Convex bodies and asymptotic invariants for powers of monomial ideals
João Camarneiro, Benjamin Drabkin, Duarte Fragoso, William Frendreiss, Daniel Hoffman, Alexandra Seceleanu, Tingting Tang, Sewon Yang
aa r X i v : . [ m a t h . A C ] J a n CONVEX BODIES AND ASYMPTOTIC INVARIANTS FORPOWERS OF MONOMIAL IDEALS
JO ˜AO CAMARNEIRO, BEN DRABKIN, DUARTE FRAGOSO, WILLIAM FRENDREISS,DANIEL HOFFMAN, ALEXANDRA SECELEANU, TINGTING TANG, SEWON YANG
Abstract.
Continuing a well established tradition of associating convex bodies tomonomial ideals, we initiate a program to construct asymptotic Newton polyhedrafrom decompositions of monomial ideals. This is achieved by forming a graded familyof ideals based on a given decomposition. We term these graded families powers sincethey generalize the notions of ordinary and symbolic powers. We introduce a novelfamily of irreducible powers.Irreducible powers and symbolic powers of monomial ideals are studied by meansof the corresponding irreducible polyhedron and symbolic polyhedron respectively.Asymptotic invariants for these graded families are expressed as solutions to linearoptimization problems on the respective convex bodies. This allows to establish alower bound on the Waldschmidt constant of a monomial ideal, an asymptotic invari-ant which can be defined using the symbolic polyhedron, by means of an analogousinvariant stemming from the irreducible polyhedron, which we introduce under thename of naive Waldschmidt constant. Introduction
This paper concerns invariants of monomial ideals which admit interpretations froma convex geometry perspective. Monomial ideals are ideals I that can be generated bymonomials in a polynomial ring R = K [ x , . . . , x n ] with coefficients in a field K .There is a well established tradition of associating convex bodies to monomial ideals.The preeminent example in this direction is the Newton polyhedron, which is the convexhull of all the exponent vectors of monomials in I . Several invariants of monomialideals can be read from their Newton polyhedron. For example the Hilbert-Samuelmultiplicity of an ideal primary to the homogeneous maximal ideal can be interpretedas the normalized volume of the complement of its Newton polyhedron in R n ≥ (see[Tei88]). A similar interpretation extends to arbitrary monomial ideals via the notionof j -multiplicity [JMn13]. For an introduction to the significance of Newton polyhedrain commutative algebra with emphasis on the role they play in integral closure werecommend [HS06, § § § Mathematics Subject Classification.
Primary 13F55, 13F20; Secondary 52B20, 14M25.
Key words and phrases. monomial ideals, irreducible decomposition, Newton polyhedron, symbolicpowers, linear programming, Waldschmidt constant.The second author was supported by the NSF RTG grant in algebra and combinatoricsat the University of Minnesota DMS–1745638. The sixth author was supported by NSFDMS–1601024. This work was completed in the framework of the 2020 Polymath program https://geometrynyc.wixsite.com/polymathreu . In this paper we focus our attention on associating convex bodies to decompositionsof a monomial ideal as an intersection of monomial ideals. Such a decomposition I = J ∩ · · · ∩ J s leads to considering on one hand a graded family of monomial ideals(1.1) I m = J m ∩ · · · ∩ J ms obtained by intersecting the powers of the components in the original decomposition.On the other hand it leads to considering a convex body C = N P ( J ) ∩ · · · ∩ N P ( J s )obtained by intersecting the Newton polyhedra of the components in the original de-composition. Our first main result shows that C can be understood as a limit of theNewton polyhedra for the family of ideals { I m } m ≥ , appropriately scaled. For thisreason we term C the asymptotic Newton polyhedron of the family { I m } . Theorem (Theorem 3.11) . If J , . . . , J s are monomial ideals and I m = J m ∩ · · · ∩ J ms ,then there is an equality of polyhedra C = N P ( J ) ∩ · · · ∩ N P ( J s ) = [ m ≥ m N P ( I m ) . The idea of associating an asymptotic Newton polyhedron to a graded family ofmonomial ideals has appeared previously in the context of Okounkov bodies attached toa graded linear series [LM09, KK12]. To our knowledge, asymptotic Newton polyhedraarising from ideal decompositions have not been studied before.Our work is motivated by the family of symbolic powers of a monomial ideal. Sym-bolic powers are a topic of sustained interest from a geometric as well as a combinatorialviewpoint. We recommend [DDSG + C which corresponds to the graded family ofsymbolic powers is know as the symbolic polyhedron . It was introduced in [CEHH17]and utilized in [BCG + irreducible polyhedra , which are easier to control than the symbolicpolyhedra. Our second main result captures the symbolic polyhedron between the twoother convex bodies discussed above. Theorem (Theorem 3.7) . For a monomial ideal I the following containments holdbetween its Newton (NP), symbolic (SP) and irreducible (IP) polyhedra: N P ( I ) ⊆ SP ( I ) ⊆ IP ( I ) . ONVEX BODIES AND ASYMPTOTIC INVARIANTS 3
We focus our efforts on invariants for graded families of ideals that can be read off therespective convex bodies by means of linear optimization. These invariants generalizethe notion of initial degree of a homogeneous ideal, by which we mean the least degreeof a nonzero element of the ideal, to an asymptotic counterpart. For symbolic powersthis asymptotic invariant is known in the literature as the
Waldschmidt constant . It hasbeen investigated in many works, among which we cite [Sko77, Wal77, HH13, BH10]and specifically for the case of monomial ideals in [CEHH17, BCG + naive Waldschmidt con-stant , which can be interpreted as the solution of a linear optimization problem onthe irreducible polyhedron and which gives an intrinsic lower bound on the Wald-schmidt constant. We make progress on obtaining further lower bounds on the naiveWaldschmidt constant reminiscent of a Chudnovsky-type inequality conjectured in[CEHH17, Conjecture 6.6]. Our results in this direction can be summarized by thefollowing inequalities, the first two of which reflect the previous theorem. Theorem (Theorem 4.28) . Let I ⊆ K [ x , . . . , x n ] be a monomial ideal with initialdegree α ( I ) = d . If d − ≡ k mod ( n ) , ≤ k < n , then the following inequalities aresatisfied by the Walshchmidt constant b α ( I ) and the naive Walshchmidt constant e α ( I ) α ( I ) ≥ b α ( I ) ≥ e α ( I ) ≥ ( n + d − − k )(2 n + d − − k ) n (2 n + d − − k ) ≥ (cid:22) α ( I ) + n − n (cid:23) . Our paper is organized as follows: in section 2 we discuss notions of powers arisingfrom decompositions of monomial ideals, in section 3 we associate asymptotic Newtonpolyhedra to the families introduced previously, in section 4 we define the asymptoticinitial degrees for our graded families, we express these invariants by means of linearoptimization, and we derive bounds on their values.2.
