Real powers of monomial ideals
Pratik Dongre, Benjamin Drabkin, Josiah Lim, Ethan Partida, Ethan Roy, Dylan Ruff, Alexandra Seceleanu, Tingting Tang
aa r X i v : . [ m a t h . A C ] J a n REAL POWERS OF MONOMIAL IDEALS
PRATIK DONGRE, BENJAMIN DRABKIN, JOSIAH LIM, ETHAN PARTIDA, ETHAN ROY,DYLAN RUFF, ALEXANDRA SECELEANU, TINGTING TANG
Abstract.
This paper concerns the exponentiation of monomial ideals. While itis customary for the exponentiation operation on ideals to consider natural powers,we extend this notion to powers where the exponent is a positive real number. Realpowers of a monomial ideal generalize the integral closure operation and highlightmany interesting connections to the theory of convex polytopes. We provide multiplealgorithms for computing the real powers of a monomial ideal. An important resultis that given any monomial ideal I , the function taking real numbers to the corre-sponding real power of I is a step function whose jumping points are rational. Thisreduces the problem of determining real powers to rational exponents. Introduction
An ideal of the polynomial ring R = K [ x , . . . , x d ] with coefficients in a field K is a monomial ideal if it is generated by monomials.We denote by N the set of non negative integers. It is customary to denote mono-mials in R by the shorthand notation x a := x a · · · x a d d , where a ∈ N d . The bijectivecorrespondence between monomials x a and lattice points a ∈ N n gives rise to con-vex geometric representations for monomial ideals, chief among which is the Newtonpolyhedron discussed in section 2.In this paper, we develop a notion of powers for monomial ideals, where the exponentsare allowed to be real numbers; see Definition 3.1. This notion arises as an algebraiccounterpart for the operation of scaling the Newton polyhedron of an ideal by a positivereal scalar. We term the resulting ideals real powers of monomial ideals.Our notion of real powers is inspired by a closely related notion of rational powers(powers with rational exponents), which can be defined for arbitrary ideals and haveappeared previously, albeit not to a great extent, in the literature. Rational pow-ers of ideals appear in [HS06, §
10. 5], [Knu06], [Rus07], [Ciu20], [Ciu], [Lew20]. Inthese works, they come up in contexts ranging from valuation theory to intersectiontheory, exhibit connections to the notion of symbolic powers and have application toestablishing the Golod property.
Mathematics Subject Classification.
Primary 13F55, 13F20.
Key words and phrases. monomial ideals, rational powers, Newton polyhedron, computational al-gebra, jumping numbers.The second author was supported by the NSF RTG grant in algebra and combinatoricsat the University of Minnesota DMS–1745638. The seventh author was supported by NSFDMS–1601024. This work was completed in the framework of the 2020 Polymath program https://geometrynyc.wixsite.com/polymathreu . The focus of this paper is twofold. First, we handle the task of computing realpowers of monomial ideals. One main result in this direction is Theorem 3.5, wherewe show that the generators of a specified real power of a monomial ideal can beconfined within a bounded convex region depending only on the exponent and theNewton polytope of the ideal. We complement this theoretical insight with a series ofalgorithms, Algorithm 1, Algorithm 2, Algorithm 3, and Algorithm 4 which exploit dif-ferent features of the problem to provide practical solutions for computing real powersof monomial ideals.Our second aim is to study continuity properties of the exponentiation functionwhere the base is a monomial ideal. Being able to do this provides motivation forworking with real powers as opposed to the more common rational version. We findthat the exponentiation function is a step function with rational jumping points. Thisleads to the conclusion that all distinct real powers of a fixed monomial ideal are givenby rational exponents. Our main results on properties for the real exponentiationfunction of a monomial ideal are contained in Proposition 5.2 (existence of right lim-its) Proposition 5.6 (left continuity), Corollary 5.7 (step function), and Theorem 5.9(jumping numbers).Our paper is organized as follows. After introducing the notions of Newton polyhe-dron and integral closure in section 2, we turn our attention to real powers of monomialideals in section 3 and present algorithms capable of computing these ideals in section 4.We end with studying continuity properties and jumping numbers for exponentiationin section 5.2.
Background on integral closure and the Newton polyhedron
Let R and R + denote the real numbers and non negative real numbers respectively.Let R = K [ x , · · · , x d ] be a polynomial ring with coefficients in a field K . Everymonomial ideal I in R has a unique minimal monomial generating set denoted G ( I ).This is a set of monomials that generates I and such that no element of G ( I ) dividesanother element of G ( I ). Definition 2.1.
For any monomial ideal I denote by L ( I ) the set of exponent vectorsof all monomials in I L ( I ) = { a | x a ∈ I } . The
Newton polyhedron of I , denoted N P ( I ), is the convex hull of L ( I ) in R d N P ( I ) = convex hull L ( I ) = convex hull( { a | x a ∈ I } ) . The
Newton polytope of I , denoted np( I ), is the convex hull of the exponent vectors ofa minimal monomial generating set for I .np( I ) = convex hull( { a | x a ∈ G ( I ) } ) . Notice that Newton polyhedra are unbounded, while Newton polytopes are boundedconvex bodies. Both are integer polyhedra, meaning that their vertices have integercoordinates. Their relationship can be described using the notion of Minkowski sum.
EAL POWERS OF MONOMIAL IDEALS 3
Definition 2.2.
The
Minkowski sum of subsets
A, B ⊆ R n is A + B = { a + b | a ∈ A, b ∈ B } . The precise relationship between the Newton polyhedron and the Newton polytopeof I , established for example in [CEHH17, Lemma 5.2], is given by the Minkowski sumdecomposition(2.1) N P ( I ) = np( I ) + R d + , where R d + = { ( a , . . . , a d ) ∈ R d | a i ≥ } denotes the positive orthant in R d .By the version of Carath´eodory’s theorem in [CEHH17, Theorem 5.2], any point a ∈ N P ( I ) is written as(2.2) a = λ t + · · · + λ d t d + c e + · · · + c d e d , with λ i , c j ≥ , P di =1 λ i = 1, t , . . . , t d ∈ np( I ), and e , . . . , e d standard basis vectorsin R d . Thus one can reformulate equation (2.1) using coordinatewise inequalities as(2.3) N P ( I ) = { a ∈ R d | a ≥ b for some b ∈ np( I ) } While the containment L ( I ) ⊆ N P ( I ) ∩ N d holds by definition, in general the sets oflattice points L ( I ) and N P ( I ) ∩ N d need not be equal. We recall below that the set oflattice points in N P ( I ) is in fact given by N P ( I ) ∩ N d = L ( I ), where I is the integralclosure of I . Definition 2.3.
