R/I n ) = cn + d for all n (cid:29) 0. Setreg-stab( I ) = min { t ≥ | reg( R/I n ) = cn + d ∀ n ≥ t } . It is of great interest to give a bound for reg-stab( I ) and reg-stab( I ). However,until now only special cases were treated. Even in the case of m -primary ideals theknown bounds on reg-stab( I ) are not explicitly defined (see [12, 8]). Explicit boundson reg-stab( I ) are given in the case of m -primary monomial ideals and some othermonomial ideals.Recall that a monomial ideal is an ideal generated by monomials. In this sectionwe can use results in Section 2 to provide an explicit bound on reg-stab( I ) for anymonomial ideal I . n order to do that we need to recall some notation and known results. Given a(column) vector α ∈ N r we write X α := X α · · · X α r r . Definition 3.2. Let I be a monomial ideal of R . We define(1) For a subset A ⊆ R , the exponent set of A is E ( A ) := { α | X α ∈ A } ⊆ N r .(2) The Newton polyhedron of I is N P ( I ) := conv { E ( I ) } , the convex hull of theexponent set of I in the space R r .The integral closure of a monomial ideal I is a monomial ideal as well. Using thenotion of Newton polyhedron, we can geometrically describe I (see [26]):(3.3) E ( I ) = N P ( I ) ∩ N r = { α ∈ N r | X n α ∈ I n for some n (cid:62) } . (3.4) N P ( I n ) = nN P ( I ) = n conv { E ( I ) } + R r + for all n (cid:62) . Fig. 1: Newton polyhedron I = ( X X , X X , X X )The hole means the pointdoes not belong to E ( I ) , or equivalently, X X ∈ ¯ I \ I. The above equalities say that (exponents of) all monomials of I form the set of integerpoints in N P ( I ) (while we do not know which points among them do not belong to I ), and the Newton polyhedron N P ( I n ) of I n is just a multiple of N P ( I ). This is oneof the reasons why it is easier to work with integral closures of powers of monomialideals than with their ordinary powers. Further, one can use linear algebra to describeNewton polyhedron. Let G ( I ) denote the minimal generating system of I consistingof monomials and d ( I ) := max {| α | := α + · · · + α r | X α ∈ G ( I ) } , the maximal generating degree of I . Let e , ..., e r be the canonical basis of R r . In orderto avoid trivial cases, we can assume that r ≥ d ( I ) ≥ N P ( I ). It is a variation of [21,Lemma 2.2]: Lemma 3.3. The Newton polyhedron N P ( I ) is the set of solutions of a system ofinequalities of the form { x ∈ R r | a i x ≥ b i , i = 1 , . . . , s } , where a i = ( a i , . . . , a ir ) are row vectors, such that each hyperplane with the equation a i x = b i defines a facet of N P ( I ) , which contains t i affinely independent points of E ( G ( I )) and is parallel to r − t i vectors of the canonical basis. Furthermore, we can hoose (cid:54) = a i ∈ N r , b i positive integers for all i = 1 , ..., s , such that a ij , b i ≤ d ( I ) t i forall i, j , where t i is the number of non-zero coordinates of a i .Proof. The first part of the lemma is [33, Lemma 6]. For the second part, we need theway to define a i and b i given in the proof of [21, Lemma 2.2]. In order to make thepresentation self-contained, let us recall it here.Let H be a hyperplane which defines the i -th facet of N P ( I ). W.l.o.g., we mayassume that H is defined by t := t i affinely independent points α , ..., α t ∈ E ( G ( I ))and is parallel to r − t vectors e t +1 , . . . , e r . Then the defining equation of H can bewritten as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · · · x t α · · · α t α t · · · α tt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Expanding this determinant in the first row, we get: a (cid:48) x + · · · + a (cid:48) t x t = b (cid:48) , where a (cid:48) i arethe (1 , i )-cofactor for i = 1 , . . . , t and b (cid:48) is the (1 , s + 1)-cofactor of this determinant.Then one can take a ij = | a (cid:48) j | for j ≤ t , a ij = 0 for t + 1 ≤ j ≤ r and b i = | b (cid:48) | . Let ( c ij )be a square submatrix of rank t (cid:48) ≤ t in the above sum. By Hadamard’s inequality, wehave (det( c