Dimension-Dependent Upper Bounds for Grobner Bases
aa r X i v : . [ c s . S C ] M a y Dimension-Dependent Upper Bounds for Gr -obner Bases
Amir Hashemi
Department of Mathematical SciencesIsfahan University of TechnologyIsfahan, 84156-83111, IranSchool of Mathematics, Institute for Research inFundamental Sciences (IPM)Tehran, 19395-5746, Iran
[email protected] Werner M. Seiler
Institut f -ur Mathematik, Universit -at Kassel,Heinrich-Plett-Straıe 40, 34132 Kassel,Germany [email protected]
ABSTRACT
We improve certain degree bounds for Gr¨obner bases of poly-nomial ideals in generic position. We work exclusively in de-terministically verifiable and achievable generic positions ofa combinatorial nature, namely either strongly stable po-sition or quasi stable position. Furthermore, we exhibitnew dimension- (and depth-)dependent upper bounds forthe Castelnuovo-Mumford regularity and the degrees of theelements of the reduced Gr¨obner basis (w.r.t. the degree re-verse lexicographical ordering) of a homogeneous ideal inthese positions.
Categories and Subject Descriptors
F.2.2 [
Analysis of Algorithms and Problem Complex-ity ]: Nonnumerical Algorithms and Problems
Keywords
Polynomial ideals, Gr¨obner bases, Pommaret bases, genericpositions, stability, degree, dimension, depth, Castelnuovo-Mumford regularity.
1. INTRODUCTION
Gr¨obner bases, introduced by Bruno Buchberger in hisPh.D. thesis (see e.g. [6, 7]), have become a powerful toolfor constructive problems in polynomial ideal theory and re-lated domains. For practical applications, in particular, theimplementation in computer algebra systems, it is importantto establish upper bounds for the complexity of determininga Gr¨obner basis for a given homogeneous polynomial ideal.Using Lazard’s algorithm [23], a good measure to estimatesuch a bound, is an upper bound for the degree of the inter-mediate polynomials during the Gr¨obner basis computation.If the input ideal is not homogeneous, the maximal degreeof the output Gr¨obner basis is not sufficient for this estima-tion. On the other hand, M¨oller and Mora [30] showed thatto discuss degree bounds for Gr¨obner bases, one can restrict
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ISSAC’17,
July 25-28, 2017, Kaiserslautern, Germany.Copyright 2012 ACM X-XXXXX-XX-X/XX/XX ... $ to homogeneous ideals. Thus upper bounds for the degreesof the elements of Gr¨obner bases of homogeneous ideals, al-low us to estimate the complexity of computing Gr¨obnerbases in general.Let us review some of the existing results in this direc-tion. Let P be the polynomial ring k [ x , . . . , x n ] where k is of characteristic zero and I ⊂ P be an ideal gener-ated by homogeneous polynomials of degree at most d withdim( I ) = D . The first doubly exponential upper boundswere proven by Bayer, M¨oller, Mora and Giusti, see [31,Chapter 38] for a comprehensive review of this topic. Basedon results due to Bayer [2] and Galligo [14, 15], M¨oller andMora [30] provided the upper bound (2 d ) (2 n +2) n +1 for anyGr¨obner basis of I . They also proved that this doubly ex-ponential behavior cannot be improved. Simultaneously,Giusti [16] showed the upper bound (2 d ) n − for the degreeof the reduced Gr¨obner basis (w.r.t. the degree reverse lex-icographic order) of I when the ideal is in generic position .Then, using a self-contained and constructive combinato-rial argument, Dub´e [10] proved the so far sharpest degreebound 2( d / d ) n − ∼ d n .In 2005, Caviglia and Sbarra [8] studied upper bounds forthe Castelnuovo-Mumford regularity of homogeneous ideals.Analyzing Giusti’s proof, they gave a simple proof of theupper bound (2 d ) n − for the degree reverse lexicographicGr¨obner basis of an ideal I in generic position (they alsoshowed that this bound holds independent of the character-istic of k ). Finally, Mayr and Ritscher [29], by following thetracks of Dub´e [10], obtained the dimension-dependent up-per bound 2(1 / d n − D + d ) D − for any reduced Gr¨obner ba-sis of I . It is worth while remarking that there are also lower bounds for the complexity: d m with m = n/ − O (1) fromthe work of Mayr and Meyer [28] and d m where m ∼ n/ I is in strongly stable position and D > d ( n − D )2 D − is a simultaneous upper bound for theCastelnuovo-Mumford regularity of I and for the maximaldegree of the elements of the Gr¨obner basis of I (with re-spect to the degree reverse lexicographic order). Further-more, we will sharpen the bound of Caviglia-Sbarra to ( d n − D +( n − D )( d − D − . We will see that neither of these boundsis always greater than the other. Finally, we will show that,if I is in quasi stable position and D ≤
1, Giusti’s boundmay be replaced by nd − n + 1 (this result was already ob-tained by Lazard [23] when the ideal is in generic position).n the recent work [21], we showed how many variants ofstable positions – including quasi stable and strongly stableposition – can be achieved via linear coordinate transforma-tions constructed with a deterministic algorithm.The article is organized as follows. In the next section, wegive basic notations and definitions. In Sections 4 ,
2. PRELIMINARIES
Throughout this article, we keep the following notations.Let P = k [ x , . . . , x n ] be the polynomial ring (where k isof characteristic zero). A power product of the variables x , . . . , x n is called term and T denotes the monoid of allterms in P . We consider non-zero homogeneous polynomi-als f , . . . , f k ∈ P and the ideal I = h f , . . . , f k i generatedby them. We assume that f i is of degree d i and that thenumbering is such that d ≥ d ≥ · · · ≥ d k >