The integral closure of an ideal I of a ring R is the set of elements y ∈ R that satisfy an equation of integral dependence of the form y n + m y n − + · · · + m n − y + m n = 0 where m i ∈ I i , n ≥ . The integral closure of I is denoted I . Remark . It is shown in [HS06] that the description is significantly simpler if I is amonomial ideal. In this case one can give an alternate definition for the integral closure(2.4) I = ( { x a | x n a ∈ I n for some n ∈ N } ) . We recall below how the integral closure of a monomial ideal I can be describedin terms of its Newton polyhedron. We also show that the minimal generators of I lie at bounded lattice distance from the Newton polytope np( I ). In the following weuse the notion of lattice (or taxicab) distance between points in a , b ∈ R d defined asdist( a , b ) = P di =1 a i − b i . Lemma 2.5.
Let I be a monomial ideal in K [ x , . . . , x d ] . Then (1) N P ( I ) ∩ N d = L ( I ) , (2) N P ( I ) = N P ( I ) , (3) if x a ∈ G ( I ) , then there is a lattice point b ∈ np( I ) ∩ N d such that a ≥ b and d X i =1 ( a i − b i ) ≤ d − . POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM
Proof.
Statement (1) is well known; see for example [HS06, Proposition 1.4.6].(2) follows from (1) by noticing that, since
N P ( I ) is an integer polyhedron we have N P ( I ) = convex hull( N P ( I ) ∩ N d ) = convex hull( L ( I )) = N P ( I ) . (3) Suppose now that x a ∈ G ( I ) is a minimal generator of I . By equation (2.3)there exists b ∈ np( I ) such that the inequality a ≥ b is satisfied coordinatewise. Since a ∈ N d , we have that a ≥ ⌈ b ⌉ := ( ⌈ b ⌉ , . . . , ⌈ b d ⌉ ). Therefore we may assume b ∈ N d by replacing b by ⌈ b ⌉ . Moreover, since x a is a minimal generator, the monomials x a /x i = x a − e i are not in I for 1 ≤ i ≤ d . Therefore we must have a i − < b i for1 ≤ i ≤ d , otherwise a − e i ≥ b which would yield x a − e i ∈ I again by (2.3). Summingup the preceding inequalities we obtain d X i =1 ( a i − < d X i =1 b i ⇐⇒ d X i =1 ( a i − b i ) < d. Since a i , b i ∈ N , the displayed inequality is equivalent to P di =1 ( a i − b i ) ≤ d − (cid:3) Real powers of monomial ideals
We now discuss powers of monomial ideals with real exponents, termed real powers,and their relationship to integral closure.
Definition 3.1.
Fix a real number r ≥
0. We define the r -th real power of a monomialideal, I , to be I r = (cid:0) { x a | a ∈ r · N P ( I ) ∩ N d } (cid:1) . When r ∈ Q we will refer to I r as the r -th rational power of I .Rational powers of monomial ideals have appeared previously in the literature underthe following definition and notation, see [HS06, Definition 10.5.1]: the r -th rationalpower of an arbitrary ideal I of a ring R for r = pq with p, q ∈ N , q = 0 is the ideal(3.1) I r := { y ∈ R | y q ∈ I p } , where I p denotes the integral closure of the p -th ordinary power of I , I p . In thefollowing we show that these two definitions agree, i.e., I r = I r whenever r ∈ Q andfurthermore for natural exponents r ∈ N the r -th real power agrees with the integralclosure of the r -th ordinary power of I , I r .Our notation for real powers deviates from that in (3.1), which is more establishedin the literature, in favor of being intentionally consistent with the notation for integralclosure, since these notions agree for r ∈ N as shown in the following lemma. Lemma 3.2.
Let I be a monomial ideal. Then (1) If r ∈ N , then the r -th real power of I is equal to the integral closure of the r -th ordinary power I r . In particular, the first rational power of I , I , is theintegral closure of I . (2) If r ∈ Q then the r -th real power of I in Definition 3.1 agrees with the r -thrational power of I , I r , in (3.1) . EAL POWERS OF MONOMIAL IDEALS 5
Proof. (1) By definition, a monomial x a is an element of the r -th real power of I ifand only if a ∈ r · N P ( I ). Noting that r · N P ( I ) = N P ( I r ), the latter condition isequivalent to a ∈ N P ( I r ). Now by Lemma 2.5 (1), we have a ∈ N P ( I r ) ∩ N d if andonly if x a is an element of the integral closure of I r .(2) Let r = pq with p, q ∈ N , q = 0 and let x a be a monomial. By (3.1), x a ∈ I r holds if and only if we have x q a ∈ I p , equivalently q a ∈ N P ( I p ) = N P ( I p ) = pN P ( I ).In turn, the last assertion is equivalent to a ∈ rN P ( I ) ∩ N d and by Definition 3.1 thisholds if and only if x a ∈ I r . (cid:3) Using Lemma 2.5, for r ∈ Q + we aim to confine the minimal generators of I r to abounded convex set, which will be obtained by Minkowski sum. In order to define thisconvex set we introduce the unit simplex in d -dimensional space, S d = { a = ( a , . . . , a d ) ∈ R d | a + · · · + a d ≤ , a ≥ ≤ i ≤ d } . In the metric space R d endowed with the lattice distance, the unit simplex is the nonnegative portion of the ball of radius one centered at the origin. Denoting the originin R d by , this observation yields an alternate description S d = { a ∈ R d | a ≥ , dist( a , ) ≤ } . Remark . Lemma 2.5 (3) can be reformulated using this notation as follows: If I isa monomial ideal and x a ∈ G ( I ), then a ∈ np( I ) + ( d − · S d .The following technical result shall prove very useful for our purposes. Lemma 3.4.