ij )) (cid:54) Π t (cid:48) i =1 (Σ t (cid:48) j =1 | c ij | ) (cid:54) Π t (cid:48) i =1 (Σ t (cid:48) j =1 | c ij | ) (cid:54) d ( I ) t (cid:48) . From that we immediately get the bounds on a ij and b i . (cid:3) Further, we need a formula for computing local cohomology modules H i m ( R/I ) givenby Takayama in [32]. Note that H i m ( R/I ) admits an Z r -grading over R . For everydegree α ∈ Z r we denote by H i m ( R/I ) α the α -component of H i m ( R/I ).Recall that a simplicial complex ∆ on a finite vertex set V is a collection of subsetsof V such that F ⊂ G and G ∈ ∆ implies F ∈ ∆ and { v } ∈ ∆ for all v ∈ V . Inthis case we also write V = V (∆). An element of ∆ is called a face, and a maximalelement of ∆ is called a facet. The set of facets of ∆ is denoted by F (∆). Clearly, ∆is uniquely defined by F (∆). In this case we also write ∆ = (cid:104)F (∆) (cid:105) .Let ∆( I ) denote the simplicial complex corresponding to the radical ideal √ I , i.e.,∆( I ) := {{ i , ..., i j } ⊆ [ r ] | X i · · · X i j (cid:54)∈ √ I } , where [ r ] := { , , ..., r } . For every α = ( α , . . . , α r ) T ∈ Z r , we define the negative partof its support to be the set supp − ( α ) := { i | α i < } . For a subset F of [ r ], let R F := R [ X − i | i ∈ F ]. Set(3.5) ∆ α ( I ) := { F ⊆ [ r ] \ supp − ( α ) | X α / ∈ IR F ∪ supp − ( α ) } . For an example, consider the case α = . Then supp − ( ) = ∅ , and∆ ( I ) = { F := { i , ..., i j } ⊆ [ r ] | / ∈ IR F } = {{ i , ..., i j } ⊆ [ r ] | X i · · · X i j (cid:54)∈ √ I } = ∆( I ) . This was mentioned in [25, Example 1.4(1)].We set (cid:101) H i ( ∅ ; K ) = 0 for all i , (cid:101) H i ( {∅} ; K ) = 0 for all i (cid:54) = − 1, and (cid:101) H − ( {∅} ; K ) = K .Using [15, Lemma 1.1] we may write Takayama’s formula as follows. emma 3.4. ([32, Theorem 2.2]) We have dim K H i m ( R/I ) α = (cid:26) dim K (cid:101) H i −| supp − ( α ) |− (∆ α ( I ); K ) if supp − ( α ) ∈ ∆( I ) , otherwise . Let Γ ⊆ ∆( I ) be a simplicial subcomplex with the vertex set V (Γ) ⊆ [ r ]. We set(3.6) a Γ ,i ( I ) = (cid:40) sup {| α | | α ∈ Z r and ∆ α ( I ) = Γ } if (cid:101) H i + | V (Γ) |− r − (Γ; K ) (cid:54) = 0 } , −∞ otherwise . Then we get the following immediate consequence of Lemma 3.4 and of the Artinianproperty of local cohomology modules: Corollary 3.5. a i ( R/I ) = max { a Γ ,i ( I ) | Γ ⊆ ∆( I ) is a simplicial subcomplex } . More-over, if (cid:101) H i + | V (Γ) |− r − (Γ; K ) (cid:54) = 0 } , then a i ( R/I ) ∈ Z . Recall that the support of a vector a ∈ R r issupp( a ) = { i | a i (cid:54) = 0 } . Using (3.4), Lemma 3.3 and the definition (3.5), one can describe ∆ α ( I n ) via a systemof linear constraints. The case α ∈ N r is given in [21, Lemma 3.1]. In the general case, α ∈ Z r , we already have Lemma 3.6. ([20, Lemma 1.2]) Assume that supp − ( α ) ∈ ∆( I ) for some α ∈ Z r . Then ∆ α ( I n ) = { F \ supp − ( α ) | supp − ( α ) ⊆ F, F ∈ ∆( I ) , and X α (cid:54)∈ I n R F } . Now we can give a generalization of [21, Lemma 3.1]. Lemma 3.7. Keep the notations in Lemma 3.3. Let α ∈ Z r , n ≥ . Assume that supp − ( α ) ∈ ∆( I ) . Then we have ∆ α ( I n ) = (cid:104) [ r ] \ (supp( a i ) ∪ supp − ( α )) | i ≤ s ; supp − ( α ) ⊆ [ r ] \ supp( a i ) and (cid:80) j (cid:54)∈ supp − ( α ) a ij α j < nb i (cid:105) . Proof. The following proof is an adaptation of that of [21, Lemma 3.1] to our case. Inorder to make the presentation self-contained, let us give the detail.Let F ∈ ∆ α ( I n ). By Lemma 3.6, F = F (cid:48) \ supp − ( α ) for some F (cid:48) ∈ ∆( I ) suchthat supp − ( α ) ⊆ F (cid:48) and X α (cid:54)∈ I n R F (cid:48) . We may assume that F (cid:48) = { p, . . . , r } for some1 ≤ p ≤ r +1 (where p = r +1 means that F (cid:48) = ∅ ). Note that X α / ∈ I n R F (cid:48) if and only if X α X γ / ∈ I n for any monomial X γ ∈ K [ X p , . . . , X r ]. Fix a monomial X γ = X mp · · · X mr with m (cid:29) 0. By Lemma 3.3, (3.3) and (3.4), there is 1 ≤ i ≤ s such that a i α + m ( a ip + · · · + a ir ) = a i · [ α + m ( e p + · · · + e r )] < nb i . Recall, by Lemma 3.3, that a ij ≥ j . Hence we must have a ip = · · · = a ir = 0,whence F (cid:48) ⊆ [ r ] \ supp( a i ), and a i α = a i [ α + m ( e p + · · · + e r )] < nb i . Since supp − ( α ) ⊆ F (cid:48) ⊆ [ r ] \ supp( a i ) = { j | a ij = 0 } , we have (cid:88) j (cid:54)∈ supp − ( α ) a ij α j = (cid:88) ≤ j ≤ r a ij α j = a i α < nb i . rom the equality F = F (cid:48) \ supp − ( α ) and the inclusion F (cid:48) ⊆ [ r ] \ supp( a i ) we also get F ⊆ [ r ] \ (supp( a i ) ∪ supp − ( α )).Conversely, let F ⊆ [ r ] \ (supp( a i ) ∪ supp − ( α )) for some 1 ≤ i ≤ s such thatsupp − ( α ) ⊆ [ r ] \ supp( a i ) and (cid:80) j (cid:54)∈ supp − ( α ) a ij α j < nb i . Note that F (cid:48) := F ∪ supp − ( α ) ⊆ [ r ] \ supp( a i ). W.l.o.g., we may assume that F (cid:48) = { p, ..., r } , ≤ p ≤ r + 1. Then a ij = 0 for all p ≤ j ≤ r , and for any X γ p · · · X γ r ∈ K [ X p , ..., X r ], we have a i ( α + (0 , ..., , γ p , ..., γ r ) T ) = (cid:88) j (cid:54)∈ supp − ( α ) a ij α j < nb i . Using (3.3) and (3.4) and Lemma 3.3, we get that X α X γ p · · · X γ r (cid:54)∈ I n , or equivalently, X α (cid:54)∈ I n R F (cid:48) . By Lemma 3.6, F ∈ ∆ α ( I n ).Now using the maximality of facets, we immediately get the statement of the lemma. (cid:3) Let Γ be a simplicial subcomplex of ∆( I ) such that Γ = ∆ α ( I n ) for some α ∈ Z r and n ≥ 1. W.l.o.g., we may assume that V (Γ) = [ r (cid:48) ], where r (cid:48) ≤ r , so that supp − ( α ) = { r (cid:48) + 1 , ..., r } . Further, assume that supp − ( α ) ⊆ [ r ] \ supp( a i ) for i = 1 , ..., ˜ s andsupp − ( α ) (cid:54)⊆ [ r ] \ supp( a i ) for i > ˜ s , where ˜ s ≤ s . Then, by Lemma 3.7, we may furtherassume that(3.7) Γ = (cid:10) [ r ] \ (supp( a i ) ∪ supp − ( α )) | i = 1 , ..., s (cid:48) (cid:11) , where s (cid:48) ≤ ˜ s . For a row vector a = ( a , ..., a r ) ∈ R r , denote a (cid:48) := ( a , ..., a r (cid:48) ) ∈ R r (cid:48) .Similarly for column vectors. Consider the following polyhedron Q Γ ,n := { x (cid:48) ∈ R r (cid:48) | a (cid:48) i x (cid:48) ≤ nb i − i = 1 , ..., s (cid:48) ) , (3.8) a (cid:48) l x (cid:48) ≥ nb l ( l = s (cid:48) + 1 , ..., ˜ s ) , and x j ≥ j = 1 , ..., r (cid:48) ) } . Together with Corollary 3.5, the following result translates the problem of computingthe a i -invariant a i ( R/I n ) into solving a finite set of integer programming problems. Theorem 3.8. Assume that (cid:101) H i + r (cid:48) − r − (Γ; K ) (cid:54) = 0 . Then a Γ ,i ( I n ) = sup { x + · · · + x r (cid:48) | x (cid:48) ∈ Q Γ ,n ∩ N r (cid:48) } + r (cid:48) − r. Proof. Let L := { α ∈ Z r | ∆ α ( I n ) = Γ } . By Lemma 3.7 and from (3.7) we must have a (cid:48) l α (cid:48) ≥ nb l for any l = s (cid:48) + 1 , ..., ˜ s and a (cid:48) i α (cid:48) < nb i for any i = 1 , ..., s (cid:48) . Since a i , α are integer vectors and b i is an integer, thelast inequality is equivalent to a (cid:48) i α (cid:48) ≤ nb i − 1. This means α (cid:48) ∈ Q Γ ,n , provided that α ∈ L . Note that the condition ∆ α ( I n ) = Γ implies supp − ( α ) = { r (cid:48) + 1 , ..., r } , so that α j ≤ − j = r (cid:48) + 1 , ..., r . Hencesup { α + · · · + α r | α ∈ L } ≤ sup { α + · · · + α r (cid:48) | α (cid:48) ∈ Q Γ ,n ∩ N r (cid:48) } − ( r − r (cid:48) ) . On the other hand, if ( β , ..., β r (cid:48) ) T ∈ Q Γ ,n ∩ N r (cid:48) , then ( β , ..., β r (cid:48) , − , ..., − T ∈ L (thenumber of − r − r (cid:48) ). So, the above inequality is in fact an equality. Now thelemma follows from (3.6). (cid:3) n this paper, we do not intend to compute the a i -invariant. We only want to studythe asymptotic behavior of a i ( R/I n ). Since there are only finitely many simplicialsubcomplexes of ∆( I ), we can restrict our problem to a fixed Γ, which satisfies the as-sumption of Theorem 3.8. Moreover, we do not need to compute any reduced homologygroup of Γ.From now on, fix Γ as above. Consider the following polyhedron: P Γ ,n := { x (cid:48) ∈ R r (cid:48) | a (cid:48) i x (cid:48) ≤ nb i ( i = 1 , ..., s (cid:48) ) , (3.9) a (cid:48) l x (cid:48) ≥ nb l ( l = s (cid:48) + 1 , ..., ˜ s ) , and x j ≥ j = 1 , ..., r (cid:48) ) } . From (3.8) it implies that Q Γ ,n ⊆ P Γ ,n . In order to apply Theorem 2.12 to the integerprogram in Theorem 3.8, we need the following result. This was shown in the proof of[21, Lemma 3.2]. For the convenience of the reader, we give a proof here. Lemma 3.9. Assume that (cid:101) H i + r (cid:48) − r − (Γ; K ) (cid:54) = 0 and Q Γ ,n ∩ N r (cid:48) (cid:54) = ∅ for some n ≥ .Then P Γ , is a bounded and full-dimensional polyhedron in R r (cid:48) .Proof. Since (cid:101) H i + r (cid:48) − r − (Γ; K ) (cid:54) = 0, there is α ∈ Q Γ ,n ∩ N r (cid:48) . Then α /n ∈ P Γ , , whence P Γ , (cid:54) = ∅ . Recall by Lemma 3.3 that a ij ≥ 0. If a ij = 0 for some j ≤ r (cid:48) and all i = 1 , ..., s (cid:48) , then α + m e j ∈ Q Γ ,n ∩ N r (cid:48) for all m ≥ 0, where by abuse of notation, e , ..., e r (cid:48) also denote the canonical basis of R r (cid:48) . Hence a Γ ,i ( R/I n ) = ∞ , a contradiction(see Corollary 3.5) . From this it follows that s (cid:48) ≥ 1, and (cid:80) ≤ i ≤ s (cid:48) a ij > j ≤ r (cid:48) .For all x (cid:48) ∈ P Γ , we have (cid:80) ≤ j ≤ r (cid:48) ( (cid:80) ≤ i ≤ s (cid:48) a ij ) x j ≤ (cid:80) ≤ i ≤ s (cid:48) b i , which yields that P Γ , is bounded.Further, let a ∗ = max i,j | a (cid:48) ij | . Then for all 0 ≤ ε , ..., ε r (cid:48) ≤ / (2 r (cid:48) a ∗ ) and any i ≤ s (cid:48) ,we have a (cid:48) i · ( α + ε e + · · · + ε r (cid:48) e r (cid:48) ) ≤ nb i − / < nb i . By (3.9) it implies that α + ε e + · · · + ε r (cid:48) e r (cid:48) ∈ P Γ ,n = nP Γ , . Hence α /n + ( ε /n ) e + · · · +( ε r (cid:48) /n ) e r (cid:48) ∈ P Γ , , which yields that P Γ , is a full-dimensional polyhedron in R r (cid:48) . (cid:3) In order to apply Theorem 2.12 to the integer program in Theorem 3.8, in the sequellet us consider the following set up (cid:101) A = a a · · · a r (cid:48) · · · · · · a s (cid:48) a s (cid:48) · · · a s (cid:48) r (cid:48) − a ( s (cid:48) +1)1 − a ( s (cid:48) +1)2 · · · − a ( s (cid:48) +1) r (cid:48) · · · · · ·− a ˜ s − a ˜ s · · · − a ˜ sr (cid:48) , (3.10) (cid:101) b = b ... b s (cid:48) − b s (cid:48) +1 ... − b ˜ s , (cid:101) c = − − , (cid:101) d = , here the number of − (cid:101) c ∈ R r (cid:48) is s (cid:48) , a ij , b i are defined in Lemma 3.3, and (cid:101) d is an r (cid:48) -vector. (In the rows starting from ( s (cid:48) + 1)-st in (cid:101) A and (cid:101) b above we have to put theminus sign in order to keep the type of constraints considered in Sections 1 and 2.)From Theorem 3.8, Corollary 3.5 and Theorem 2.12 we immediately get that each a i ( R/I n ) is a quasi-linear function for n ≥ 0. This fact was proved in [19, Theorem4.1] by a different method. However, our main aim is to provide a bound, from where a i ( R/I n ) becomes a quasi-linear function. In order to do that, we first determine anumber N ∗ such that a Γ ,i ( R/I n ) is a quasi-linear function for all n ≥ N ∗ . Moreover,some additional information on the coefficients of linear functions of a Γ ,i ( R/I n ) are alsogiven. Lemma 3.10. In the setting of (3.10), let D, D (cid:48) and ∆ be the maximum absolutevalue of the subdeterminants of the matrices (cid:101) A , [ (cid:101) A (cid:101) b ] and [ (cid:101) A (cid:101) b (cid:101) c ] , respectively. Set N ∗ := ( r + 1) D (cid:48) ( D − 1) + ( r + 2)∆ − D. Assume that a Γ ,i ( I n ) is finite for some n ≥ . Set ψ := max { x + · · · + x r (cid:48) | x (cid:48) ∈ P Γ , } . Then there are linear functions f , ..., f t − of the form f j ( n ) = ψn + β j , where t ≥ ,such that a Γ ,i ( R/I n ) = f j ( n ) whenever n ≡ j mod t for all n ≥ N ∗ .Moreover, ψ is a non-negative rational number with positive denominator at most D , and − r D ≤ β j ≤ ≤ j ≤ t − . Proof. Since a Γ ,i ( I n ) is finite, by Theorem 3.8, (cid:101) H i + r (cid:48) − r − (Γ; K ) (cid:54) = 0 and Q Γ ,n ∩ N r (cid:48) (cid:54) = ∅ .By Lemma 3.9, P Γ , is a bounded