0. We alsoset d = d . Furthermore, we denote by R = P / I the cor-responding factor ring and by D its dimension. Finally, weuse throughout the degree reverse lexicographic order with x n ≺ · · · ≺ x .The leading term of a polynomial f ∈ P , denoted byLT( f ), is the greatest term (with respect to ≺ ) appear-ing in f and its coefficient is the leading coefficient of f and we denote it by LC( f ). The leading monomial of f is the product LM( f ) = LC( f )LT( f ). The leading ideal of I is defined as LT( I ) = h LT( f ) | f ∈ Ii . For thefinite set F = { f , . . . , f k } ⊂ P , LT( F ) denotes the set { LT( f ) , . . . , LT( f k ) } . A finite subset G ⊂ I is called a Gr¨obner basis of I w.r.t. ≺ , if LT( I ) = h LT( G ) i . We referto [1] for more details on Gr¨obner bases.Given a graded P -module X and a positive integer s , wedenote by X s the set of all homogeneous elements of X ofdegree s . To define the Hilbert regularity of an ideal, re-call that the Hilbert function of I is defined by HF I ( t ) =dim k ( R t ); the dimension of R t as a k -linear space. Froma certain degree on, this function of t is equal to a poly-nomial in t , called Hilbert polynomial , and denoted by HP I (see [9] for more details on this topic). The Hilbert regular-ity of I is hilb( I ) = min { m | ∀ t ≥ m, HF I ( t ) = HP I ( t ) } .Finally, recall that the Hilbert series of I is the power seriesHS I ( t ) = P ∞ s =0 HF I ( s ) t s . Proposition
There exists a univariate polynomial p ( t ) with p (1) = 0 such that HS I ( t ) = p ( t ) / (1 − t ) D . Fur-thermore, hilb( I ) = max { , deg( p ) − d + 1 } . For a proof of this result, we refer to [13, Thm. 7, page130]. It follows immediately from Macaulay’s theorem thatthe Hilbert function of I is the same as that of LT( I ) andthis provides an effective method to compute it using Gr¨obnerbases, see e.g. [18].Let us state some auxiliary results on regular sequences.Recall that a sequence of polynomials f , . . . , f k ∈ P iscalled regular if f i is a non-zero divisor on the ring P / h f , . . . , f i − i for i = 2 , . . . , k . This is equivalent to the condition that f i does not belong to any associated prime of h f , . . . , f i − i . Itcan be shown that the Hilbert series of a regular sequence f , . . . , f k is equal to Q ki =1 (1 − t d i ) / (1 − t n ), see e.g. [25].The converse of this result is also true, see [13, Exercise 7,page 137]. In addition, these conditions are equivalent tothe statement that D = n − k . Lemma ([25, Prop. 4.1, page 108]) There exist ho-mogeneous polynomials g , . . . , g n − D ∈ P such that the fol-lowing conditions hold: (1) deg( g i ) = d i for each i , (2) g i ≡ λ i f i mod h f i +1 , . . . , f k i for some = λ i ∈ k for i = 1 , . . . , n − D , (3) g , . . . , g n − D is regular sequence in P . Definition
The depth of the homogeneous ideal I is defined as the maximal integer λ such that there exists aregular sequence of linear forms y , . . . , y λ on P / I . Definition
The homogeneous ideal I is m -regular ,if its minimal graded free resolution is of the form −→ M j P ( e rj ) −→ · · ·· · · −→ M j P ( e j ) −→ M j P ( e j ) −→ I −→ with e ij − i ≤ m for each i, j . The Castelnuovo-Mumfordregularity of I is the smallest m such that I is m -regular;we denote it by reg( I ) . For more details on the regularity, we refer to [32, 12, 3,5]. It is well-known that in generic coordinates reg( I ) isan upper bound for the degree of the Gr¨obner basis w.r.t.the degree reverse lexicographic order. This upper bound issharp, if the characteristic of k is zero (see [3]). A good mea-sure to estimate the complexity of the computation of theGr¨obner basis of I is the maximal degree of the polynomialswhich appear in this computation (see [22, 23, 16]). Definition
We denote by deg( I , ≺ ) the maximal de-gree of the elements of the reduced Gr¨obner basis of the non-zero homogeneous ideal I w.r.t. the term order ≺ . Theorem ([25, Prop. 4.8, page 117]) If I is zero-dimensional, then deg( I , ≺ ) ≤ d + · · · + d n − n + 1 . We conclude this section with a brief review of the theoryof Pommaret bases. Suppose that f ∈ P and LT( f ) = x α with α = ( α , . . . , α n ). We call max { i | α i = 0 } the class of f , denoted by cls( f ). Then the multiplicative variables of f are X P ( f ) = { x cls( f ) , . . . , x n } . Furthermore, x β is a Pommaret divisor of x α , written x β | P x α , if x β | x α and x α − β ∈ k [ x cls( f ) , . . . , x n ]. Definition
Let
H ⊂ I be a finite set such that noleading term of an element of H is a Pommaret divisor ofthe leading term of another element. Then H is called a Pommaret basis of I for ≺ , if I = M h ∈H k [ X P ( h )] · h. (1)One can easily show that any Pommaret basis is a (gen-erally non-reduced) Gr¨obner basis of the ideal it generates.The main difference between Gr¨obner and Pommaret basesconsists of the fact that by (1) any polynomial f ∈ I has a unique involutive standard representation. If an ideal I pos-sesses a Pommaret basis H , then reg( I ) equals the maximaldegree of an element of H , cf. [34, Thm. 9.2]. The mainrawback of Pommaret bases is however that they do notalways exist. Indeed, a given ideal possesses a finite Pom-maret basis, if and only if the ideal is in quasi stable position – see [34, Prop 4.4]. Definition
A monomial ideal J in P is called quasistable , if for any term m ∈ J and all integers i, j, s with ≤ j < i ≤ n and s > such that x si | m , there exists anexponent t ≥ such that x tj m/x si ∈ J . A homogeneous ideal I is in quasi stable position , if LT( I ) is quasi stable. In the sequel, we will use the following notations: given anideal I in quasi stable position, we write H = { h , . . . , h s } for its Pommaret basis. Furthermore, for each i we set m i =LT( h i ) and it is then easy to see that { m , . . . , m s } forms aPommaret basis of LT( I ). Remark
Since any linear change of variables is a k -linear automorphism of P preserving the degree, it fol-lows trivially that the dimensions over k of the homogeneouscomponents of a homogeneous ideal I or of its factor ring R remain invariant. Hence the Hilbert function and thereforealso the Hilbert series, the Hilbert polynomial and the Hilbertregularity of I do not change. The same is obviously truefor the Castelnuovo-Mumford regularity. In addition, due tothe special form of the Hilbert series of the ideal generatedby a regular sequence, we conclude that any regular sequenceremains regular after a linear change of variables and hencethe depth is invariant, too. Finally, we note that almost alllinear changes of variables transform a given homogeneousideal into quasi stable position (which is thus a generic po-sition) [34]. It follows that to study any of the mentionedinvariants of I , w.l.o.g. we may assume that I is in quasistable position.