Let x a be a minimal generator of I r , where r = pq is a positive rationalnumber. Then there exists a minimal generator of I p , x b , such that q a − b ∈ d ( q − · S d .Proof. To show q a − b ∈ d ( q − · S d we will prove the equivalent statements q a ≥ b and d X i =1 ( qa i − b i ) ≤ d ( q − . Suppose to the contrary that for all minimal generators, x b , of I p we have d X i =1 ( qa i − b i ) ≥ d ( q −
1) + 1 . Applying the pigeon-hole principle, we find that there must exist i ∈ { , . . . , d } suchthat qa i − b i ≥ q . Rewriting, we get that q ( a i − ≥ b i .Now let x a be a minimal generator of I r . By Lemma 3.2 (2), we obtain x q a ∈ I p .Thus there exists a minimal generator x b ∈ G ( I p ) such that x b divides x q a . Thisimplies b ≤ q a , that is, b i ≤ qa i for all 1 ≤ i ≤ d . By assumption exists i such that b i ≤ q ( a i − a ′ = a − e i and with this notation we find b ≤ q ( a , . . . , a i − , a i − , a i , . . . , a d ) = q a ′ . Thus x b divides x q a ′ and x q a ′ is an element of I p . Applying Lemma 3.2 (2) again, thisyields that, x a ′ ∈ I r , which contradicts that x a is a minimal generator of I r . (cid:3) POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM
We are now able to describe a bounded convex set which contains the minimalgenerators of a rational power for a monomial ideal. The following result constitutesthe basis for our Minkowski algorithm described in Algorithm 1. See also Example 4.2for an illustration of the convex set C ( I, r ) defined below.
Theorem 3.5.
Let I be a monomial ideal in K [ x , . . . , x d ] . If r = pq is a positiverational number and x a ∈ G ( I r ) , then a is in the following bounded convex set (3.2) C ( I, r ) = r · np( I ) + (cid:18) d − q (cid:19) · S d . Moreover, if a ∈ C r ( I ) , then x a ∈ I r and thus I r = ( { x a | a ∈ C ( I, r ) ∩ N d ) } .Proof. By Lemma 3.4, there exists a minimal generator of I p , x b , such that q a − b ∈ d ( q − · S d and from Remark 3.3 applied to the monomial ideal I p we have that b ∈ np( I p ) + ( d − · S d = p · np( I ) + ( d − · S d . Combining the displayed statements, we obtain q a ∈ p · np( I ) + ( d − · S d + d ( q − · S d ⇐⇒ a ∈ pq · np( I ) + d − q · S d + d ( q − q · S d ⇐⇒ a ∈ r · np( I ) + (cid:18) d − q (cid:19) · S d . Finally, since S d ⊆ R d + , we have that C ( I, r ) ⊆ r · N P ( I ) by (2.1). Thus if a ∈ C ( I, r ),then a ∈ r · N P ( I ) which yields x a ∈ I r according to Definition 3.1. The identity I r = ( x a | a ∈ C ( I, r )) follows from the previous assertions. (cid:3)
Remark . While the previous theorem does not require the rational number r = pq to have gcd( p, q ) = 1, in applications is desirable to work with the reduced form of r in order to obtain the smallest possible region C ( I, r ).4.
Algorithms for computing real powers
Several algorithms are proposed below for computing real powers of monomial ideals.Our algorithms rely on several auxiliary computational tasks, which are highly nontrivial, but can be performed currently by computer algebra systems such as [GS]or [tt]. Specifically, we assume that independent routines are used to compute theNewton polyhedron or polytope for a given monomial ideal. For this reason, we takethese convex bodies as input for our algorithms. For Algorithm 1 we additionallyassume the existence of a routine that finds all the lattice points in a bounded convexpolytope. This task is discussed in detail in [DLHTY04].
EAL POWERS OF MONOMIAL IDEALS 7
Minkowski Algorithm.
Our first algorithm uses the ideas presented in Theorem 3.5and illustrated in Example 4.2 to confine the generators of a real power I r within aconvex region of bounded lattice distance from the Newton polytope np( I ). Algorithm 1:
Minkowski Sum algorithm
Input: the Newton polytope np( I ) of an ideal I , a rational number r = pq ∈ Q + Output: a list of monomial generators for the ideal I r /* Scaled newton polytope of I */ scalednp := r · np ( I ) /* Bounded convex set, as given by Theorem 3.5 */ d := dimension of the polynomial ring containing I simplex := d -dimensional simplex with vertices at { , ( d − q ) e , . . . , ( d − q ) e d } . C := minkowskiSum(scalednp, simplex) /* Find all lattice points and their monomial counterpart */ exponentVectors := latticePoints( C ) Initialize generators := ∅ for b in exponentVectors do generators := append( x b , generators) /* Return the possibly non minimal monomial generators */ Return generators.
Proposition 4.1. If I is a monomial ideal of a d -dimensional polynomial ring and r ∈ R + , then Algorithm 1 returns a not necessary minimal set of monomial generatorsfor I r .Proof. This follows from the assertion I r = ( { x a | a ∈ C ( I, r ) } ) ∩ N d ) of Theorem 3.5.In Algorithm 1 the set C ( I, r ), termed C , is constructed according to equation (3.2). (cid:3) Example 4.2.
Consider the ideal I = ( xy , x y , x y ) and the rational number r = .Then one can determine that I / = ( x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y )based on identifying the lattice points in the convex region C (cid:18) I, (cid:19) = 43 · np( I ) + 53 · S given by Theorem 3.5. Note that I / is minimally generated by G ( I / ) = { x y , x y , x y } .Thus, Algorithm 1 does not in general identify the minimal generators, but rather apossibly non minimal set of generators for I r . In the Figure 1, the region C ( I, ) isshaded in darker blue, while the rest of the scaled polyhedron · N P ( I ) is shaded inlighter blue.4.2. Hyperrectangle Algorithm.
The next algorithms depend on the notion of thehyperrectangle of a scaled Newton polyhedron, which is defined below.
POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM x y x y r · NP ( I )Convex RegionMinimal GeneratorsInterior Points Figure 1.