and full-dimensional polyhedron in R r (cid:48) . Hence ψ is finite. Applying Theorem 2.12 to the setting (3.10), we then get the existence of f j ( n ) = ψn + β j , 0 ≤ j ≤ t − 1, such that a Γ ,i ( R/I n ) = f j ( n ) whenever n ≡ j mod t for all n ≥ N ∗ .Let ψ n := max { x + · · · + x r (cid:48) | x (cid:48) ∈ P Γ ,n } , Ψ n := max { x + · · · + x r (cid:48) | x (cid:48) ∈ Q Γ ,n } ,M n := max { x + · · · + x r (cid:48) | x (cid:48) ∈ Q Γ ,n ∩ N r (cid:48) } . Note that ψ = ψ reaches its value at a vertex of P Γ , . From that all properties of ψ follow.It is left to give bounds on β j . By Theorem 3.8, a Γ ,i ( R/I n ) = M n + r (cid:48) − r . Note that ψ n = ψn . Since Q Γ ,n ⊆ P Γ ,n , we have M n ≤ Ψ n ≤ ψ n = ψn . Hence β j ≤ r (cid:48) − r ≤ j ≤ t − β j is the following: Since Ψ n is attained ata vertex of Q Γ ,n , we can compute an optimal solution for it. Well-known results inInteger Programming say that this solution cannot be too far from an optimal integersolution for M n . ndeed, by [28, Theorem 17.2], there is an optimal solution v for Ψ n and an optimalsolution u for M n such that | v j − u j | ≤ rD for all j ≤ r . Note that (cid:101) d T v = Ψ n . Hence M n = (cid:101) d T u = (cid:101) d T v + (cid:101) d T ( u − v ) = Ψ n + (cid:101) d T ( u − v ) ≥ Ψ n − r D. On the other hand, Ψ n reaches its value at a vertex of Q Γ ,n . This vertex has the form v (cid:48) = n z I , (cid:101) b + z I , (cid:101) c for some I ∈ I (see (2.1) and (2.2)). HenceΨ n = n [( z I , (cid:101) b ) + · · · + ( z I , (cid:101) b ) r (cid:48) ] + β (cid:48) , where β (cid:48) = ( z I , (cid:101) c ) + · · · + ( z I , (cid:101) c ) r (cid:48) . By Proposition 1.3, we must have Ψ n = ψn + β (cid:48) . ByCramer’s rule, | ( z I , (cid:101) c ) j | ≤ | D j | , where D j is the determinant of a matrix obtained from (cid:101) A ( i , ..., i k ; j , ..., j k ) by replacing the j -th column by (˜ c i , ..., ˜ c i k ) T . Expanding D j bythis column, we see that | ( z I , (cid:101) c ) j | ≤ kD ≤ r (cid:48) D . Hence β (cid:48) ≥ − ( r (cid:48) ) D , which yields a Γ ,i ( R/I n ) = M n + r (cid:48) − r ≥ ψn − ( r (cid:48) ) D − r D + r (cid:48) − r ≥ ψn − r D. This implies β j ≥ − r D , as required. (cid:3) We are now ready to state the main result on a i -invariants. Theorem 3.11. In the setting of (3.10), let D, D (cid:48) and ∆ be the maximum absolutevalue of the subdeterminants of the matrices (cid:101) A , [ (cid:101) A (cid:101) b ] and [ (cid:101) A (cid:101) b (cid:101) c ] , respectively. Then(i) The invariant a i ( R/I n ) is a quasi-linear function of the same slope for all n ≥ max { ( r + 1) D (cid:48) ( D − 1) + ( r + 2)∆ − D, r D } . (ii) In particular, a i ( R/I n ) is a quasi-linear function of the same slope for all n ≥ N † := 2 r r/ d ( I ) r .Proof. (i) The intuition of the proof is quite simple. By Lemma 3.10, each finite a Γ ,i ( R/I n ) is a quasi-linear function of n for all n (cid:29) 0. By choosing the product ofperiods of all such a Γ ,i ( R/I n ) as a new period τ of all a Γ ,i ( R/I n ), and hence also aperiod of a i ( R/I n ), by Corollary 3.5, we see that in each equivalence class modulo τ , a i ( R/I n ) is the maximum of a finitely many linear functions. As already mentionedat the end of the proof of Proposition 1.3, after the largest coordinate of intersectionpoints of corresponding lines, there is one line lying above all other ones. This is thegraph of a i ( R/I n ).For the detail, set N ∗ := ( r + 1) D (cid:48) ( D − 1) + ( r + 2)∆ − D. Let n ≥ max { N ∗ , r D } be any integer. By Corollary 3.5,(3.11) a i ( R/I n ) = max { a Γ ,i ( R/I n ) | Γ ⊆ ∆( I ) is a simplicial subcomplex } . Assume that Γ and Γ are two simplicial subcomplexes in the right hand side of (3.11)such that both a Γ ,i ( R/I n ) and a Γ ,i ( R/I n ) are finite. By Lemma 3.10, a Γ ,i ( R/I n ) = α n + β ( n ) ,a Γ ,i ( R/I n ) = α n + β ( n ) , where β ( n ) and β ( n ) are periodic functions (of the same period τ ). If α ≥ α and β ( n ) ≥ β ( n ), then