3. IMPROVING GIUSTI’S UPPER BOUND
In 1984, Giusti [16] established the upper bound (2 d ) n − for deg( I , ≺ ) in the case that the coordinates are in genericposition. The key point of Giusti’s proof is the use of thecombinatorial structure of the generic initial ideal in charac-teristic zero. Later on, Mora [31, Ch. 38], by a deeper analy-sis of Giusti’s proof, improved this bound to ( d +1) ( n − D )2 D − λ where λ is the depth of I . In this section, we improve Mora’sbound by following his general approach and correcting someflaws in his method. Our presentation seems to be simplerthan the ones by Mora and Giusti.We first note that for a given ideal in quasi stable position,we are able to reduce the number of variables by the depthof the ideal to obtain a sharper bound for deg( I , ≺ ). A novelproof `a la Pommaret of this result is given below. Proposition
Suppose that U ( n, d, D ) is a functiondepending in n, d and D so that deg( I , ≺ ) ≤ U ( n, d, D ) forany ideal I which is in quasi stable position and is gen-erated by homogeneous polynomials of degree at most d in n variables. Then, deg( I , ≺ ) ≤ U ( n − λ, d, D − λ ) where depth( I ) = λ . Proof.
Let t be the maximal class of the elements in H .It is shown in [34, Prop 2.20] that in quasi-stable positionthe variables x t +1 , . . . , x n define a regular sequence on R andthat thus λ = n − t (note that this reference distinguishesbetween depth( I ) and depth( R ) with the two related by depth( R ) = depth( I ) −
1; what we call here depth( I ) cor-responds to depth( R ) in [34]). By definition of t , no leadingterm of an element of H is divisible by any of these vari-ables. Thus ˜ H = H| x t +1 = ··· = x n =0 is the Pommaret basisof the ideal ˜ I = I| x t +1 = ··· = x n =0 in k [ x , . . . , x t ] and hencedeg( I , ≺ ) = deg(˜ I , ≺ ). This entails our claim. Corollary
As a similar statement to Prop. 3.1,suppose that R ( n, d, D ) is a function depending in n, d and D such that reg( I ) ≤ R ( n, d, D ) . Then, reg( I ) ≤ R ( n − λ, d, D − λ ) . Proof.
This claim follows by the same argument as inthe proof of Prop. 3.1 and using the facts that for each f inthe Pommaret bases H the corresponding element ˜ f ∈ ˜ H hasthe same degree as f and in quasi stable position reg( I ) =reg(˜ I ) is given by the maximal degree of the elements of H and ˜ H .To state the refined version of Giusti’s bound, we need torecall the crystallisation principle . Let A = ( a ij ) ∈ GL( n, k )be an n × n invertible matrix. By A. I we mean the idealgenerated by the polynomials A.f with f ∈ I where A.f = f ( P ni =1 a i x i , . . . , P ni =1 a in x i ). The following fundamentaltheorem is due to Galligo [14]. Theorem
There exists a non-empty Zariski open sub-set
U ⊂
GL( n, k ) such that LT( A. I ) = LT( A ′ . I ) for allmatrices A, A ′ ∈ U . Definition
The monomial ideal
LT( A. I ) with A ∈U and U as given in Theorem 3.3 is called the generic initialideal of I (w.r.t. ≺ ) and is denoted by gin( I ) . Suppose that I = h f , . . . , f k i and that for some s ∈ N we have deg( f i ) ≤ s for all i and gin( I ) has no minimalgenerator in degree s + 1. Then, the crystallisation principle(CP) states that for each m in the generating set of gin( I )we have deg( m ) ≤ s , see [17, Prop 2.28]. Note that thisprinciple holds only in characteristic zero and it has beenproven only for generic initial ideals and for lexicographicideals (see [17, Thm. 3.8]).Giusti’s proof consists in applying this property alongwith an induction on the number of variables. One cru-cial fact in this direction is that CP also holds for a genericinitial ideal modulo the last variable. Below, we will showthat both properties remain true for arbitrary strongly stable ideals. Definition
A monomial ideal J is called stronglystable , if for any term m ∈ J we have x j m/x i ∈ J for all i and j such that j < i and x i divides m . A homogeneousideal I is in strongly stable position , if LT( I ) is stronglystable. Proposition
Let I be in strongly stable position.Then, CP holds for LT( I ) . Proof.