Computing ( xy , x y , x y ) / using the Minkowski algorithm. Definition 4.3.
Given a monomial ideal I of a d -dimensional polynomial ring and r ∈ R + , define the set of scaled vertices of I with respect to r to be V ( I, r ) = {⌈ r a ⌉ := ( ⌈ ra ⌉ , . . . , ⌈ ra d ⌉ ) | a ∈ L ( I ) } .Let α = ( α , . . . , α d ) ∈ V ( I, r ). Define(4.1) min( V , i ) = min α ∈V α i and max( V , i ) = max α ∈V α i . Finally, set the hyperrectangle of r · N P ( I ) to be the following sethype( I, r ) = { c = ( c , . . . , c d ) | c i ∈ [min( V , i ) , max( V , i )] } = d Y i =1 [min( V , i ) , max( V , i )] . We now see that the generators for the r -th real power of I are among the set oflattice points in hype( I, r ). Lemma 4.4.
Let I be a monomial ideal and let r ∈ R + . Denote the set of latticepoints in hype( I, r ) by S ( I, r ) . Then (1) ⌈ r · np( I ) ⌉ := { ( ⌈ p ⌉ , . . . , ⌈ p d ⌉ ) | p ∈ r · np( I ) } ⊆ S ( I, r )(2) I r is generated by a subset of the lattice points in hype( I, r ) , more precisely I r = ( { x a | a ∈ r · N P ( I ) ∩ hype( I, r ) ∩ N d } ) . Proof. (1) Every point in p ∈ r · np( I ) is a convex combination of the vertices of thispolytope, which are in the set V = { r a | x a ∈ r · G ( I ) } . Since every coordinate p i of p is a convex combination of i -th coordinates of elements in V we obtain that p i ∈ [min a ∈ V a i , max a ∈ V a i ] for 1 ≤ i ≤ d . Thus ⌈ p i ⌉ ∈ [min( V , i ) , max( V , i )], whichsettles the claim.(2) Temporarily denote J := ( x a | a ∈ r · N P ( I ) ∩ S ( I, r )). Then J ⊆ I r followsfrom Definition 3.1. Now let a ∈ N d be such that x a ∈ I r and thus a ∈ r · N P ( I ) ∩ N d .From (2.1) we know r · N P ( I ) = r · np( I ) + r · R d + = r · np( I ) + R d + , EAL POWERS OF MONOMIAL IDEALS 9 thus there exists b ∈ r · np( I ) such that a ≥ b . Since a ∈ N d it follows that a ≥ ⌈ b ⌉ =( ⌈ b ⌉ , . . . , ⌈ b d ⌉ ], where ⌈ b ⌉ ∈ ⌈ r · np( I ) ⌉ . From part (1) it follows that ⌈ b ⌉ ∈ S ( I, r )and from ⌈ b ⌉ ≥ b we deduce ⌈ b ⌉ ∈ r · np( I ) hence ⌈ b ⌉ ∈ r · N P ( I ). We have thusshown that ⌈ b ⌉ ∈ r · N P ( I ) ∩ S ( I, r ), hence x ⌈ b ⌉ ∈ J and since a ≥ ⌈ b ⌉ we deduce x a ∈ J . Thus the containment I r ⊆ J has been established. (cid:3) Based on the previous result we produce the following algorithm.
Algorithm 2:
Hyperrectangle algorithm
Input: the Newton polyhedron
N P ( I ) of an ideal I , a real number r ∈ R + Output: a list of monomial generators for the ideal I r d := dimension of the polynomial ring containing I candidates := hype( I, r ) ∩ N d Initialize generators := ∅ for b in candidates do if b in r · N P ( I ) then generators := append( x b , generators) Return generators.
Proposition 4.5. If I is a monomial ideal and r ∈ R + , then Algorithm 2 returns anot necessary minimal set of monomial generators for I r .Proof. This follows from part (2) of Lemma 4.4. (cid:3)
Example 4.6.
Figure 2 illustrates the set of lattice in the hyperrectangle hype( I, )for the ideal I = ( xy , x y , x y ). These are marked in solid yellow, solid purple andhollow black. The set of generators returned by Algorithm 2 corresponds to the yellowand purple lattice points, while the minimal generator correspond to the purple points. x y x y r · NP ( I )Boundary of hype( I, r )Minimal GeneratorsInterior PointsExterior Points
Figure 2.
Computing ( xy , x y , x y ) / using the Hyperrectangle al-gorithm (left) and Improved Hyperrectangle algorithm (right)In general, for fixed I and r , the two convex sets C ( I, r )) and hype(
I, r ) whereAlgorithm 1 and Algorithm 2, respectively, look for a set of generators for I r are in-comparable. For an illustration consider Figure 1 in Example 4.2, where the set C ( I, r )) is shaded in darker blue and Figure 2 where the set hype( I, r ) is the marked by theorange boundary. Note that there are no containments between the sets C ( I, r )) andhype(
I, r ) in this example. In general one does not expect a containment between thecorresponding sets of lattice points sets C ( I, r )) ∩ N d and hype( I, r ) ∩ N d either. How-ever, the cardinality of the former set is typically smaller than the latter. We addressthis shortcoming in the next Algorithm 3.The exponent vectors for minimal generators of I r are in C ( I, r )) ∩ hype( I, r ) ∩ N d . However, as illustrated by Figure 1 and Figure 2, the exponents for the minimalgenerators of I r can form a proper subset of C ( I, r )) ∩ hype( I, r ) ∩ N d .The next variant improves on the hyperrectangle algorithm by reducing some redun-dancies in the traversal of lattice points. Using the while-loop on the final coordinate,the improved hyperrectangle algorithm stops looking for other generators after it findsa lattice point that is inside r · N P ( I ). Note that the improved hyperrectangle algo-rithm optimizes traversal of the set hype( I, r ) ∩ N d only on the last coordinate, so thebenefits of using this algorithm over the hyperrectangle algorithm is more apparent inlow dimensional rings. Algorithm 3:
Improved Hyperrectangle algorithm
Input: the Newton polyhedron
N P ( I ) of an ideal I , a real number r ∈ R + Output: a list of monomial generators for the ideal I r d := dimension of the polynomial ring containing I startPoints := { b ∈ hype( I, r ) | b d = min( V , d ) } Initialize generators := ∅ for b in startPoints do while b not in r · N P ( I ) and b d ≤ max( V , d ) do b := b + (0 , . . . , , /* ‘‘move up’’ */ if b in r · N P ( I ) then generators := append( x b , generators) /* Return the possibly non minimal monomial generators */ Return generators.