Decompositions of monomial ideals and notions of powers
Let N denote the set of nonnegative integers. For vectors a = ( a , . . . , a n ) ∈ N n weuse the shorthand notation x a := x a · · · x a n n and thus any monomial ideal is describedby a finite set of vectors a , . . . , a ℓ ∈ N n as I = ( x a , . . . , x a n ) . An ideal J is called irreducible if whenever there is a decomposition J = J ∩ J ,with J , J ideals, then J = J or J = J . An irreducible decomposition of an ideal I is an expression I = J ∩ J ∩ · · · ∩ J s where J i are irreducible ideals for 1 ≤ i ≤ s .Such a decomposition is called irredundant if none of the J i can be omitted from thisexpression. Emmy Noether showed in [Noe21] that every ideal I in a noetherian ringadmits an irredundant irreducible decomposition. Moreover, although the numberof components in any irredundant irreducible decomposition of I is the same, thecomponents themselves are in general not unique.A monomial ideal J is irreducible if and only if it is generated by pure powers fora subset of the variables, i.e., J = ( x a i , · · · , x a t i t ). By contrast to arbitrary ideals, POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM any monomial ideal has a unique irredundant decomposition into irreducible monomialideals. Irreducible decompositions are special cases of primary decompositions. Bothfor monomial and for arbitrary ideals they possess the advantage of being much moreeasily computable in an algorithmic fashion; see [MS05, § I , the symbolic powers retain only the components of the ordinary powers whoseradicals are contained in some associated prime of I . When I is a radical ideal and K is a field of characteristic 0, the m -th symbolic power of I encodes the polynomialfunctions vanishing on the variety cut out by I to order at least m . Definition 2.1.
Let R be a noetherian ring and I an ideal in R . The m -th symbolicpower of I is the ideal I ( m ) = \ P ∈ Ass(
R/I ) I m R p ∩ R. Recall that the set of associated primes, denoted Ass( I ), of an ideal I in a noetherianring is finite. We view it as a poset with respect to containment. A minimal elementof this poset is called a minimal prime of I and the non minimal elements are calledembedded primes.We note that the symbolic powers of monomial ideals admit an alternate descriptionwhich is even more closely related to their primary decomposition. Lemma 2.2 ([HHT07, Lemma 3.1], [CEHH17, Theorem 3.7]) . If I is a monomial idealwith monomial primary decomposition I = Q ∩ Q ∩ · · · ∩ Q s , set Max( I ) to denote theset of maximal elements in the poset of associated primes of I and for each P ∈ Max( I ) denote Q ⊆ P = \ √ Q i ⊆ P Q i . Then the symbolic powers of I can be expressed as follows I ( m ) = \ P ∈ Max( I ) ( Q ⊆ P ) m . Remark . The above Lemma employs the decomposition I = T P ∈ Max( I ) Q ⊆ P . Wewill call this a combined primary decomposition for I . The ideals Q ⊆ P are uniquelydetermined by I and P and are independent of the primary decomposition in thestatement of Lemma 2.2. This follows from the identity Q ⊆ P = IR P ′ ∩ R , where P ′ isthe prime monomial ideal generated by the variables of R that are not in P . Example 2.4. If I is a monomial ideal with no embedded primes and I = Q ∩ · · ·∩ Q s is a primary decomposition, then the symbolic powers of I are given for all integers m ≥ I ( m ) = Q m ∩ Q m ∩ · · · ∩ Q ms . In this paper we introduce a notion of irreducible powers for monomial ideals, whichparallels the behavior in Example 2.4.
ONVEX BODIES AND ASYMPTOTIC INVARIANTS 5
Definition 2.5.
Let I be a monomial ideal with a monomial irreducible decompositiongiven by I = J ∩ J ∩ · · · ∩ J s . For integers m ≥
1, the m -th irreducible power of I isthe ideal I { m } = J m ∩ J m ∩ · · · ∩ J ms . It is easy to see that the definition above does not depend on the choice of irreducibledecomposition, nor on whether it is irredundant.
Remark . If I is a square-free monomial ideal then the irredundant irreducible de-composition of I coincides with the combined primary decomposition thus the symbolicpowers and irreducible powers of square-free monomial ideals coincide.More generally if the components in the irredundant irreducible decomposition of I have distinct radicals, then the symbolic powers and irreducible powers coincide.One similarity between the symbolic and irreducible powers is that they both formgraded families. A graded family of ideals { I m } m ∈ N is a collection of ideals that satisfies I a · I b ⊆ I a + b for all pairs a, b ∈ N . Lemma 2.7.
The irreducible powers of a monomial ideal form a graded family, i.e.,any nonnegative integers a, b give rise to a containment I { a } · I { b } ⊆ I { a + b } . Proof.
The containment follows easily from Definition 2.5. (cid:3)
In many ways, the irreducible powers of monomial ideals resemble closely the sym-bolic powers of square-free monomial ideals. A similarity between irreducible powers ofmonomial ideals and symbolic powers of square-free monomial ideals is that their asso-ciated primes are among the associated primes of I . This is not the case for symbolicpowers of arbitrary ideals; see Remark 2.9. Lemma 2.8.