The following arguments are inspired by [31, page728]. Let us consider an integer s ≥ d . Suppose that we arecomputing a Gr¨obner basis of I using Buchberger’s algo-rithm and by applying the normal strategy. In addition, as-sume that we have already computed the set G = { g , . . . g t } up to degree s (this set will be enlarged to a Gr¨obner basisof I ), and there is no new polynomial of degree s + 1 to bedded into G . Note that we have chosen s ≥ d to be surethat G generates I . To prove the assertion, it suffices toshow that G is a G¨obner basis of I .We introduce the set M s = h LT( G ) i s ∩ T . We nowclaim that for each pair of terms x α = x α · · · x α n n = x β = x β · · · x β n n in it either deg(lcm( x α , x β )) = s + 1 or thereexists a further term x γ ∈ M s \ { x α , x β } such that • x γ | lcm( x α , x β ), • deg(lcm( x γ , x α )) < deg(lcm( x α , x β )), • deg(lcm( x γ , x β )) < deg(lcm( x α , x β )).If this claim is true, then Buchberger’s second criterion im-plies that it suffices to consider those pairs { g i , g j } withdeg(lcm(LT( g i ) , LT( g j ))) = s + 1. If for each such pairthe corresponding S-polynomial reduces to zero, then G isa Gr¨obner basis and we are done. Otherwise, there existsa new generator of degree s + 1 contradicting the made as-sumptions.For proving the made claim, it suffices to show that, ifdeg(lcm( x α , x β )) > s + 1, then there exists a term x γ ∈ M s \ { x α , x β } satisfying the above conditions. Let j be aninteger such that α j = β j and α j +1 = β j +1 , . . . , α n = β n .W.l.o.g., we may assume that α j > β j . Since x α and x β have the same degree, there is an index i < j such that β i > α i . The strongly stable position of I implies that M s is a strongly stable set. Therefore the term x γ = x i x α /x j satisfies x γ ∈ M s \ { x α , x β } and x γ | lcm( x α , x β ). Fur-thermore, deg(lcm( x γ , x α )) = s + 1 < deg(lcm( x α , x β )) anddeg(lcm( x γ , x β )) = deg(lcm( x α , x β )) − Example
One should note that strong stability of theleading term ideal does not imply that it is the generic initialideal, as the following example due to Green [17] shows: I = h x x , x x + x , x i ⊂ k [ x , x , x ] . Its leading term ideal LT( I ) = h x x , x x , x , x x , x i is strongly stable, but wefind gin( I ) = h x , x x , x , x x i 6 = LT( I ) . Nevertheless, itis clear that both LT( I ) and gin( I ) satisfy CP. As a consequence of the proof of this proposition, we caninfer a generalization of CP.
Corollary
Suppose we know in advance that I isin strongly stable position. Let us fix an integer t (not nec-essarily greater than d ). Suppose that we are computing aGr¨obner basis for I using Buchberger’s algorithm and ap-plying the normal strategy. Assume that we have treatedall S-polynomials of degree at most t and G t is the set ofall polynomials computed so far. If all S-polynomials of de-gree t + 1 reduce to zero, then any critical pair { f, g } with max { deg( f ) , deg( g ) } ≤ t is superfluous. In particular, G t isa Gr¨obner basis for hI ≤ t i . In the sequel, for an index i we denote by I i the ideal I| x i = ··· = x n =0 ⊂ k [ x , . . . , x i − ]. Since we assume that ≺ isthe degree reverse lexicographic term order, strongly stableposition of I entails that I i is in strongly stable position,too, for any index i . The essence of Giusti’s approach con-sists of finding, by repeated evaluation, relations betweendeg( I , ≺ ) and deg( I i , ≺ ) for i = n, . . . , n − D + 1. For thispurpose, we introduce some further notations for an ideal I in strongly stable position. We denote by N ( I ) the setof all terms m / ∈ LT( I ). If dim( I ) = 0, then we define F ( I ) = N ( I ). Otherwise we set F ( I ) = { τ x an ∈ N ( I ) | τ ∈ F ( I n ) and deg( τ x an ) < deg( I , ≺ ) } . Since I is in stronglystable position, N ( I ) is strongly stable for the reverse or-dering of the variables. More precisely, if x α ∈ N ( I ) with α i >
0, then we claim that x j x α /x i ∈ N ( I ) for any j > i .Indeed, otherwise it belonged to LT( I ) and thus – sinceLT( I ) is strongly stable – x α ∈ LT( I ) which is a contradic-tion. Lemma