Example 4.7.
Figure 2 illustrates the set of generators for the ideal ( xy , x y , x y ) / returned by the improved hyperrectangle algorithm. The set of lattice points consideredby this algorithm are marked in solid yellow and purple and hollow black. The set ofgenerators returned by Algorithm 3 corresponds to the yellow and purple lattice points,while the minimal generator correspond to the purple lattice points only. Comparedto Figure 2, fewer non minimal generators are returned.4.3. Staircase Algorithm.
The algorithms presented in the previous sections (Algorithm 1,Algorithm 2, and Algorithm 3) have one common disadvantage in that they return pos-sibly non minimal sets of generators for the real powers of monomial ideals. The nextalgorithm, termed the staircase algorithm, traverses lattice points near the boundaryof the Newton polyhedron. The traversal is designed so that, in the 2-dimensional case,the minimal generators are found.
EAL POWERS OF MONOMIAL IDEALS 11
A benefit of the following algorithm is to improve upon the runtime of Algorithm 1and Algorithm 3. Algorithm 1 is slow in practice because of lattice points identificationin step 5, while Algorithm 3 may be inefficient because a large number of operationscould be performed to check if lattice points are in or outside r · N P ( I ). To alleviatethis issue, the staircase algorithm optimizes the traversal of lattice points on the finaltwo coordinates. The algorithm uses the notation in equation (4.1). Algorithm 4:
Staircase algorithm
Input: the Newton polyhedron
N P ( I ) of an ideal I , a real number r ∈ R + Output: a list of monomial generators for the real power I r Initialize generators := ∅ d := dimension of the polynomial ring containing I if d = 1 then Return { x min( V , } else startPoints := (cid:8) a ∈ hype( I, r ) | a d − = min( V , d − , a d = max( V , d ) (cid:9) for a in startPoints do b := a while a in hype( I, r ) do if a in r · N P ( I ) then b := a a := a − (0 , . . . , , /* ‘‘move down’’ */ else if b in r · N P ( I ) then generators := append( x b , generators) b := a a := a + (0 , . . . , , /* ‘‘move right’’ */ if b in r · N P ( I ) then generators := append( x b , generators) Return generators.
Example 4.8.
Figure 3 shows the set of lattice points considered by the staircasealgorithm within hype( I, ) for the ideal I = ( xy , x y , x y ). While all the latticepoints along the path of the algorithm are considered, only the minimal generatorscorresponding to the purple lattice points are returned.We are now ready to show the validity of Algorithm 4. We utilize terminology thatis consistent with the visual descriptions in Figure 3. We call the path of the algorithm P ( I, r ) the set of values taken by the variable a in Algorithm 4 for fixed inputs I, r .This set is the disjoint union of two subsets: the exterior path and the interior path x y r · NP ( I )Boundary of hype( I, r )Path of algorithmMinimal GeneratorsInterior PointsExterior Points
Figure 3.
Computing ( xy , x y , x y ) / using the Staircase algorithmdefined below: P ext ( I, r ) = { a ∈ P ( I, r ) \ r · N P ( I ) }P int ( I, r ) = { a ∈ P ( I, r ) ∩ r · N P ( I ) } . Proposition 4.9. If I is a monomial ideal of a d -dimensional polynomial ring and d ∈ { , } , then Algorithm 4 returns a minimal set of monomial generators for I r . If d ≥ then Algorithm 4 returns a not necessary minimal set of monomial generatorsfor I r .Proof. In the case d = 1, every monomial ideal J ⊆ K [ x ] is principal, minimallygenerated by x m , where m = min { a | x a ∈ J } . Applying this to J = I r for which case m = min( V ,
1) yields G ( I r ) = { x min( V , } , i.e., the output of Algorithm 4 in step 4.For the case d = 2, first notice that because of the succession of down moves andright moves, the interior path P int ( I, r ) is a disjoint union of vertical strips of the form s a,b,c := { γ = ( γ , γ ) | γ = a, γ ∈ [ b, c ] ∩ N } , where b = min { b ′ | ( a, b ′ ) ∈ r · N P ( I ) } by step 12 of the algorithm; see Figure 3 foran illustration. Moreover, the interior path contains one lattice point for each value ofthe x -coordinate in [min( V , , max( V , P int ( I, r ) = e [ i =min( V , s i,b i ,c i we must have c min( V , = max( V ,
2) and b i = c i +1 + 1 for each i ≤ e −
1, where e is themaximum x coordinate of any point on the interior path. In particular, if i < j thenthe inequality b i > c j holds.Let x a ∈ G ( I r ). By Lemma 4.4 it follows that a = ( a , a ) ∈ hype( I, r ), so a ∈ [min( V , , max( V , b ∈P int ( I, r ) with b = a . We claim that b = a . If not, then a < b since x a is aminimal generator (i.e., a lies “left” of b ), and for this reason a = b ≤ c b < b a (i.e a lies “below” the strip with x -coordinate a ). Since a = ( a , a ) ∈ r · N P ( I ) and EAL POWERS OF MONOMIAL IDEALS 13 b a = min { b ′ | ( a , b ′ ) ∈ r · N P ( I ) } , this yields a contradiction. We have shown that G ( I r ) ⊆ { x a | a ∈ P int ( I, r ) } . In the notation of (4.2), the algorithm returns the set { x i x b i | min( V , ≤ i ≤ e } .Each of the monomials x i x j with j ∈ ( b i , c i ] ∩ N are not in G ( I r ) since they aredivisible by x i x b i . Thus G ( I r ) is contained in the returned set. Moreover, the returnedset consists of minimal generators since no two of its elements are comparable under thedivisibility relation. In fact, this proof shows that the case d = 2 of the algorithm givesa minimal set of generators for the ideal generated by the monomials with exponentsin a given convex set (in our application to real powers, this convex set is r · N P ( I )).We use this to approach the higher dimensional cases.The case d > d = 2 by the following analysis. By virtueof Lemma 4.4 we have the identity I r = (cid:0) { x a | a ∈ hype( I, r ) ∩ r · N P ( I ) ∩ N d (cid:1) = X γ ∈ Q d − i =1 [min( V ,i ) , max( V ,i )] x γ · · · x γ d − d − · I γ,r , where I γ,r := ( { x ad − x bd | ( γ , . . . , γ d − , a, b ) ∈ r · N P ( I ) } ) is an ideal in a 2-dimensionalpolynomial ring. According to the case d = 2, steps 7–19 of the algorithm append theset x γ · · · x γ d − d − · G ( I γ,r ) to the generators list. The union of these sets generates I r bythe above displayed identity. (cid:3) Example 4.10.