Let I be a monomial ideal. Then for each integer m ≥ there arecontainments I m ⊆ I ( m ) ⊆ I { m } and Ass( I { m } ) ⊆ Ass( I ) .Proof. The containments I m ⊆ I ( m ) ⊆ I { m } follow from the definition of symbolicpowers Definition 2.1 for the former and from Lemma 2.2 for the latter. In detail, if I = J ∩ · · · ∩ J s is an irredundant irreducible decomposition, then for each 1 ≤ i ≤ s there exists a prime P i ∈ Max( I ) such that √ J i ⊆ P i . Then we see from Lemma 2.2that Q ⊆ P i ⊆ J i and so we deduce I ( m ) = \ P ∈ Max( I ) Q m ⊆ P ⊆ s \ i =1 Q m ⊆ P i ⊆ s \ i =1 J mi = I { m } . Now let I = J ∩ · · · ∩ J s be an irredundant irreducible decomposition with p i = √ J i .Since each irreducible ideal J i is generated by a regular sequence of pure powers of thevariables, it follows that Ass( J mi ) = Ass( J i ) = { p i } for each i and thus we obtainAss( I { m } ) = Ass( J m ∩ · · · ∩ J ms ) ⊆ { P , . . . , P s } = Ass( I ) . (cid:3) POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM
Remark . Lemma 2.8 reveals that the irreducible powers of monomial ideals enjoy aproperty that the symbolic powers of monomial ideals which possess embedded primesdo not enjoy. Specifically, it is not in general true that the associated primes of thesymbolic powers are restricted to a subset of Ass( I ). For example, the ideal I = ( x, y ) ∩ ( x, z ) ∩ ( x, w ) ∩ ( y, z ) ∩ ( y, w ) ∩ ( z, w ) ∩ ( x, y, z, w ) has the property that Ass( I (2) ) contains the primes ( x, y, z ) , ( x, y, w ) , ( x, z, w ) and( y, z, w ) in addition to the associated primes of I .3. Convex bodies associated to powers of monomial ideals
The symbolic and irreducible polyhedra of a monomial ideal.
In thissection we define new convex bodies associated to decompositions of monomial ideals.In our main cases of interest these convex bodies will be polyhedra. A polyhedroncan be defined in two different manners, either as convex hulls of a set of points inEuclidean space or as a finite intersection of half spaces. All polyhedra considered inthis section will be unbounded.
Definition 3.1.
For a monomial ideal I , the Newton polyhedron of I , denoted N P ( I ),is the convex hull of the exponent vectors for all the monomials in IN P ( I ) = convex hull { a ∈ N n | x a ∈ I } . One of the useful properties of Newton polyhedra is that they scale linearly upontaking ordinary powers of ideals, namely the following identity holds for all m ∈ N : N P ( I m ) = mN P ( I ) . The situation becomes more complicated upon considering Newton polyhedra forthe symbolic powers or for the irreducible powers, as taking Newton polyhedra doesnot commute with intersections of ideals. Specifically, there is always a containment
N P ( J ∩ · · · ∩ J s ) ⊆ N P ( J ) ∩ · · · ∩ N P ( J s ) , but this rarely becomes an equality. However, we shall see that there is an asymptoticsense in which Newton polyhedra can be taken to commute with intersections of ideals.To elaborate on this, we introduce two more convex bodies, one corresponding to eachof the notions of symbolic and irreducible powers introduced in the previous section.Following [CEHH17, Definition 5.3], which in turn takes inspiration from Lemma 2.2,we define a symbolic polyhedron associated to a monomial ideal. Definition 3.2.
The symbolic polyhedron of a monomial ideal I with primary decom-position I = Q ∩ · · · ∩ Q s is SP ( I ) = \ P ∈ Max( I ) N P ( Q ⊆ P ) where Q ⊆ P = \ √ Q i ⊆ P Q i . Similarly, with inspiration taken from Definition 2.5, we introduce a new convexbody termed the irreducible polyhedron.
ONVEX BODIES AND ASYMPTOTIC INVARIANTS 7
Definition 3.3.
The irreducible polyhedron of a monomial ideal I with irreducibledecomposition I = J ∩ · · · ∩ J s is IP ( I ) = N P ( J ) ∩ · · · ∩ N P ( J s ) . Remark . If I is a square-free monomial ideal or more generally an ideal such that theradicals of the irredundant irreducible components are distinct, then IP ( I ) = SP ( I ).We supplement the description of the irreducible polyhedron in Definition 3.3 byproviding equations for hyperplanes supporting the facets of the polyhedron, which weterm bounding hyperplanes. We term the linear inequalities describing a polyhedronas an intersection of half spaces its bounding inequalities.Establishing the bounding inequalities for the symbolic polyhedron of an arbitrarymonomial ideal is generally an infeasible task. However, the analogous task is consid-erably easier for the irreducible polyhedron. Lemma 3.5.
The bounding inequalities for the irreducible polyhedron of a monomialideal I are read off a monomial irreducible decomposition I = J ∩ · · · J s as follows: iffor each ≤ i ≤ s we have J i = ( x a i i , . . . , x a ihi ih i ) , where x ij ∈ { x , . . . , x n } , then setting y ij = y k if and only if x ij = x k yields that IP ( I ) is the set of points y = ( y , . . . , y n ) ∈ R n which satisfy the system of inequalities (3.1) a y + · · · + a h y h ≥ ... a s y s + · · · + a shs y sh s ≥ y , . . . , y n ≥ . Proof.
For each irreducible component J i we have that N P ( J i ) is the complementwithin the positive orthant of R n of a simplex with vertices given by the origin and theexponent vectors of the minimal monomial generators x a i i , . . . , x a ihi ih i for J i , that is, N P ( J i ) = ( a i y i + · · · + a ihi y ih i ≥ y , . . . , y d ≥ . Equation (3.1) collects together all the inequalities of each
N P ( J i ) according to Definition 3.3. (cid:3) We next give an account of the containments between the three polyhedra discussedabove. This is based upon observing that more refined decompositions of an ideal willyield larger polyhedra. We make this precise in the following lemma.
Lemma 3.6.
Assume given two collections of monomial ideals I , . . . , I t and J , . . . , J s such that the latter refines the former, that is, for each ≤ j ≤ s there exists ≤ i j ≤ t such that I i j ⊆ J j . Then there is a containment of polyhedra N P ( I ) ∩ · · · ∩ N P ( I t ) ⊆ N P ( J ) ∩ · · · ∩ N P ( J s ) . POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM
Proof.
Employing the hypothesis that for each 1 ≤ j ≤ s there exists 1 ≤ i j ≤ t such that I i j ⊆ J j , we deduce that N P ( I j i ) ⊆ N P ( J i ). Thus we obtain the desiredcontainments N P ( I ) ∩ · · · ∩ N P ( I t ) ⊆ N P ( I j i ) ∩ · · · ∩ N P ( I j s ) ⊆ N P ( J ) ∩ · · · ∩ N P ( J s ) . (cid:3) With this key ingredient in hand we established the containments between the threetypes of polyhedra considered in this paper.
Theorem 3.7.
For any monomial ideal I the following containments hold: N P ( I ) ⊆ SP ( I ) ⊆ IP ( I ) . Proof.