Suppose that I is in strongly stable position.Then the following statements hold. ( a ) deg( I , ≺ ) ≤ max { d, deg( I n , ≺ ) } + F ( I n ) , ( b ) F ( I ) ≤ (cid:0) max { d, F ( I n ) } (cid:1) .(Here X denotes the cardinality of a finite set X .) Proof. ( a ) Let G be the reduced Gr¨obner basis of I for ≺ . Because of our use of the degree reverse lexico-graphic term order, we easily see that G | x n =0 is the reducedGr¨obner basis of I n for ≺ . Let G ′ ⊂ G be the subset ofall polynomials in G of maximal degree. We distinguishtwo cases. If LT( G ′ ) ∩ k [ x , . . . , x n − ] = ∅ , then obviouslydeg ( I , ≺ ) = deg( I n , ≺ ) and the assertion is proved.Otherwise, CP (applicable by Prop. 3.6) implies that foreach degree max { d, deg( I n , ≺ ) } < i ≤ deg( I , ≺ ) there ex-ists a polynomial g i ∈ G with deg( g i ) = i (note that ifdeg( I , ≺ ) = d then ( a ) holds and we are done). Thus,we can write LT( g i ) in the form x a i n τ i with a i > τ i ∈ k [ x , . . . , x n − ]. We claim that τ i ∈ F ( I n ). Writ-ing τ i = x α i i · · · x α ik i k where i < · · · < i k , we may con-clude by the assumed reducedness of G that τ i / ∈ LT( I )and by the strong stability of LT( I ) that x α i i · · · x α ik + a i i k ∈ LT( I ). Hence there exists an integer a > x α i i · · · x α ik + a − i k / ∈ LT( I ) and x α i i · · · x α ik + ai k ∈ LT( I ).It follows that there exists a generator g ∈ G ∩ I n suchthat its leading term LT( g ) = x β i i · · · x β ik i k divides the lat-ter term. We must have β ℓ ≤ α ℓ for each ℓ < i k and β i k = α i k + a by definition of a . Furthermore, the strongstability of LT( I ) implies that deg( g ) > deg( τ i ), as other-wise another generator g ′ ∈ G existed with LT( g ′ ) | τ i . Thusdeg( τ i ) < deg( I n , ≺ ). If we write τ i = ¯ τ i x α ik i k , then thereonly remains to show that ¯ τ i ∈ F ( I i k ), as τ i ∈ N ( I n ) isa trivial consequence of τ i ∈ N ( I ). If dim( I i k ) = 0, thisfollows immediately from F ( I i k ) = N ( I i k ). Otherwise werepeated the same arguments as above.Thus for each i with max { d, deg( I n , ≺ ) } < i ≤ deg( I , ≺ )there exists a generator g i ∈ G such that LT( g i ) = x a i n τ i and τ i ∈ F ( I n ). Since G is reduced, the terms τ i are pairwisedifferent. Hence deg( I , ≺ ) − max { d, deg( I n , ≺ ) } ≤ F ( I n )and this proves ( a ).To show ( b ), we introduce for each degree δ ∈ N the subset F δ ( I ) = { x δn τ | x δn τ ∈ F ( I ) } . By definition, x δn τ ∈ F δ ( I )implies τ ∈ F ( I n ) and thus F δ ( I ) ≤ F ( I n ). Since weused in the proof of ( a ) CP, the claims proven there aretrue only for polynomials of degree at least d . Thus inthe sequel we shall replace F ( I n ) by max { d, F ( I n ) } .We observe that the maximal δ such that x δn τ ∈ F ( I ) ismax { d, F ( I n ) } and thus F ( I ) ≤ max { d, F ( I n ) }− X δ =0 max { d, F ( I n ) } which immediately yields the inequality in ( b ). emark Mora [31, Thm. 38.2.7] presented anotherversion of this lemma. Instead of our set F ( I ) , he defined ˜ F ( I ) = { τ x an ∈ N ( I ) | τ ∈ N ( I n ) , deg( τ x an ) < deg ( I , ≺ ) } which differs only in the condition on τ . Assuming the equal-ity ˜ F ( I ) = ˜ F ( I n ) where ˜ F ( I ) contains the elements of ˜ F ( I ) with a = 0 , he proved the following two properties: ( a ) deg( I , ≺ ) ≤ deg( I n , ≺ ) + F ( I n ) , ( b ) F ( I ) ≤ (cid:0) F ( I n ) (cid:1) .However, in general these assertions are not correct – noteven for an ideal in generic position. Indeed, in general wehave only ˜ F ( I n ) ⊆ ˜ F ( I ) and if dim( I ) > and deg ( I ≺ ) < deg ( I , ≺ ) then equality does not hold. As a concrete ex-ample consider I = h x , x x i ⊂ k [ x , x ] . We performa generic linear change x = ay + by and x = cy + dy with parameters a, b, c, d ∈ k . The leading term idealof the new ideal is then h y , y y i . This show that I =gin( I ) and therefore the original coordinates for I are al-ready generic. We have I = h x i , F ( I ) = { , x } and deg( I , ≺ ) = 2 . Furthermore, we have ˜ F ( I ) = { x } ∪{ x i , x i x | i = 0 , . . . , } and F ( I ) = 23 . Thus,
12 =deg( I , ≺ ) deg( I n , ≺ ) + F ( I n ) = 2 + 2 = 4 and
23 = F ( I ) ( F ( I n )) = 4 . In the case that I is a zero-dimensional ideal, we canderive explicit upper bounds for deg( I , ≺ ) and F ( I ) usingthe following well-known lemma. We include an elementaryproof for the sake of completeness. Lemma
Let I be a zero-dimensional ideal. Then ( a ) deg( I , ≺ ) ≤ d + · · · + d n − n + 1 , ( b ) F ( I ) ≤ d · · · d n . Proof. ( a ) was already proven in Thm. 2.6. We presentnow an elementary proof for ( b ). The assumption dim( I ) =0 implies that F ( I ) = dim k ( P / I ) and this dimension isequal to the sum of the coefficients of the Hilbert seriesof I (which is of course a polynomial here). We may as-sume w.l.o.g. that the first n generators f , . . . , f n form aregular sequence (Lem. 2.2). Thus the Hilbert series of I ′ = h f , . . . , f n i is HS I ′ ( t ) = Q ni =1 (1 + · · · + t d i − ) anddim k ( P / I ′ ) is at most HS I ′ (1) = d · · · d n . We obviouslyhave dim k ( P / I ) ≤ dim k ( P / I ′ ) and this proves the asser-tion.We state now the main result of this section. Theorem
If the ideal I is in strongly stable posi-tion, then F ( I ) ≤ d ( n − D )2 D and deg( I , ≺ ) ≤ max (cid:8) ( n − D + 1)( d −
1) + 1 , d ( n − D )2 D − (cid:9) . Proof.