We give a visual illustration of using Algorithm 4 to compute theintegral closure of I = ( y , y z , x y , x z ), that is, I in Figure 4. In 3-dimensionalspace, the path of the algorithm is a disjoint union of paths, each corresponding to anideal in a 2-dimensional ring as shown in the proof of Proposition 4.9. xy Boundary of r · N P ( I )Interior PointsMinimal GeneratorsExterior Points Figure 4.
Computing ( y , y z , x y , x z ) using the Staircase algorithm Continuity and jumping numbers for exponentiation
In this section we analyze how the real powers of monomial ideals vary with theexponent. To be precise, for a fixed monomial ideal I we consider continuity propertiesfor the exponentiation function of base I exp : R + → T , exp( r ) = I r whose domain is R + with its Euclidean topology and whose codomain is the set T = { I r | r ∈ R + } endowed with the discrete topology.We start with two elementary properties enjoyed by the family of real powers of thefixed ideal. Lemma 5.1. If I is a monomial ideal and r, s ∈ R + then (1) if s ≥ r ≥ , then the containment I s ⊆ I r holds, (2) I s · I r ⊆ I s + r .Proof. Assertion (1) is clear from Definition 3.1. To clarify assertion (2), note thatmonomials in I s · I r correspond to lattice points in the Minkowski sum s · N P ( I ) + r · N P ( I ) = ( s + r ) · N P ( I ) . (cid:3) Part (2) of Lemma 5.1 shows that the real powers of a fixed monomial ideal form a graded family , although this terminology is more commonly used for families indexedby a discrete set. Property (1) of Lemma 5.1 allows to define for each r ∈ R themonomial ideal I >r = [ s>r I s . We show that this ideal can be understood as a limit in T , meaning that a sequenceof real powers of I where the exponents approach a real number r from the right muststabilize to I >r . Proposition 5.2.
Let I be a monomial ideal and let { t n } n ∈ N be a non-increasingsequence of non-negative real numbers with r = lim n →∞ t n . Then I t n = I >r for n sufficiently large.Proof. A non-increasing sequence of non-negative numbers { t n } n ∈ N gives an ascendingchain of ideals I t ⊆ I t ⊆ · · · ⊆ I r by Lemma 5.1 (1). Since the polynomial ring isNoetherian, any such chain must in fact stabilize, i.e. there exists N ≫ I t n = I t m for m, n ≥ N . We show that the stable value of this chain is I >r . Indeed,from the definition of I >r one deduces the containment I t N = ∞ [ n =0 I t n ⊆ [ s>r I s = I >r . Conversely, for each s > r , there exists n ≥ N such that s > t n , hence one has thecontainments I s ⊆ I t N = I t n for all s > r and consequently I t N ⊆ I >r . (cid:3) EAL POWERS OF MONOMIAL IDEALS 15
To distinguish those real numbers r for which the function exp : R + → T , exp( r ) = I r is right discontinuous, we term them jumping numbers. Definition 5.3. A jumping number for I is a real number r ∈ R + for which the realpowers of I are not equal to I r when we approach r from the right, i.e. I r = I >r . Example 5.4. I = R but I r isa proper ideal for any r > Example 5.5.
For I = ( x , x y, xy ) we have that is not a jumping number while is a jumping number. This is because for small values of ε > · N P ( I ) ∩ N = (cid:18)
13 + ε (cid:19) · N P ( I ) ∩ N , while 12 · N P ( I ) ∩ N = (cid:18)
12 + ε (cid:19) · N P ( I ) ∩ N because the point (2 ,
0) belongs to the leftmost set but not the rightmost. In fact, forthe ideal I in this example, we have ( x , xy ) = I / = I > / = I / = I > / = ( x , xy ). x y x y Figure 5.
Comparing · N P ( I ) and · N P ( I )To verify that right continuity is a special characteristic to study, we show that theexponentiation function is a left continuous function. Proposition 5.6.
The function exp : R + → T , exp( r ) = I r is left continuous.Proof. Fix r ∈ R + and consider the set A r = R d + \ r · N P ( I ). Since each point a ∈ A r lies at a positive Euclidean distance from any point in r · N P ( I ), and since varying r changes this distance continuously, it follows that for all ε > < ε < ε each point a ∈ A r lies at a positive Euclidean distance from any pointin ( r − ε ) · N P ( I ) as well. Equivalently we have A r ∩ ( r − ε ) · N P ( I ) = ∅ whichyields A r = A r − ε and thus r · N P ( I ) ∩ N d = ( r − ε ) · N P ( I ) ∩ N d and I r = I r − ε for0 < ε < ε . (cid:3) We now show that the real exponentiation function of a monomial ideal is a stepfunction.
Corollary 5.7.