Let I = J ∩ · · · ∩ J s be a monomial irreducible decomposition and note thatit is also a primary decomposition. Hence the combined primary decomposition I = T P ∈ Max( I ) Q ⊆ P can be computed using Q ⊆ P = T √ J i ⊆ P J i according to Remark 2.3.This shows that the irreducible decomposition refines the combined decomposition inthe sense that for each 1 ≤ j ≤ s there exists P j ∈ Max( I ) such that Q ⊆ P j ⊆ J j .Indeed, this is the case for each P ∈ Max( I ) such that p J j ⊆ P and such a primeexists by finiteness of the poset Ass( I ).Now we apply Lemma 3.6 to obtain the second desired containment SP ( I ) = \ P ∈ Max( I ) N P ( Q ⊆ P ) ⊆ s \ j =1 N P ( J j ) = IP ( I ) . The remaining containment,
N P ( I ) ⊆ SP ( I ) can be deduced by applying Lemma 3.6to the trivial decomposition I = I and its refinement I = T P ∈ Max( I ) Q ⊆ P . (cid:3) Asymptotic Newton polyhedra for graded families of monomial ideals.
Let { I m } m ≥ denote a graded family of monomial ideals. By definition, such a familysatisfies containments I a · I b ⊆ I a + b for each pair a, b ∈ N . We define a convex body cap-turing the asymptotics of each such family. This construction bears some resemblanceto the Newton-Okounkov bodies of [LM09, KK12]. A similar construction appears in[May14] but for a different family of monomial ideals. Definition 3.8.
Given a graded family of monomial ideals I := { I m } m ≥ , the limitingbody associated to this family is C ( I ) = [ m →∞ m N P ( I m ) . If the limiting body is a polyhedron, we call it the asymptotic Newton polyhedron associ-ated to the family I . For an example of non polyhedral limiting body see Remark 3.14. Example 3.9.
For the family of ordinary powers { I m } m ∈ N of a monomial ideal, thesequence m N P ( I m ) is constant, each term being equal to N P ( I ). Thus the asymptoticNewton polyhedron associated to the family of ordinary powers of I is none other thanthe Newton polyhedron of I itself. ONVEX BODIES AND ASYMPTOTIC INVARIANTS 9
Lemma 3.10.
The limiting body for a graded family of monomial ideals is a convexbody.Proof.
Let I = { I m } m ≥ be a graded family of monomial ideals. This implies that( I m ) k ⊆ I mk for all k ≥ m N P ( I m ) ⊆ mk N P ( I mk ). Now let a , b ∈ P ( I )and suppose a ∈ a N P ( I a ) and b ∈ b N P ( I b ). Then by the preceding argument a , b are points of the same convex body ab N P ( I ab ), which is a subset of C ( I ). Thus anyconvex combination of a , b is also in ab N P ( I ab ) and hence in C ( I ). (cid:3) We consider special types of graded families arising from decompositions into mono-mial normal ideals. In this scenario the limiting body is a polyhedron that can bedescribed explicitly.
Theorem 3.11.
Let I be a monomial ideal equipped with a decomposition into mono-mial ideals I = J ∩ · · · ∩ J s . Consider the graded family I = { I m } m ≥ where I m = J m ∩ · · · ∩ J ms . Then the asymptotic Newton polyhedron of this family can be described equivalently as C ( I ) = N P ( J ) ∩ · · · ∩ N P ( J s ) . Proof.
Let Q = T si =1 N P ( J i ). First we see that for each m ≥ m N P ( I m ) ⊆ Q . Indeed, since I m = J m ∩ · · · ∩ J ms , we have N P ( I m ) ⊆ T si =1 N P ( J mi ) = m · Q . This yields the inclusion C ( I ) ⊆ Q .Next, for the opposite containment, we will show that for each 1 ≤ i ≤ s everypoint of Q ∩ Q n is in C ( I ). This is enough to guarantee the containment Q ⊆ C ( I ),since all the vertices of the former polyhedron have rational coordinates. Thus assume a ∈ Q ∩ Q n and hence a ∈ N P ( J i ) ∩ Q n for 1 ≤ i ≤ s . Fix i and let v , . . . , v t be thevertices of the polyhedron N P ( J i ). Since these correspond to a subset of the monomialgenerators of J i we notice that v j ∈ Z n for 1 ≤ j ≤ t and N P ( J i ) = convex hull { v , . . . , v t } + R n ≥ . By a version of Cartheodory’s theorem for unbounded polyhedra [CEHH17, Theorem5.1] we can write a = n X j =1 λ j v i j + n X j =1 c j e j , where λ j , c j ≥ P nj =1 λ j = 1. Let m be the leastcommon multiple of the denominators of the rational numbers λ j , c j for 1 ≤ j ≤ n .Multiplying the equation displayed above by m we deduce the identity m a = n X j =1 mλ j v i j + n X j =1 mc j e j , where P nj =1 mλ j = m and mλ j ∈ N for 1 ≤ j ≤ t . This yields that x m a ∈ J mi andsince the argument holds for each i , we deduce that x m a ∈ T si =1 J mi = I m . Based onthis we see that a ∈ m N P ( I m ) ⊆ Q , as desired. (cid:3) The previous theorem allows us to identify the symbolic and irreducible polyhe-dra as asymptotic Newton polyhedra for the graded families of symbolic powers andirreducible powers of a monomial ideals respectively.
Corollary 3.12.
Let I be a monomial ideal. Then the asymptotic Newton polyhedronof the family of symbolic powers { I ( m ) } m ≥ is the symbolic polyhedron SP ( I ) .Proof. This follows by applying Theorem 3.11 to the family of symbolic powers, whichis defined in terms of the decomposition I = T P ∈ Max( I ) Q ⊆ P with Q ⊆ P = IR P ∩ R .Together with Definition 3.2, this result yields the claim. (cid:3) Corollary 3.13.
Let I be a monomial ideal. Then the asymptotic Newton polyhedronof the family { I { m } } m ≥ of irreducible powers is the irreducible polyhedron IP ( I ) .Proof. By Definition 2.5, we are in the setting of Theorem 3.11 where the family ofirreducible powers is defined in terms of a monomial irreducible decomposition I = T si =1 J i . Thus Theorem 3.11 and Definition 3.3 yield the desired conclusion. (cid:3) Remark . Limiting bodies for arbitrary graded families of monomial ideals can failto be polyhedral. Consider for example, the family I of monomial ideals I m ⊆ k [ x, y ]such that x a y b ∈ I m if and only if ab ≥ m . Then C ( I ) = { ( a, b ) | ab ≥ , a ≥ , b ≥ } is a non-polyhedral convex region in R .4. Asymptotic invariants for families of monomial ideals
Asymptotic initial degrees and linear optimization.