We proceed by induction over D = dim( I ). Inthis proof without loss of generality, we may assume that d ≥
2. If D = 0, the assertions follow immediately fromLem. 3.11. For D >
0, we exploit that then dim( I ) =dim( I n ) + 1 and that we may consider I n as an ideal in k [ x , . . . , x n − ]. Lem. 3.9 now entails that F ( I ) ≤ max { d, F ( I n ) } ≤ (cid:0) d ( n − − ( D − D − (cid:1) = d ( n − D )2 D and thus the first inequality.For the second inequality, Thm. 5.3, which will be provenin the last section, provides the starting point for the induc-tion, as it immediately implies our claim for D ≤
1. For D ≥
2, we obviously have ( n − − ( D −
1) + 1)( d −
1) + 1 ≤ d ( n − D )2 D − and 2 d ( n − − ( D − D − ≤ d ( n − − ( D − D − . Wecan thus rewrite the induction hypothesis asdeg ( I n , ≺ ) ≤ max (cid:8) ( n − − ( D −
1) + 1)( d −
1) + 1 , d ( n − − ( D − D − (cid:9) ≤ d ( n − D )2 D − . Again by Lem. 3.9, we can also estimatedeg( I , ≺ ) ≤ max { d, deg( I n , ≺ ) } + F ( I n ) ≤ d ( n − D )2 D − + d ( n − − ( D − D − = 2 d ( n − D )2 D − proving the second assertion. Example
Let us consider the values n = 2 , d = 2 and D = 0 . The above theorem states deg( I , ≺ ) ≤ = 4 .Consider the ideal I = h x , x x + x i . By performinga generic linear change of coordinates, we get gin( I ) = h x x , x , x i . Therefore F ( I ) = 4 ≤ and deg ( I , ≺ ) =3 ≤ confirming the accuracy of the presented upper bounds.It should be noted that for such a zero-dimensional idealTheorem 2.6 provides the best upper bound for deg ( I , ≺ ) ,namely d + · · · + d n − n + 1 which is equal to the exact value for this example. Using Prop. 3.1, we obtain even sharper bounds depend-ing on both the dimension and the depth of I . We continueto write dim( I ) = D and depth( I ) = λ . It is well-knownthat we always have D ≥ λ (a simple proof using Pommaretbases can be found in [34] after Prop. 3.19). If D = λ ,then R is Cohen-Macaulay . In this case, a nearly optimalupper bound for deg( I , ≺ ) exists. Recall that a homoge-neous ideal I ⊂ P is in
Nœther position , if the ring exten-sion k [ x n − D +1 , . . . , x n ] ֒ → P / I is integral. Alternatively,Nœther position can be defined combinatorially as a weak-ened version of quasi stable position (see [21, Thm. 4.4]). Theorem ([25, Prop. 4.8, page 117]) Let R be aCohen-Macaulay ring with I in Nœther position. Then, deg( I , ≺ ) ≤ d + · · · + d n − D − ( n − D ) + 1 . For the rest of this section, we thus assume that R is notCohen-Macaulay, i. e. that D > λ . Corollary If I is in strongly stable position and D > , then F ( I ) ≤ d ( n − D )2 D − λ − and deg( I , ≺ ) ≤ d ( n − D )2 D − λ − . The maximal degree of an element of the Pommaret basisof an ideal in quasi stable position equals the Castelnuovo-Mumford regularity [34, Cor. 9.5]. If the ideal is even instable position, then the Pommaret basis coincides with thereduced Gr¨obner basis [26, Thm. 2.15]. These considerationsimply now immediately the following two results.
Corollary
If the ideal I is in strongly stable po-sition and D > , then reg( I ) ≤ d ( n − D )2 D − λ − . Corollary
Let the ideal I be in quasi stable posi-tion, H its Pommaret basis and D > . If we write deg( H ) for the maximal degree of an element of H , then deg ( I , ≺ ) ≤ deg ( H ) ≤ d ( n − D )2 D − λ − . . IMPROVING THE UPPER BOUND OFCAVIGLIA-SBARRA In 2005, Caviglia and Sbarra [8] gave a simple proof forthe upper bound (2 d ) n − for deg( I , ≺ ) when the coordi-nates are in generic position by analyzing Giusti’s proof andexploiting some properties of quasi stable ideals. We willnow improve this bound to a dimension dependent bound.As a by-product, we will show that the notion of genericitythat one needs here is strongly stable position.We begin with a quick review of the approach of Cavigliaand Sbarra [8]. For any monomial ideal J ⊂ P let G ( J )be its unique minimal generating set. We write deg i ( J ) =max { deg i ( u ) | u ∈ G ( J ) } where deg i denotes the degree inthe variable x i . Slightly changing our previous notation, wenow denote by J i the ideal J | x i +1 = ··· = x n =0 ⊂ k [ x , . . . , x i ].It follows immediately from the definition of a quasi stableideal that deg i ( J i ) = deg i ( J ). We note that two distinctterms in G ( J ) must differ already in the first n − G ( J ). Hence G ( J ) ≤ Q n − i =1 (deg i ( J ) + 1).Assume that I is in quasi stable position and I satisfiesCP w.r.t. d . CP implies that deg( I , ≺ ) − d +1 ≤ G (LT( I ))and hence deg( I , ≺ ) ≤ d − Q n − i =1 (deg i (LT( I ))+1). Quasistability of LT( I ) implies that deg i (LT( I )) = deg( I i , ≺ ) andthereby deg( I , ≺ ) ≤ d − Q n − i =1 (deg( I i , ≺ ) + 1).Set B = d and for i ≥ B i = d − Q i − j =1 ( B j + 1). If we assume that for each index 1 ≤ i < n the reduced ideal I i satisfies CP w.r.t. d , then by the con-siderations above deg( I i , ≺ ) ≤ B i . In particular, B = 2 d and deg( I , ≺ ) ≤ B n . One easily sees that the B i satisfy therecursion relation B i = d − B i − + 1)( B i − − d + 1) = B i − − ( d − B i − for all i ≥
2. Since we may suppose that d ≥
2, we have B i ≤ B i − . Thus, for all i ≥ B i ≤ (2 d ) i − and therefore B n = deg( I , ≺ ) ≤ (2 d ) n − .We summarize the above discussion in the next theorem. Theorem ([8]) Suppose that I is in quasi stable po-sition and that the ideals I , . . . , I n − , I satisfy CP w.r.t. d .Then deg( I , ≺ ) ≤ reg( I ) ≤ (2 d ) n − . Proof.