Let j < j ′ be two consecutive jumping numbers for I . Then thefunction exp : R + → T , exp( r ) = I r is constant on ( j, j ′ ] and I j = I j ′ .Proof. Since j < j ′ are consecutive jumping, meaning there is no jumping numberin ( j, j ′ ), the exponentiation function is continuous on ( j, j ′ ) by a combination ofProposition 5.2 and Proposition 5.6 and left continuous at j . Since T carries thediscrete topology, this continuity is equivalent to the function being constant on ( j, j ′ ].However, the exponentiation function is right discontinuous at j by the definition ofjumping number, thus I j is distinct from the common value of the exponentiationfunction on ( j, j ′ ], that is, I j = I j ′ . (cid:3) Our next aim is to show that the jumping numbers for monomial ideals are rational.Towards this end recall that any polyhedron admits a description as a finite intersectionof half spaces. We term the linear inequalities describing a polyhedron as an intersectionof half spaces its bounding inequalities. In particular, if I is a monomial ideal in apolynomial ring of dimension d then there is a d × s matrix A and such that(5.1) N P ( I ) = { x ∈ R d + | A x ≥ } , where is the vector in R s with all coordinates equal to 1. Since N P ( I ) is an in-teger polyhedron, the matrix A has rational entries and since N P ( I ) is closed underincreasing coordinates, according to (2.1) the entries of A are non negative. Moreover,scaling the Newton polyhedron amounts to scaling the constant term of the boundinginequalities, that is, r · N P ( I ) = { x ∈ R d + | A x ≥ r · } . Setting A = [ a ij ], the bounded facets of the Newton polyhedron are supported onhyperplanes H i with equation P dj =1 a ij x j = 1. Each bounded facet F i of N P ( I ) is thuscut out by a system formed by one equation and several inequalities of the form(5.2) F i = ( x | d X j =1 a ij x j = 1 , min( F i , j ) ≤ x j ≤ max( F i , j ) for 1 ≤ j ≤ d ) , where min( F i , j ) and max( F i , j ) respectively denote the smallest and largest value ofthe j th coordinate of any point in F i . Proposition 5.8.
Given a monomial ideal I with bounded facets F i , ≤ i ≤ s for N P ( I ) described as in (5.2) above, the following are equivalent (1) r ∈ R + is a jumping number for I (2) for some ≤ i ≤ s there exists a lattice point a ∈ r · F i ∩ N d (3) for some ≤ i ≤ s there exists an integer solution to the system of equationsand inequalities that describes r · F i , namely (5.3) (P dj =1 a ij x j = r,r min( F i , j ) ≤ x j ≤ r max( F i , j ) for ≤ j ≤ d. Proof. (2) ⇔ (3) is clear.(1) ⇒ (2) We show the contrapositive. Assume that r ∈ R + is such that the unionof the bounded facets of r · N P ( I ) contains no lattice point. Since each unbounded EAL POWERS OF MONOMIAL IDEALS 17 facet of NP(I) is a Minkowski sum of an ( < d dimensional) orthant and a face of oneof the bounded facets F i by [Gr¨u03, page 317], this implies that the unbounded facescontain no lattice points either. Since each face varies continuously with respect toscaling by r (as illustrated by the continuity of the functions in (5.3)), and the set oflattice points is discrete, hence closed in the Euclidean topology, it follows that thereexists ε such that for each 0 < ε < ε and 1 ≤ i ≤ s there are no lattice points on( r + ε ) · ∂ ( N P ( I )), where ∂ ( N P ( I )) denotes the boundary of N P ( I ). Based on theequality r · N P ( I ) \ ( r + ε ) · N P ( I ) = [ s ∈ [ r,r + ε ) s · ∂ ( N P ( I ))we see that there are no lattice points in r · N P ( I ) \ ( r + ε ) · N P ( I ) for 0 < ε < ε . Itfollows that I r = I r + ε for 0 < ε < ε and thus r is not a jumping number for I .(3) ⇒ (1) Let a be an integer solution to (5.3). Since this implies b ∈ r · F i ⊆ r · N P ( I ), we see that x a ∈ I r . Since a attains equality in the first equation of(5.3) it follows that a satisfies a ij x j < ( r + ε ) · for any ε >
0. Thus we conclude a ( r + ε ) · N P ( I ) and x a / ∈ I r + ǫ for all ǫ > x a / ∈ I >r . Consequently r is a jumping number. (cid:3) From the above characterization we obtain that jumping numbers control the be-havior of all real powers of a given monomial ideal and are all rational numbers.
Theorem 5.9.
Let I be a monomial ideal. (1) All jumping numbers for I are rational. (2) All distinct real powers of I are given by rational exponents, i.e., for each r ∈ R + there exists r ′ ∈ Q so that I r = I r ′ . Moreover r ′ can be taken to be a jumpingnumber for I . (3) If r is a jumping number of I then nr is a jumping number for all n ∈ N . (4) If v is a vertex of N P ( I ) , then for all n ∈ N the number r n = n gcd( v , ··· ,v d ) is ajumping number of I . (5) The set of jumping numbers is a finite union of numerical semigroups scaledby reciprocals of positive integers, both of which can be computed from the facet(in)equalities of
N P ( I ) in (5.3) .Proof. (1) follows since Proposition 5.8 (3) yields that there is an integer solution toone of the equalities in the system A x ≥ r · . Since the entries of A are rationalnumbers, this makes r a rational combination of integers, hence r ∈ Q .(2) If r ∈ Q + is a jumping number, set r ′ = r . If r is not a jumping number, let r ′ = inf { u | u > r and u is a jumping number for I } . Notice first that r ′ is in fact the minimum of the set above, equivalently r ′ ∈ Q isa jumping number for I . Indeed, if this is not the case, then there is a sequence ofpairwise distinct jumping numbers { u n } n ∈ N converging to r ′ from the right. Since wehave assumed r ′ is not a jumping number, the exponential function with base I is rightcontinuous at r ′ , thus it must be the case that I u n = I r ′ for n ≫
0. This contradicts thatthe numbers u n are distinct jumping numbers, since distinct jumping numbers yield distinct real powers by Corollary 5.7. Another application of Corollary 5.7 togetherwith the observation that r is not a jumping number yields that the exponentiationfunction is constant on [ r, r ′ ], thus we conclude there is an equality I r = I r ′ .(3) follows since the condition on integer solutions to the system (5.3) in Proposition 5.8is preserved upon scaling the system by any natural number.(4) Each vertex v of N P ( I ) furnishes an integer solution to the system of (in)equalities(5.3) corresponding to each facet F i such that v ∈ F i . Scaling by r n we see that r n · v ∈ N d is an integer solution to the analogous system (P dj =1 a ij x j = r n ,r n min( F i , j ) ≤ x j ≤ r n max( F i , j ) for 1 ≤ j ≤ d. Proposition 5.8 yields that r n is a jumping number for I .For (5), for each 1 ≤ i ≤ d , let a ij ∈ Q + be the entries in the i -th row of the matrix A in Proposition 5.8 and let g i be the least common multiple of the denominators of theserational numbers. Then the equality in (5.3) can be written as P dj =1 ( g i a ij ) x j = g i r and having a non negative integer solution to this equation is equivalent to g i r ∈ S i ,where S i is the semigroup generated by the integers g i a ij for 1 ≤ j ≤ d . We considera (usually proper) subsemigroup of this set given by S i = { s ∈ N | s = d X j =1 ( g i a ij ) x j for some x j ∈ N such that u j s < x j < v j s, ∀ j } . With this notation, Proposition 5.8 can be rephrased to say that the set of jumpingnumbers for I is J = s [ i =1 g i S i . (cid:3) In regards to item (1) of Theorem 5.9 we observe that every nonnegative rationalnumber is a jumping number. Indeed if r = pq with p, q ∈ N , q = 0 then r is a jumpingnumber of I = ( x ) p .Item (2) of Theorem 5.9 yields a new description for the image of the exponentiationfunction with base I T = { I r | r ∈ Q is a jumping number for I } . Moreover, the elements of the set T listed above are pairwise distinct by Corollary 5.7.We end with a worked out example which illustrates the jumping numbers and realpowers of a particular monomial ideal using the criterion in Proposition 5.8 and itapplies to Theorem 5.9 (5). Example 5.10.