In this section we de-fine asymptotic invariants for graded families of monomial ideals which are derivedfrom their initial degree. For a homogeneous ideal I the initial degree, denoted α ( I ),is the least degree of a non zero element of I . Definition 4.1.
For a graded family of ideals I = { I m } m ≥ define the asymptoticinitial degree of the family to be α ( I ) = lim m →∞ α ( I m ) m . Remark . The existence of the limit in Definition 4.1 is ensured by Farkas’s lemma[Far02] by means of the subadditivity of the sequence of initial degrees { α ( I m ) } m ≥ .In turn, the subadditivity arises from the graded family property, as the containments I a I b ⊆ I a + b give rise to inequalities α ( I a + b ) ≤ α ( I a ) + α ( I b ) for all integers a, b ≥ Definition 4.3.
Let I be a homogeneous ideal. The asymptotic initial degree of thefamily of symbolic powers { I ( m ) } m ≥ is termed the Waldschmidt constant of I anddefined as follows b α ( I ) = lim m →∞ α ( I ( m ) ) m . ONVEX BODIES AND ASYMPTOTIC INVARIANTS 11
Applying the definition for asymptotic initial degree of the family of irreduciblepowers yields a novel invariant.
Definition 4.4.
Let I be a monomial ideal. The asymptotic initial degree of the familyof irreducible powers { I { m } } m ≥ is termed the naive Waldschmidt constant of I anddefined as follows e α ( I ) = lim m →∞ α ( I { m } ) m . We now show that asymptotic initial degrees for families of monomial ideals aresolutions to an optimization problem. Note that the initial degree of a monomial ideal I can be expressed as the solution of a linear programming problem in the followingmanner:(4.1) α ( I ) = min { y + · · · + y n | ( y , · · · , y n ) ∈ N P ( I ) } . This is because the optimal solution is attained at a vertex of
N P ( I ) and the verticesof N P ( I ) correspond to a subset of the minimal generators of I . We see below that theasymptotic initial degree for a graded family of monomial ideals can also be expressedas an optimization problem. Moreover, the feasible set is the limiting body of thefamily as defined in Definition 3.8. Theorem 4.5.
Let I = { I m } m ≥ be a graded family of monomial ideals. Then α ( I ) isthe solution of the following optimization problem minimize y + · · · + y n subject to ( y , · · · , y n ) ∈ C ( I ) ,where C ( I ) denotes the closure of C ( I ) in the Euclidean topology of R n .Proof. Recall from Remark 4.2 the alternate definition α ( I ) = inf m ≥ α ( I m ) m . From(4.1) we deduce α ( I m ) = min { y + · · · + y n | ( y , · · · , y n ) ∈ N P ( I m ) } , hence there areequalities α ( I m ) m = min { y + · · · + y n | ( y , · · · , y n ) ∈ m N P ( I m ) } . Now passing to the infimum and denoting the solution of the optimization problem inthe statement of the theorem by β , we deduce α ( I ) = inf m ≥ α ( I m ) m = inf m ≥ (cid:26) min { y + · · · + y n | ( y , · · · , y n ) ∈ m N P ( I m ) } (cid:27) = inf { y + · · · + y n | ( y , · · · , y n ) ∈ [ m ≥ m N P ( I m ) = C ( I ) } = min { y + · · · + y n | ( y , · · · , y n ) ∈ C ( I ) } = β. (cid:3) Applying this theorem, we are able to recover a result relating the Waldschmidtconstant to the symbolic polyhedron from [CEHH17, Corollary 6.3] and [BCG + Corollary 4.6.
The Waldschmidt constant of a monomial ideal I is the solution to thefollowing linear optimization problem with feasible region given by its symbolic polyhe-dron: minimize y + · · · + y n subject to ( y , · · · , y n ) ∈ SP ( I ) . Corollary 4.7.
The naive Waldschmidt constant of a monomial ideal I is the solutionto the following linear optimization problem with feasible region given by its irreduciblepolyhedron: minimize y + · · · + y n subject to ( y , · · · , y n ) ∈ IP ( I ) . From the containments in Theorem 3.7 and the above two corollaries we deduceinequalities relating the various asymptotic initial degrees.
Proposition 4.8.
For any monomial ideal I there is an inequality e α ( I ) ≤ b α ( I ) ≤ α ( I ) .Proof. Theorem 3.7 gives
N P ( I ) ⊆ SP ( I ) ⊆ IP ( I ) and taking the minimum value ofthe sum of the coordinates of any point in these convex bodies turns containments intoreverse inequalities. These minimum values are e α ( I ) for SP ( I ) and b α ( I ) for IP ( I ) byCorollary 4.6 and Corollary 4.7 respectively and α ( I ) for N P ( I ) by equation (4.1). (cid:3) Under special circumstances, we may also deduce equality between the asymptoticinvariants discussed above.
Proposition 4.9. If I is a monomial ideal whose irredundant irreducible componentshave distinct radicals, then b α ( I ) = e α ( I ) . In particular, this equality holds when I issquare-free.Proof. The equality follows from Corollary 4.6 and Corollary 4.7 after noticing that SP ( I ) = IP ( I ) under the given hypothesis, according to Remark 2.6. (cid:3) Lower bounds on asymptotic initial degrees.
Proposition 4.8 establishesthat the initial degree of I is an upper bound for both e α ( I ) and b α ( I ). This upperbound is attained, for example, when I is an irreducible monomial ideal, hence acomplete intersection, and thus I { m } = I ( m ) = I m for each integer m ≥ e α ( I ) and b α ( I ). Theseare formulated in terms of the initial degree of I and an invariant termed big-height ,which is defined as follows:big-height( I ) = max { ht( P ) | P ∈ Ass( I ) } . For the Waldschmidt constant the following lower bounds are either known or conjec-tured to be true. An inequality similar to Proposition 4.10 first appeared in [Sko77,Wal77] and was proven in the generality given here in [HH13].
Proposition 4.10 (Skoda bound) . For any homogeneous ideal I the following inequal-ity holds b α ( I ) ≥ α ( I )big-height( I ) . ONVEX BODIES AND ASYMPTOTIC INVARIANTS 13
The following conjecture proposing a stronger bound has been formulated in [CEHH17,Conjecture 6.6].