We mentioned already above that for any ideal inquasi stable position deg( I , ≺ ) ≤ reg( I ), since the regularityequals the maximal degree of an element of the Pommaretbasis of I . As the regularity remains invariant under linearcoordinate transformations, we may w.l.o.g. assume that I is even in strongly stable position where deg( I , ≺ ) = reg( I )and where Prop. 3.6 entails that also I , . . . , I n − , I satisfyCP w.r.t. d . Now the assertion follows from the considera-tion above.We derive now a dimension dependent upper bound fordeg( I , ≺ ). Theorem
Suppose that I is in strongly stable posi-tion and D = dim( I ) ≥ . Then deg( I , ≺ ) = reg( I ) ≤ (cid:0) d n − D + ( n − D )( d − (cid:1) D − . Proof.
Since I is in strongly stable position, the ideal I n − D ⊂ k [ x , . . . , x n − D ] is zero-dimensional [34, Prop 3.15].According to Lem. 3.11, deg( I n − D ) ≤ ( n − D )( d −
1) + 1.Hence the maximal degree of a term in G (LT( I )) which de-pends only on x , . . . , x n − D is at most this bound. We shallnow construct an upper bound for the degree of the terms in G (LT( I )) containing at least one of the remaining vari-ables x n − D +1 , . . . , x n . Following the approach of Cavigliaand Sbarra, we first look for an upper bound for the num-ber of these terms.Consider a term m = x α · · · x α n n ∈ G (LT( I )) with α i > i ≥ n − D + 1. It is clear that x α · · · x α n − D n − D belongs to the complement of LT( I n − D ). Since the ideal I n − D ⊂ k [ x , . . . , x n − D ] is zero-dimensional, Lem. 3.11 en-tails that dim k (cid:0) k [ x , . . . , x n − D ] / I n − D (cid:1) ≤ d n − D . Hence thenumber of terms x α · · · x α n − D n − D is at most d n − D . On theother hand, for any index n − D + 1 ≤ i ≤ n we have α i ≤ deg i (LT( I )) ≤ deg( I i , ≺ ). Furthermore, we know thattwo distinct term in G (LT( I )) differ already in their first n − G (LT( I )) containing at least one of the variables x n − D +1 , . . . , x n is at most d n − D Q n − i = n − D +1 (cid:0) deg( I i , ≺ ) + 1 (cid:1) .The strongly stability of I implies that CP holds for LT( I )w.r.t. ( n − D )( d − ≥ d by Prop. 3.6. Hence deg ( I , ≺ ) − (cid:0) ( n − D )( d −
1) + 1 (cid:1) + 1 must be less than or equal to thenumber of terms in G (LT( I )) containing at least one of thevariables x n − D +1 , . . . , x n leading to the estimatedeg( I , ≺ ) ≤ d n − D n − Y i = n − D +1 (cid:0) deg( I i , ≺ )+1 (cid:1) +( n − D )( d − . Set B n − D +1 = d n − D + ( n − D )( d −
1) and recursively B j = d n − D Q j − i = n − D +1 ( B i +1)+( n − D )( d −
1) for n − D +2 ≤ j ≤ n .One easily verifies that these numbers satisfy the recursionrelation B j = (cid:0) B j − − ( n − D )( d − (cid:1) ( B j − +1)+( n − D )( d −
1) = B j − − (cid:0) ( n − D )( d − − (cid:1) B j − . We may again assumethat d ≥
2, and therefore B j ≤ B j − for n − D + 2 ≤ j ≤ n .This implies that B j ≤ ( d n − D +( n − D )( d − j − n + D − andin particular we have B n ≤ ( d n − D + ( n − D )( d − D − . Remark
Let us compare the dimension dependentbounds A ( n, d, D ) = 2 d ( n − D )2 D − derived in Thm. 3.12 and B ( n, d, D ) = 2(1 / d n − D + d ) D − due to Mayr and Ritscher[29] with C ( n, d, D ) = ( d n − D +( n − D )( d − D − obtainednow. Obviously, all three bounds describe essentially thesame qualitative behaviour, although they are derived withfairly different approaches. However, the bound B ( n, d, D ) of Mayr and Ritscher has almost always the best constants.But there are some cases where one of the other bounds isbetter. For example, in the case of a hypersurface, i.e. for D = n − , A ( n, d, D ) is smaller than B ( n, d, D ) . For somecurves of low degree, i.e. for D = 1 and small values of d , C ( n, d, D ) is smaller than B ( n, d, D ) . Some concrete in-equalities are: • A (5 , , < C (5 , , , • A (3 , , > C (3 , , , • A (5 , , < B (5 , , , • A (5 , , > B (5 , , , • B (4 , , > C (4 , , , • B (5 , , < C (5 , , .Hence no bound is always the best one. gain an application of Prop. 3.1 yields immediately animproved bound depending on both the depth and the di-mension of I . Corollary
Under the assumptions of Thm. 4.2, onehas deg( I , ≺ ) = reg( I ) ≤ ( d n − D + ( n − D )( d − D − λ − . It should be noted that in positive characteristic it is notalways possible to achieve strongly stable position by linearcoordinate transformations (see [21] for a more detailed dis-cussion). Nevertheless, following [8], we state the followingconjecture.
Conjecture
The upper bound for the Castelnuovo-Mumford regularity of I in Cor. 4.4 holds independently ofthe characteristic of k .
5. LAZARD’S UPPER BOUND
Finally, in this section we study Lazard’s upper bound [23]for the degree of Gr¨obner bases for both homogeneous andnon-homogeneous ideals. We provide a simple proof for hisresults and generalize Giusti’s bound to non-homogeneousideals. Note that for Lazard [23] dimension was always theone as projective variety, whereas we use throughout thispaper the one as affine variety which is one higher. In thesequel, we always set d i = 1 for any i > k . Theorem ([23, Thm. 2]) Assume that dim( I ) ≤ .Then we have deg (gin( I ) , ≺ ) ≤ d + · · · + d r − r + 1 where r = n − λ . We showed in [20] that many properties of gin( I ) also holdfor lt( I ) provided I is in quasi stable position. Along theselines, we shall now prove that in Lazard’s upper bound wecan replace gin( I ) by I , if I is in quasi stable position. Forthis, we need the next proposition also due to Lazard, whichis the key point in the proof of the above theorem. Proposition 5.2 ([22, Thm. 3.3]).