The monomial ideal I = ( x , x y , x y , y ) has Newton polyhedrondepicted in Figure 6 with vertices at (9 , , (4 , , (2 , , (0 , EAL POWERS OF MONOMIAL IDEALS 19 x y N P ( I ) p ≥ p + 2 q ≥ p + q ≥ p + 5 q ≥ q ≥ Figure 6.
The Newton polyhedron of ( x , x y , x y , y )We show that the jumping numbers of I are the elements of the following set(5.4) J = (cid:26) , i , j , k , | i, j, k ∈ N , i ≥ , j ≥ , k ∈ { , , , , , , } or k ≥ (cid:27) . The bounded faces of the Newton polyhedron F , F , F are shown in Figure 6 to-gether with the corresponding bounding inequalities for N P ( I ). Using the respectiveequations and the values min( F ,
1) = 2 , max( F ,
1) = 4 , min( F ,
2) = 3 , max( F ,
2) =5, min( F ,
1) = 0 , max( F ,
1) = 2 , min( F ,
2) = 5 , max( F ,
2) = 8 , min( F ,
1) = 4,max( F ,
1) = 9 , min( F ,
2) = 0 , max( F ,
2) = 3 and the criterion in Proposition 5.8(3), it follows that the jumping numbers are r ∈ Q + for which either one of the follow-ing three systems has integer solutions p, q ∈ N p + q = 7 r, r ≤ p ≤ r, r ≤ q ≤ r or3 p + 2 q = 16 r, ≤ p ≤ r, r ≤ q ≤ r or3 p + 5 q = 27 r, r ≤ p ≤ r, ≤ q ≤ r. The sets of such numbers r can be shown with the help of a software system to formthe union of the following three scaled semigroups J = 17 S ∪ S ∪ S , where S = 2 N + 3 N , S = 2 N , S = 3 N + 11 N + 19 N . Writing the the elements of eachsemigroup S , S , S explicitly yields the set displayed in equation (5.4) above.We list below the rational powers of I for exponents r ∈ (0 , I r = ( y, x ) r ∈ (0 , ]( y, x ) r ∈ ( , ]( y , xy, x ) r ∈ ( , ]( y , xy, x ) r ∈ ( , ]( y , xy, x ) r ∈ ( , ]( y , xy , x y, x ) r ∈ ( , ]( y , xy , x y, x ) r ∈ ( , ]( y , xy , x y, x ) r ∈ ( , ]( y , xy , x y, x ) r ∈ ( , ]( y , xy , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y , x y, x ) r ∈ ( , ]( y , xy , x y , x y , x y , x y , x y, x ) r ∈ ( , EAL POWERS OF MONOMIAL IDEALS 21
References [CEHH17] Susan M Cooper, Robert JD Embree, Huy T`ai H`a, and Andrew H Hoefel,
Symbolic pow-ers of monomial ideals , Proceedings of the Edinburgh Mathematical Society (2017),no. 1, 39–55.[Ciu] C˘at˘alin Ciuperc˘a, Integral closure of strongly golod ideals , Nagoya Math. J., 1–13.[Ciu20] ,
Derivations and rational powers of ideals , Arch. Math. (Basel) (2020), no. 2,135–145. MR 4055142[DLHTY04] Jes´us A. De Loera, Raymond Hemmecke, Jeremiah Tauzer, and Ruriko Yoshida,
Effectivelattice point counting in rational convex polytopes , J. Symbolic Comput. (2004), no. 4,1273–1302. MR 2094541[Gr¨u03] Branko Gr¨unbaum, Convex polytopes , vol. 2, Springer-Verlag New York, 2003.[GS] Daniel R. Grayson and Michael E. Stillman,
Macaulay2, a software system for researchin algebraic geometry , Available at .[HS06] Craig Huneke and Irena Swanson,
Integral closure of ideals, rings, and modules , vol. 13,Cambridge University Press, 2006.[Knu06] Allen Knutson,
Balanced normal cones and Fulton-MacPherson’s intersection theory ,Pure Appl. Math. Q. (2006), no. 4, Special Issue: In honor of Robert D. MacPherson.Part 2, 1103–1130. MR 2282415[Lew20] Jamess Lewis, Limit behavior of the rational powers of monomial ideals , arXiv:2009.05173(2020).[Rus07] David E. Rush,
Rees valuations and asymptotic primes of rational powers in Noetherianrings and lattices , J. Algebra (2007), no. 1, 295–320. MR 2290923[tt] 4ti2 team, . Indian Institute of Information Technology, Nagpur
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