Conjecture 4.11 (Chudnovsky bound) . For a monomial ideal I the following inequal-ity is obeyed: b α ( I ) ≥ α ( I ) + big-height( I ) − I ) . The Chudnovsky bound in Conjecture 4.11 is known to hold true for square-freemonomial ideals cf. [BCG +
16, Theorem 5.3].We now proceed to convey lower bounds for the asymptotic irreducible degree e α ( I ),by analogy to the bounds discussed above for b α ( I ). First we prove a Skoda-type lowerbound. Theorem 4.12.
Let I be a monomial ideal. Then e α ( I ) ≥ α ( I )big-height( I ) .Proof. We proceed by adapting the proof of [BCG +
16, Theorem 5.3].Let I be a monomial ideal with big-height e and irredundant irreducible decom-position I = J ∩ · · · ∩ J s . We know from Corollary 4.7 that e α ( I ) is the minimumvalue of y + · · · + y n over IP ( I ) and from Lemma 3.5 that, if for each i = 1 , . . . , sJ i = ( x a i i , . . . , x a ihi ih i ), then the bounding inequalities for this polyhedron are IP ( I ) = a y + · · · + a h y h ≥ · · · a s y s + · · · + a shs y sh s ≥ y , . . . , y n ≥ . To establish the claim, if suffices to show that, for every t ∈ IP ( I ), we have t + · · · + t n ≥ α ( I )big-height( I ) = α ( I ) e which implies by taking infimums that e α ( I ), the minimal value of the sum of coordinatesof any point in IP ( I ), will satisfy the desired inequality.We find a subset of the components of t whose sum is greater or equal to α ( I ) /e .To start, consider a bounding inequality corresponding to an irreducible component J i . This takes the form 1 a i y i + · · · + 1 a ih i y ih i ≥ , where h i ≤ big-height( I ) is the height of the monomial prime ideal √ J i . The displayedinequality implies that for y = t at least one of the terms is greater or equal to e , i.e., t ij ≥ a ij e for some 1 ≤ j ≤ h i and some integer a ij ≥ . Now, suppose we have found t k , t k , . . . , t k m such that t k ≥ a k e , ..., t k m ≥ a km e , butwe have a k + a k + · · · + a k m < α ( I ). Consider the monomial x a k k x a k k · · · x a km k m . By theassumption, it has degree smaller than α ( I ), so it’s not an element of I . Therefore, thereis some component J i that does not contain this monomial. Repeating the previous argument, from the corresponding inequality we obtain t k m +1 ≥ a ij e for some indices i, j . There are two possibilities depending on whether k m +1 ∈ { k , . . . , k m } or not:(1) t k m +1 is not one of t k , t k , . . . , t k m . Then we set a k m +1 := a ij and we observethat a k + a k + · · · + a k m + a k m +1 > a k + a k + · · · + a k m . (2) t k m +1 is one of t k , t k , . . . , t k m , say t k m +1 = t k ℓ . Since the monomial x a k k x a k k · · · x a km k m is not contained in J i , it must be that a ij > a k ℓ . Therefore, we can replace theinequality t k j ≥ a kℓ e by the stronger inequality t k j = t k m +1 ≥ a ij e . Updating thevalue of a k ℓ to a k ℓ := a ij , this increases the value of the sum a k + a k + · · · + a k m .Since in either case the value of the sum a k + a k + · · · + a k m or a k + a k + · · · + a k m +1 increases, we see that iterating this procedure eventually results in positive integers a k , . . . a k m such that a k + a k + · · · + a k m ≥ α ( I )as well as in a corresponding set of coordinates of t that satisfy the desired inequality t k + · · · + t k m ≥ a k + · · · + a k m e ≥ α ( I ) e . (cid:3) We remark that the direct analogue of the Chudnovsky bound in Conjecture 4.11fails for g α ( I ), as shown by the following example. Example 4.13.
Consider the ideal I = ( x , xy, y ) = ( x , y ) ∩ ( x, y ) ⊆ k [ x, y ]. Theinitial degree is α ( I ) = 2, the big height is big-height( I ) = 2 and the naive Waldschmidtconstant is e α ( I ) = 4 /
3. The value of the last invariant follows by observing that IP ( I ) = N P ( x , y ) ∩ N P ( x, y ) has vertices at (2 , , (0 ,
2) and (2 / , / e α ( I ) = 43 <
32 = α ( I ) + big-height( I ) − I ) . However, there are many ideals for which the expression in the Chudnovsky conjec-ture Conjecture 4.11 does indeed provide a lower bound on e α ( I ). In the next sectionwe give a modified Chudnovsky-type lower bound for e α ( I ) that applies to all monomialideals I .4.3. Powers of the maximal ideal.
In this section we determine the naive Wald-schmidt constant for the powers of the homogeneous maximal ideal. We will later usethis to deduce a Chudnovsky-type lower bound on the naive Waldschmidt constant ofideals primary to the homogeneous maximal ideal.In the following, we denote by m n the homogeneous maximal ideal ( x , . . . , x n ) ofthe polynomials ring R = k [ x , . . . , x n ]. We start by establishing the irredundantirreducible decompositions for the ordinary powers of m n . ONVEX BODIES AND ASYMPTOTIC INVARIANTS 15
Notation . For a positive integer s we denote by P n ( s ) be the set of partitions of s into n nonempty parts P n ( s ) = ( ( a , . . . , a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ N , a i ≥ , n X i =1 a i = s ) . Proposition 4.15.
Given an integer d ≥ , the irredundant irreducible decompositionof the ideal m dn = ( x , . . . , x n ) d is (4.2) m dn = \ ( a ,...,a n ) ∈ P n ( d + n − ( x a , . . . , x a n n ) Proof.
Let x b = x b · · · x b n n ∈ T ( a ,...,a n ) ∈ P n ( d + n − ( x a , . . . , x a n n ), and suppose x b m dn .Then there are inequalities n X i =1 b i < d and thus d − n X i =1 b i ≥ . Let a i = ( b i + 1 1 ≤ i < nb n + d − P ni =1 b i i = n which implies a i ≥ b i + 1 for all i from 1 to n . From this, we have an equality n X i =1 a i = n − X i =1 ( b i + 1) + b n + d − n X i =1 b i = n X i =1 b i + n − d − n X i =1 b i = d + n − a , . . . , a n ) ∈ P n ( d + n − a i > b i for all i , x b ( x a , . . . , x a n n ),a contradiction. As a result, we obtain the containment(4.3) \ ( a ,...,a n ) ∈ P n ( d + n − ( x a , . . . , x a n n ) ⊆ m dn Now take x b ∈ m dn , and suppose x b T ( a ,...,a n ) ∈ P n ( d + n − ( x a , . . . , x a n n ). Then thereis some Q = ( x c , . . . , x c n n ) with ( c , . . . , c n ) ∈ P n ( d + n −
1) such that x b Q . Thisimplies that c i > b i for all 1 ≤ i ≤ n , so c i ≥ b i + 1. But then we deduce d + n − n X i =1 c i ≥ n X i =1 ( b i + 1) = n + n X i =1 b i = n + d > d + n − , which is of course a contradiction. Hence(4.4) m dn ⊆ \ ( a ,...,a n ) ∈ P n ( d + n − ( x a , . . . , x a n n )Combining (4.3) and (4.4), we obtain our desired result. (cid:3) Having determined the irredundant irreducible decomposition of m dn , we deduce thebounding inequalities for the irreducible polyhedron from Lemma 3.5. Corollary 4.16.