we may delete a Γ ,i ( R/I n ) in the right hand side of (3.11). o, we may now assume that α > α and β ( n ) > β ( n ). Note that(3.12) α n + β ( n ) ≥ α n + β ( n ) if n ≥ β ( n ) − β ( n ) α − α . By Lemma 3.10 one can write α i = α i α i , ( i = 1 , α i is an integer, and α i ≤ D is a positive integer. Then α − α = α α − α α α α ≥ D . On the other hand, again by Lemma 3.10, β ( n ) − β ( n ) ≤ r D . Hence β ( n ) − β ( n ) α − α ≤ r D . Since n ≥ max { N ∗ , r D } , from (3.12) we get a Γ ,i ( R/I n ) ≥ a Γ ,i ( R/I n ), so that wecan also delete a Γ ,i ( R/I n ) in the right hand side of (3.11). In both cases, the biggerslope is the slope of the maximum of two quasi-linear functions. Using this fact, wecan conclude that for each fixed j ≤ τ we can find a subcomplex ∆ j of ∆( I ) such that a i ( R/I n ) = a ∆ j ,i ( R/I n ) for all n ≥ max { N ∗ , r D } such that n ≡ j mod τ .Now (i) follows from Lemma 3.10.(ii) Denote by b ∗ := max {| b | , ..., | b s |} and a ∗ := max {| a ij || i ≤ s, j ≤ r } . Note that the matrix [ (cid:101) A (cid:101) b (cid:101) c ] has a rank at most r (cid:48) + 2 ≤ r + 2. Using the Laplaceexpansion along the last column of a determinant of [ (cid:101) A (cid:101) b (cid:101) c ] we can conclude that∆ ≤ ( r + 2) D (cid:48) . Similarly, D (cid:48) ≤ ( r + 1) b ∗ D . Then N ∗ ≤ ( r + 1) b ∗ D + ( r + 2) ( r + 1) b ∗ D. Applying Hadamard’s inequality to a subdeterminant of (cid:101) A and using Lemma 3.3, thereis a ( t × t )-submatrix of (cid:101) A such that D ≤ (˜ a i j + · · · + ˜ a i j t ) · · · (˜ a i t j + · · · + ˜ a i t j t ) ≤ [ r ( a ∗ ) ] r ≤ r r d ( I ) r , whence D ≤ r r/ d ( I ) r . Since b ∗ ≤ d ( I ) r by Lemma 3.3, it implies N ∗ < ( r + 1) d ( I ) r r r d ( I ) r + ( r + 2) ( r + 1) r r/ d ( I ) r d ( I ) r = d ( I ) r r r d ( I ) r ( r + 1)[( r + 1) + ( r +2) r r/ d ( I ) r ] < d ( I ) r r r d ( I ) r ( r + 1)[ r + 1 + 1] (since r, d ( I ) ≥ , whence(3.13) N ∗ < ( r + 1)( r + 2) r r d ( I ) r < r r/ d ( I ) r . We also have 2 r D ≤ r r/ d ( I ) r . Therefore (ii) now follows from (i). (cid:3) From the definition (3.2) and Theorem 3.11 we can only conclude that reg( I n ) isa quasi-linear function for n (cid:29) 0. However, more than twenty years ago, using to-tally different methods, Cutkosky-Herzog-Trung [9] and independently Kodiyalam [24]proved that reg( R/I n ) is a linear function of n for n (cid:29) 0. Moreover, if I is a monomialideal, the behavior of reg( I n ) is much more understood, as the following result shows. emma 3.12. ([19, Theorem 4.10]) Let I be a non-zero monomial ideal of R . Thenthere are a positive integer p and a nonnegative integer ≤ e ≤ dim( R/I ) such that reg( I n ) = pn + e for all n (cid:29) . Moreover pn ≤ reg( I n ) ≤ pn + dim( R/I ) for all n > . In the following result we can give an upper bound for the stability index of reg( I n ). Theorem 3.13. Let I be a non-zero monomial ideal of R = K [ X , ..., X r ] of maximalgenerating degree d ( I ) . Then there are a positive integer p ≤ d ( I ) and a non-negativeinteger ≤ e ≤ dim R/I such that reg( I n ) = pn + e for all n ≥ ( r + 1)( r + 2) r r d ( I ) r , and pn ≤ reg( I n ) ≤ pn + dim( R/I ) for all n ≥ . In particular reg-stab( I ) ≤ ( r + 1)( r + 2) r r d ( I ) r . Proof. The existence of p and 0 ≤ e ≤ dim( R/I ), such that reg( I n ) = pn + e for all n (cid:29) pn ≤ reg( I n ) ≤ pn + dim( R/I ) for all n ≥ I ). The main idea is to find in eachequivalence class of n modulo a period t an index j and a simplicial complex Γ suchthat a Γ ,j ( R/I n ) + j = reg( R/I n ) = pn + e − , for all n (cid:29) N be a number such thatreg( I n ) = pn + e, for all n ≥ N. Keep the notation in Theorem 3.11. Let N ∗ := ( r + 1) D (cid:48) ( D − 1) + ( r + 2)∆ − D. By enlarging periods we can assume that all eventually quasi-linear functions a Γ ,i ( R/I n )have the same period t for some t ≥ 1. Since reg( I n ) = 1 + max { a i ( R/I n ) + i | ≤ i ≤ dim R/I } , from Corollary 3.5 and Lemma 3.10 it implies that if a Γ ,i ( R/I