Assume again that dim( I ) ≤ . Then dim k ( P / I ) ℓ = dim k ( P / I ) ℓ +1 for each ℓ ≥ d + · · · + d n − n + 1 . Thus, under the assumptions of this proposition, we cansay that hilb( I ) ≤ d + · · · + d n − n + 1. Theorem
Suppose that I is in quasi stable positionand dim( I ) ≤ . Then, deg( I , ≺ ) ≤ d + · · · + d r − r + 1 where r = n − λ . Proof.
It suffices to show that deg( I , ≺ ) ≤ d + · · · + d n − n + 1, since then the desired inequality follows imme-diately from Prop. 3.1. As I is in quasi stable position,we have the inequality deg( I , ≺ ) ≤ max { hilb( I ) , hilb( I ′ ) } where I ′ = ( I + h x n i ) ∩ k [ x , . . . , x n − ] is an ideal in the ring k [ x , . . . , x n − ] [19, Thm. 4.17], [36, Thm. 4.7]. Obviously, I ′ is generated by f | x n =0 , . . . , f k | x n =0 and dim( I ) ≤ I is in quasi stable position) entailsdim( I ′ ) ≤
1. These arguments show that, by Prop. 5.2,hilb( I ) ≤ d + · · · + d n − n + 1 and hilb( I ′ ) ≤ d + · · · + d n − − ( n −
1) + 1 which proves the assertion.
Example
Lazard [23, Conj. 3] conjectured that theconclusion of Theorem 5.1 remained true, if one replaces gin I by I . Mora claimed that the following ideal (see theAppendix of [23]) provided a counter-example. Consider the homogeneous ideal I = h x x t − − x t , x t +11 − x x t − x , x t x − x t x i in the polynomial ring P = k [ x , . . . , x ] . Thus wehave d = t, d = d = t + 1 . One can show that the poly-nomial x t +13 − x t x appears in the Gr¨obner basis of I andhence deg( I , ≺ ) ≥ t + 1 . For simplicity we restrict to thecase t = 4 where we obtain LT( I ) = h x x , x x , x , x x , x x , x x , x i . Thus we find here deg( I , ≺ ) = 17 > d + d + d − .But as dim( I ) = 2 , I does not yield a counter-example toLazard’s conjecture. However, if we consider I ′ = I| x =0 ⊂ k [ x , x , x ] , then we find that I ′ has dimension and that LT( I ′ ) is generated by the same terms as LT( I ) . I ′ isnot in quasi stable position, as no pure power of x be-longs to LT( I ′ ) . Hence I ′ represents a counter-example toLazard’s conjecture. This example shows furthermore that inThm. 5.3 it is not possible to drop the assumption of quasistable position. Remark
We gave above a direct proof for Thm. 5.3.However, we can provide a more concise proof using Thm. 5.1and Pommaret bases. Indeed, from Thm. 5.1 it follows that reg( I ) ≤ d + · · · + d r − r + 1 where r = n − λ , as reg( I ) =deg(gin( I ) , ≺ ) . Since the ideal I is in quasi stable posi-tion, it possesses a finite Pommaret basis H where reg( I ) is the maximal degree of the elements of H and therefore deg( I , ≺ ) ≤ d + · · · + d r − r + 1 . These considerations alsoyield immediately the following corollary. Corollary If dim( I ) ≤ , then reg( I ) ≤ d + · · · + d r − r + 1 where r = n − λ . Finally, we present an affine version of Thm. 5.3. We dropnow the assumption that the polynomials f , . . . , f k gener-ating I are homogeneous. Let x n +1 be an extra variableand ˜ f the homogenization of f using x n +1 . We further de-note by ˜ I the ideal generated by ˜ f , . . . , ˜ f k (note that ingeneral this is not equal to the homogenization of I ). Thenext proposition may be considered as a generalization ofLazard’s upper bound [23, Thm. 2] to ideals in quasi stableposition. Proposition
Assume that ˜ I is in quasi stable po-sition, that dim(˜ I ) ≤ and that depth(˜ I ) = λ . Then, deg( I , ≺ ) ≤ d + · · · + d r − r + 1 where r = n + 1 − λ . Proof.
By Thm. 5.3, hilb(˜ I ) ≤ reg(˜ I ) ≤ d + · · · + d r − r +1 where r = n +1 − λ . Hence dim k ( k [ x , . . . , x n +1 ] / ˜ I ) ℓ =dim k ( k [ x , . . . , x n +1 ] / ˜ I ) ℓ +1 for all degrees ℓ ≥ d + · · · + d r − r + 1. Therefore, we have dim k ( P / I ) ≤ ℓ = dim k ( P / I ) ≤ ℓ +1 for each ℓ ≥ d + · · · + d r − r + 1 and this observation impliesthat the reduced Gr¨obner basis of I contains no element ofdegree greater than d + · · · + d r − r + 1.We conclude this paper by mentioning that it is easy tosee that for a homogeneous ideal with dim( I ) ≤
1, beingin quasi stable position is equivalent to being in Nœtherposition. This implies that in Thm. 5.3 and Prop. 5.7 onecan replace “quasi stable position” by “Nœther position”.
Acknowledgments.
The research of the first author was in part supported bya grant from IPM (No. 94550420). The work of the sec-nd author was partially performed as part of the H2020-FETOPEN-2016-2017-CSA project SC (712689). The au-thors would like to thank the anonymous reviewers for theirvaluable comments.
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