The irreducible polyhedron of of the ideal m dn is bounded by the in-equalities ( a y + · · · + a n y n ≥ for ( a , . . . , a n ) ∈ P n ( d + n − y i ≥ for ≤ i ≤ n. Next we give closed formulas for the naive Waldschmidt constant for the powers ofthe maximal ideal. We first single out the case when this value is an integer.
Proposition 4.17.
Suppose d ≡ n . Then the naive Waldschmidt constant of m dn is e α ( m dn ) = d + n − n ∈ N . Proof. If d ≡ n , then d + n − n ; in other words, d + n − n is an integer, say m . The ideal ( x m , . . . , x mn ) is in the irreducible decomposition of m dn by Proposition 4.15. The bounding inequality corresponding to N P ( x m , . . . , x mn )1 m y + · · · + 1 m y n ≥ ⇒ y + · · · + y n ≥ m indicates that e α ( m dn ) ≥ m . Consider the vector ( mn , . . . , mn ) in R n that clearly has sumof coordinates m . For each component ( x a , . . . , x a n n ) in the irreducible decompositionthere is an identity 1 a (cid:16) mn (cid:17) + · · · + 1 a n (cid:16) mn (cid:17) = mn n X i =1 a i . Note that the value n P ni =1 1 a i is the inverse of the harmonic mean of the set a , . . . , a n and the arithmetic mean for this set is m . Hence the inequality relating these meansyields 1 a (cid:16) mn (cid:17) + · · · + 1 a n (cid:16) mn (cid:17) ≥ m (cid:18) m (cid:19) = 1 . Therefore the point ( mn , . . . , mn ) is part of the Newton polyhedron of each irreduciblecomponent of m dn , i.e., ( mn , . . . , mn ) ∈ IP ( m dn ). Since it was shown before that the leastvalue of the sum of coordinates of points in this polyhedron is at least m , and the pointidentified above has sum of coordinates exactly m , we conclude that e α ( m dn ) = m . (cid:3) Remark . Note that the right hand side in the equality displayed in Proposition 4.17matches the Chudnovsky lower bound α ( m n )+big-height( m n ) − m n ) ; see Conjecture 4.11.Before we continue our analysis, we state a simple fact that will become useful later.The proof is omitted, since it is a direct verification. Lemma 4.19. If x, y ∈ R ≥ are such that x ≥ y − , then x + y ≥ x − + y +1 . An interesting consequence of the above lemma is presented below.
Proposition 4.20.
Fix an integer s > . The minimum value of the function f ( a ) = a + · · · + a n , where the tuple ( a , . . . , a n ) ranges over P n ( s ) is attained by a partitionwhere the parts differ by at most one, that is, | a i − a j | ≤ for all ≤ i < j ≤ n . ONVEX BODIES AND ASYMPTOTIC INVARIANTS 17
Proof.
The result follows by noticing that modifying a partition in a manner thatdecreases the difference between the parts results in an increase of the objective function f . Indeed, Lemma 4.19 insures that if ( a , . . . , a n ) ∈ P n ( s ) has two parts a i , a j suchthat | a i − a j | >
1, then the partition ( a ′ , . . . , a ′ n ) ∈ P n ( s ) obtained by setting a ′ k = a k whenever k
6∈ { i, j } , a ′ i = max { a i , a j } − a ′ j = min { a i , a j } + 1 satisfies f ( a ) = n X ℓ =1 a ℓ ≥ n X ℓ =1 a ′ ℓ = f ( a ′ ) . (cid:3) Now we turn to the determination of e α ( m dn ) for arbitrary values of d . Theorem 4.21.
Suppose d is a positive integer and d − ≡ k mod n , ≤ k < n .Then e α ( m dn ) = ( n + d − − k )(2 n + d − − k ) n (2 n + d − − k ) . Proof.
First note that if k = 0, then the formula becomes e α ( m dn ) = (2 n + d − − n + d − − n (2 n + d − − n + d − n , which is in accordance with Proposition 4.17. Therefore, let us consider the case k > d − an + k and let a = (cid:6) n + d − n (cid:7) and b = (cid:4) n + d − n (cid:5) , with explicit expressions a = n + d − n − k ) n , b = n + d − − kn . Note that a and b are positive integers. We define the balanced partition of n + d − n -tuple where k of the elements are a and n − k of the elements are b . Note that this partition is in P n ( n + d −
1) since these elements sum to n + d − (cid:0) nk (cid:1) such irreducible components,namely for each permutation σ in the symmetric group on n elements the correspondingirreducible component is J σ = ( x aσ (1) , . . . , x aσ ( k ) , x bσ ( k +1) , . . . , x bσ ( n ) ) . The bounding inequalities for IP ( m dn ) corresponding to the component J σ is1 a (cid:0) y σ (1) + · · · + y σ ( k ) (cid:1) + 1 b (cid:0) y σ ( k +1) + · · · + y σ ( n ) (cid:1) ≥ σ and utilizing the symmetry ofthe coefficients yields( y + y + · · · + y n ) (cid:18)(cid:18) n − k − (cid:19) a + (cid:18) n − k (cid:19) b (cid:19) ≥ (cid:18) nk (cid:19) whence we deduce that any point y = ( y , . . . , y n ) ∈ IP ( I ) satisfies(4.5) y + y + · · · + y n ≥ (cid:0) nk (cid:1)(cid:0) n − k − (cid:1) a + (cid:0) n − k (cid:1) b := β. From Corollary 4.7 we now deduce the inequality e α ( m dn ) ≥ β .Next consider the vector e y ∈ R n having each component e y i = β/n . We showthat e y ∈ IP ( m dn ) by verifying that this vector satisfies the bounding inequalities inCorollary 4.16. Given ( a , . . . , a n ) ∈ P n ( d + n −
1) there is an equality1 a e y + · · · + 1 a n e y n = (cid:18) a + · · · + 1 a n (cid:19) · βn and by Proposition 4.20 we can compare the sum of the reciprocals for the partition( a , . . . , a n ) to that of the balanced partition as follows1 a + · · · + 1 a n ≥ (cid:18) n − k − (cid:19) a + (cid:18) n − k (cid:19) b = (cid:0) nk (cid:1) β . Altogether, the previous two displayed equations yield the inequality1 a e y + · · · + 1 a n e y n ≥ (cid:0) nk (cid:1) β · βn = (cid:0) nk (cid:1) n ≥ ≤ k ≤ n − . Since we have shown e y satisfies the bounding inequalities for the irreducible polyhedronof m dn , it follows that e y ∈ IP ( m dn ) and thus e α ( m dn ) ≤ e y + · · · + e y n = β. This finishes the proof demonstrating that the following equalities hold; the last arisingfrom the definition of β in (4.5) by direct computation e α ( m dn ) = β = ( n + d − − k )(2 n + d − − k ) n (2 n + d − − k ) . (cid:3) We note a lower bound that extends Remark 4.18.