n ) is finite,then for all n ≥ N ∗ , a Γ ,i ( R/I n ) = α Γ ,i n + β Γ ,i ; k if n ≡ k mod t, (3.14) such that α Γ ,i ≤ p, and if α Γ ,i n = p then β Γ ,i ; k + i + 1 ≤ e. Fix an integer m ≥ N ∗ . Let n = m + tN . Again by Corollary 3.5, there are 0 ≤ j ≤ dim R/I and a simplicial subcomplex Γ such thatreg( R/I n ) = a Γ ,j ( R/I n ) + j. By the choice of t and N , it implies a Γ ,j ( R/I n ) + j = pn + e. On the other hand, by Lemma 3.10, there exist a non-negative rational number α ≤ p with positive denominator at most D and β ≤ 0, such that a Γ ,j ( R/I n ) = αn + β, for all n ≥ N ∗ and n ≡ m (mod t ). Hence pn + e = αn + β + j + 1 . f α < p , then p − α ≥ /D , whence r + 1 ≥ β + j + 1 − e = ( p − α ) n ≥ ( N ∗ + tN ) /D > N ∗ /D ≥ r + 1 , a contradiction. So α = p and β + j + 1 = e . This means a Γ ,j ( R/I n ) = pn + e − j − , for all n ≥ N ∗ and n ≡ m (mod t ). In particular a Γ ,j ( R/I m ) + j + 1 = pm + e. Combining with (3.14) we can conclude that reg( I m ) = pm + e . Since m can be anynumber bigger or equal N ∗ , we get that reg( I n ) = pn + e for all n ≥ N ∗ . Using theestimation of N ∗ in (3.13), we get the desired upper bound for reg-stab( I ). (cid:3) Remark 3.14. Fix r ≥ 4. The monomial ideal I ⊂ R given in [33, Proposition 17]has depth( R/I n ) > n ≤ n = O ( d ( I ) r − ) and depth( R/I n ) = 0 for n (cid:29) 0. Thismeans a ( R/I n ) are finite for all n ≤ n and a ( R/I n ) = −∞ for n (cid:29) 0, i.e., a ( R/I n )becomes linear only if n is at least O ( d ( I ) r − ). Unfortunately, this example does notshow that the bound on reg-stab( I ) should also be at least O ( d ( I ) r − ).Finally, we consider the Castelnuovo-Mumford regularity of the so-called symbolicpowers of a square-free monomial ideal. Recall that a monomial ideal is said to besquare-free if its generators are products of different variables. In this case one canwrite I = ∩ si =1 p i , where p i = ( X t i , ..., X t imi ) is the prime ideal generated by a subset of variables (i.e., { t i , ..., t im i } ⊆ [ r ]). Then the symbolic n -th power of I is defined by I ( n ) = ∩ si =1 p ni . Note that ∆( I ( n ) ) = ∆( I ) = (cid:104) F , ..., F s (cid:105) , where F i := [ r ] \ { t i , ..., t im i } . For an arbitrary homogeneous ideal J , the definition of the symbolic power J ( n ) ismore complicated, and very little is known about reg( J ( n ) ), see, e.g., [18, Section 2].However, for the case of square-free monomial ideals, as a consequence of [19, Theorem4.1 and Theorem 4.7], we get Corollary 3.15. If I is a square-free monomial ideal of R , then for all i ≤ dim R/I , a i ( R/I ( n ) ) is a quasi-linear function of the same slope for all n (cid:29) . In the case of symbolic powers, the following result plays the role of Lemma 3.7. Lemma 3.16. ([20, Lemma 1.3] and also [25, Lemma 2.1]) Let I be a square-freemonomial ideal of R . Assume that supp − ( α ) ∈ ∆( I ) for some α ∈ Z r . Then for all n ≥ , we have ∆ α ( I ( n ) ) = (cid:42) F \ supp − ( α ) | F ∈ F (∆( I )) , supp − ( α ) ⊆ F and (cid:88) j / ∈ F α j ≤ n − (cid:43) . sing this lemma, Corollary 3.5 and (3.6), one can again translate the problem ofdefining a place from where a i ( R/I ( n ) ) becomes a quasi-linear function to the study ofthe asymptotic behavior of a family of integer programs. From that we get the followingresult on the index of stability of the Castelnuovo-Mumford regularity reg( I ( n ) ). Theorem 3.17. Let I be a square-free monomial ideal of R . For any i ≤ dim R/I , a i ( R/I ( n ) ) is a quasi-linear function of the same slope for all n ≥ r r/ . In particu-lar, reg( I ( n ) ) is a quasi-linear function of the same slope for all n ≥ r r/ .Proof. The proof of this theorem is similar to that of Theorem 3.11. Fix a simplicialsubcomplex Γ of ∆( I ) such that Γ = ∆ α ( I ( n ) ) for some n ≥ α ∈ Z r with supp − ( α ) ∈ ∆( I ). W.l.o.g., we may assume that supp − ( α ) = { r (cid:48) + 1 , ..., r } , i.e., V (Γ) = [ r (cid:48) ], where r (cid:48) ≤ r , and that supp − ( α ) ⊆ F i