Corollary 4.22.
Let d, n be positive integers. Then the following inequality holds b α ( m dn ) ≥ (cid:22) d + n − n (cid:23) , with equality taking place if and only if d ≡ n ) .Proof. In view of Theorem 4.21, setting d − ≡ k (mod n ) where 0 ≤ k ≤ n −
1, theclaim is equivalent to the following easily verified inequality( n + d − − k )(2 n + d − − k ) n (2 n + d − − k ) ≥ d + n − − kn = (cid:22) d + n − n (cid:23) . (cid:3) In view of the result above, we make a conjecture regarding the naive Waldschmidtconstant that parallels Conjecture 4.11.
Conjecture 4.23.
Let I be a monomial ideal. Then the following inequality holds e α ( I ) ≥ (cid:22) α ( I ) + big-height( I ) − I ) (cid:23) . ONVEX BODIES AND ASYMPTOTIC INVARIANTS 19
We prove this conjecture for the case when I has maximum possible big-height,namely big-height( I ) = n . The importance of determining the value of the naiveWaldschmidt constant for the powers of the homogeneous maximal ideal earlier in thissection becomes apparent in the next result because this provides lower bounds for thenaive Waldschmidt constant of arbitrary ideals. Theorem 4.24.
Let I be a monomial ideal in K [ x , . . . , x n ] with α ( I ) = d . Then theinequality e α ( m dn ) ≤ e α ( I ) holds.Remark . We note that the analogue of the above theorem fails when replacing thenaive Waldschmidt constant with the Waldschmidt constant. That is, if α ( I ) = d , theinequality b α ( m dn ) ≤ b α ( I ) need not hold. This can be seen taking I = ( x x , x x , x x ),an ideal which satisfies the containment I ⊆ m , but yields b α ( I ) = < b α ( m ) = 2.It is nevertheless true that for square-free monomial ideals I ⊆ J one has b α ( I ) ≥ b α ( J ); see [DFMS19, Lemma 3.10]. Our proof for Theorem 4.24 draws inspiration fromthis result. Before giving the proof, we require some additional preparation. Definition 4.26.
For an ideal I denoteIrr( I ) := { J | J is irreducible and I ⊆ J } . For any monomial ideal I , the set Irr( I ) is a poset with respect to containment whichhas finitely many minimal elements. Moreover, J , . . . , J s are the minimal elements ofIrr( I ) with respect to containment if and only if I = J ∩ · · · ∩ J s is the irredundantirreducible decomposition of I . Lemma 4.27. If I ⊆ I ′ are ideals, then the following hold: (1) Irr( I ) ⊆ Irr( I ′ ) , (2) if J ′ is a minimal element of Irr( I ′ ) with respect to containment then there existsa minimal element J ∈ Irr( I ) with respect to containment such that J ⊆ J ′ , (3) e α ( I ) ≥ e α ( I ′ ) .Proof. The containment Irr( I ) ⊆ Irr( I ′ ) follows from Definition 4.26 and the fact that I ⊆ I ′ .Suppose J ′ is minimal in Irr( I ′ ). Consider the set S = { J ∈ Irr( I ) | J ⊆ J ′ } . Thisset a not empty subset of Irr( I ) since J ′ ∈ S . Thus it has a minimal element withrespect to containment, let’s call it J . Moreover, since S is a lower interval of the posetIrr( I ), we deduce that J is in fact a minimal element of Irr( I ).Now let I = J ∩ · · · ∩ J s and I ′ = J ′ ∩ · · · ∩ J ′ t be the irredundant irreducibledecompositions for I and I ′ respectively. From the second assertion of this lemma, forevery j ∈ { , , . . . , t } there exists an i j ∈ { , . . . , s } such that J i j ⊆ J ′ j . From thiswe deduce N P ( J i j ) ⊆ N P ( J ′ j ) for each j and these containments combine to show thefollowing IP ( I ) = s \ i =1 N P ( J i ) ⊆ t \ j =1 N P ( J i j ) ⊆ t \ j =1 N P ( J ′ j ) = IP ( I ′ ) . Having established the containment IP ( I ) ⊆ IP ( I ′ ) above, we deduce from this con-tainment and Corollary 4.7 the desired inequality e α ( I ) ≥ e α ( I ′ ). (cid:3) Proof of Theorem 4.24.
Theorem 4.24 follows from part 3 of Lemma 4.27 applied to I ′ = m dn . Note that the containment I ⊆ I ′ = m dn is ensured by the hypothesis α ( I ) = d . (cid:3) The following consequence of Theorem 4.24 establishes a lower bound on the naiveWaldschmidt constant applicable to all monomial ideals.
Theorem 4.28.
Let I ⊆ K [ x , . . . , x n ] be a monomial ideal with α ( I ) = d . If d − ≡ k mod ( n ) , ≤ k < n , then the following inequalities hold α ( I ) ≥ b α ( I ) ≥ e α ( I ) ≥ ( n + d − − k )(2 n + d − − k ) n (2 n + d − − k ) ≥ (cid:22) α ( I ) + n − n (cid:23) . Proof.
This follows from Proposition 4.8, Theorem 4.24 and Theorem 4.21. (cid:3)
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