for all i ≤ ˜ s and supp − ( α ) (cid:54)⊆ F i forall i > ˜ s , where ˜ s ≤ s . By Lemma 3.16 we may further assume thatΓ = (cid:104) F ∩ [ r (cid:48) ] , ..., F s (cid:48) ∩ [ r (cid:48) ] (cid:105) , where s (cid:48) ≤ ˜ s . Assume that[ r ] \ F i = { t i , ..., t im i } ∩ [ r (cid:48) ] = { t i , ..., t im (cid:48) i } . Similar to Q Γ ,n (see (3.8)) and P Γ ,n (see (3.9)), we consider the following polyhedron Q (cid:48) Γ ,n defined by(3.15) Q (cid:48) Γ ,n x t + x t + · · · + x t m (cid:48) ≤ n − , · · · x t s (cid:48) + x t s (cid:48) + · · · + x t s (cid:48) m (cid:48) s (cid:48) ≤ n − ,x t ( s (cid:48) +1)1 + x t ( s (cid:48) +1)2 + · · · + x t ( s (cid:48) +1) m ( s (cid:48) +1) ≥ n, · · · x t ˜ s + x t ˜ s + · · · + x t ˜ sm (cid:48) ˜ s ≥ n,x j ≥ j = 1 , ..., r (cid:48) ) , and the polyhedron P (cid:48) Γ ,n defined by P (cid:48) Γ ,n x t + x t + · · · + x t m (cid:48) ≤ n, · · · x t s (cid:48) + x t s (cid:48) + · · · + x t s (cid:48) m (cid:48) s (cid:48) ≤ n,x t ( s (cid:48) +1)1 + x t ( s (cid:48) +1)2 + · · · + x t ( s (cid:48) +1) m ( s (cid:48) +1) ≥ n, · · · x t ˜ s + x t ˜ s + · · · + x t ˜ sm (cid:48) ˜ s ≥ n,x j ≥ j = 1 , ..., r (cid:48) ) . Then, using Lemma 3.16, one can see that all similar results to those of Theorem3.8 to Theorem 3.11(i) hold for I ( n ) , where one should replace I n , P Γ ,n , Q Γ ,n by I ( n ) , P (cid:48) Γ ,n , Q (cid:48) Γ ,n , respectively, and the matrix (cid:101) A and the vectors (cid:101) b , (cid:101) c are now defined y (3.15). In this new setting, all entries of (cid:101) A, (cid:101) b , (cid:101) c are either ± D ≤ r r , whence D ≤ r r/ and2 r D ≤ r r/ . Using the Laplace expansion along the last column of a determinant of [ (cid:101) A, (cid:101) b , (cid:101) c ] and[ (cid:101) A, (cid:101) b ] we get ∆ ≤ ( r + 2) D (cid:48) and D (cid:48) ≤ ( r + 1) D . Set N ∗ = ( r + 1) D (cid:48) ( D − 1) + ( r + 2)∆ − D. Then N ∗ < ( r + 1) D ( D − 1) + ( r + 2) ( r + 1) D ≤ ( r + 1) r r/ ( r r/ − 1) + ( r + 2) ( r + 1) r r/ . If r = 2 , r r/ .Assume that r ≥ 4. Then N ∗ < ( r + 1) r r + ( r + 2) ( r + 1) r r/ = ( r + 1) r r [ r + 1 + ( r +2) r r/ ] ≤ ( r + 1) r r [ r + 1 + ( r +2) r < ( r + 1) r r ( r + 4) < r r/ . By a similar statement to Theorem 3.11(i) we get that a i ( R/I ( n ) ) is a quasi-linearfunction of the same slope for all n ≥ r r/ . (cid:3) In a recent preprint [10], L. X. Dung et al. were able to construct an example ofsquare-free monomial ideal I for which reg( I ( n ) ) is not an eventually linear function of n . We would like to conclude this section with Question 3.18. Assume that I is a square-free monomial ideal. Is there a number n = O ( r k ), where k does not depend on r , such that a i ( R/I ( n ) ) and reg( I ( n ) ) becomequasi-linear functions for all n ≥ n ?The above question has an affirmative answer for a special class of square-free mono-mial ideals. Let I = ∩ si =1 ( X t i , ..., X t imi ) , and a ij = (cid:40) j = t il ( l = 1 , ..., m i ) , . If the s × r -matrix A = ( a ij ) is totally unimodular, i.e., all subdeterminants of M areeither ± a i ( R/I ( n ) ) and reg( I ( n ) ) are linearfunctions of n for all n ≥ r . In this situation, all polyhedra P (cid:48) Γ ,n and Q (cid:48) Γ ,n are integral,that means their vertices are integer points, so that the optimal value of the integerprogram with the set of feasible solutions P (cid:48) Γ ,n ∩ N r (cid:48) is already a linear function of n .The main efforts in [17] are devoted to the corresponding integer program with the setof feasible solutions Q (cid:48) Γ ,n ∩ N r (cid:48) . Note that applying Corollary 2.1 and Theorem 3.11 weonly can conclude that all a i ( R/I ( n ) ), and hence also reg( I ( n ) ), are linear functions of n for all n ≥ ( r + 1)( r + 2) . cknowledgment. The author would like to thank professors Martin Gr¨otschel, Je-sus De Loera, Alexander Barvinok and Dr. Hoang Nam Dung for their consultationon Linear and Integer Programming. 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