Asymptotic Degree of Random Monomial Ideals
AAsymptotic Degree of Random Monomial Ideals
Lily Silverstein
California State Polytechnic University, Pomona [email protected]
Dane Wilburne
The MITRE Corporation [email protected] Jay Yang
University of Minnesota - Twin Cities [email protected]
Abstract
One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In thispaper we derive the probabilistic behavior of degree with respect to the versatile Erd˝os-R´enyi-typemodel for random monomial ideals defined in [10].We study the staircase structure associated to a monomial ideal, and show that in the random casethe shape of the staircase diagram is approximately hyperbolic, and this behavior is robust acrossseveral random models. Since the discrete volume under this staircase is related to the summatoryhigher-order divisor function studied in number theory, we use this connection and our control overthe shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal.Another way to compute the degree of a monomial ideal is with a standard pair decomposition.This paper derives bounds on the number of standard pairs of a random monomial ideal indexedby any subset of the ring variables. The standard pairs indexed by maximal subsets give a count ofdegree, as well as being a more nuanced invariant of the random monomial ideal.
1. Introduction
One way to understand a complicated class of mathematical objects is to study random instances.This approach has proven to be particularly fruitful in combinatorics, where, for example, the theoryof random graphs has a long and rich history (e.g., [17, 15, 1]). There is also a robust literature onthe properties of random simplicial complexes (e.g., [23, 6, 21, 4, 2]). On the algebraic side, the studyof random groups has received much attention (e.g., [18]). Work now considered classical includesthe study of random varieties, defined by random coefficients on a fixed Newton polytope support,as in [20, 22, 28] and the references therein. The field of smooth analysis studies how algorithmic Approved for Public Release; Distribution Unlimited. Public Release Case Number 20-0202. Thesecond author’s affiliation with The MITRE Corporation is provided for identification purposes only,and is not intended to convey or imply MITRE’s concurrence with, or support for, the positions,opinions, or viewpoints expressed by the author. c (cid:13) a r X i v : . [ m a t h . A C ] S e p Asymptotic Degree performance varies under random perturbations of the problem input. Contributions to algebraicgeometry using smooth analysis include [3] and [5]. More recently, Ein, Erman, and Lazarsfeld [11](see also [13, 12]), studied the Betti numbers of modules defined by uniformly random Boij-S¨oderbergcoefficients [14].Now, a new program is underway in the field of commutative algebra, centering on the studyof random monomial ideals [27, 31, 10, 9, 16]. Monomial ideals are the simplest polynomial ideals,yet they are general enough to capture the entire range of possible values for many importantinvariants of ideals. In [10], the authors described thresholds for the dimension of Erd˝os-R´enyi-typerandom monomial ideals; we extend those results by describing in detail what happens along thephase transitions. In [9], the authors explained the asymptotic and threshold behavior of projectivedimension, genericity, and certain simplicial resolutions for random monomial ideals. In 2018, Ermanand Yang studied the Stanley-Reisner ideals associated to random flag complexes [16], using theprobabilistic method to exhibit concrete examples of the asymptotic behavior of syzygies describedin [14].In this paper, we advance this program by describing the asymptotic degree of Erd˝os-R´enyi-type random monomial ideals. Specifically, a random monomial ideal I in the polynomial ring R = K [ x , . . . , x n ] is produced by randomly selecting its generators independently, with probability p ∈ (0 ,
1) each, from the set of all monomials in R of positive degree no more than D . The resultingdistribution on generating sets induces a probability distribution on the set of all monomial ideals inthe ring R , which we denote by I ( n, D, p ). In this paper, the asymptotic behavior of I ∼ I ( n, D, p )will always refer to the case where n is fixed, D → ∞ , and p = p ( D ) is a function of D .In the specific case where n = 2 and the dimension of the ideal is 0, we can also view this asgiving a random partition by taking the complement of the staircase diagram. Random partitionsand tableaux have been extensively studied, for instance in [30, 26], and yield similar pictures tothose we will discuss later. However, there is not a clear relationship between our model of a randommonomial ideal and any of the studied random partition models.If I is a zero-dimensional monomial ideal of R , its degree equals the number of standard monomialsof R/I , which equals the number of integer lattice points under the “staircase” defined by thegenerators of I (see, e.g., [7, 24] and Figure 1). The key observation in this paper is contained inProposition 2.1, which establishes that for n fixed and D (cid:29)
0, the staircase of a random monomialideal
I ∼ I ( n, D, p ) is “approximately hyperbolic.” More specifically, as D tends to infinity, themultidegrees ( α , . . . , α n ) of the minimal generators and first syzygies of I (the “outside” and “inside”corners of the staircase, respectively) will all be contained, with probability one, in regions boundedby hyperboloids of the form n (cid:89) i =1 ( α i + 1) = d ( D ) . (1.1)See Remark 1.4 for appropriate choices of the functions d ( D ).When I is zero-dimensional, its degree is bounded by the number of lattice points under thesehyperboloids. Let Z ( n, d ) denote the number of integer lattice points in the region bounded aboveby (cid:81) ni =1 ( α i + 1) = d , and below by the coordinate axes. That is, Z ( n, d ) = { α ∈ Z n ≥ : n (cid:89) i =1 ( α i + 1) ≤ d } . The value Z ( n, d ) is also given by the classical number theory problem of computing the summatoryhigher-order divisor function , which is equivalent, as a function of d , to the quantity { α ∈ Z n> : Asymptotic Degree (cid:81) ni =1 α i ≤ d } . By standard arguments from multiplicative number theory (see, e.g., [25, Theorem7.6]), the summatory higher-order divisor function is asymptotically equal to Z ( n, d ) = d (log d ) n − ( n − (cid:0) c (log d ) n − (cid:1) . (1.2)We use this to obtain new asymptotic results in random commutative algebra; such as the following: Theorem 1.1.
Let
I ∼ I ( n, D, p ) , and suppose p = D − k for k ∈ (0 , n ) , not an integer. Let s = (cid:98) k (cid:99) .Then there exist constants C , C > such that asymptotically almost surely as D → ∞ ,1. Z ( n − s, D k − s − (cid:15) ) < deg( I ) < Z ( n − s, D k − s + (cid:15) ) , and2. C D k − s − (cid:15) (log D ) n − < deg( I ) < C D k − s + (cid:15) (log D ) n − .Proof. The first statement is Theorem 3.3 with the specific choices f s = D k − s − (cid:15) , h s = D k − s + (cid:15) . Thesecond statement follows from using Equation 1.2 to give the asymptotics of the first statement.Here and throughout the paper we use the term asymptotically almost surely or a.a.s. to meanthat an event occurs with probability 1 in the limit as D → ∞ . Remark 1.2.
In Theorem 1.1, we impose the condition k (cid:54)∈ Z in order to present the simplestversion of the results in Theorem 3.3. When k is not an integer, by [10] we know the dimension of I is (cid:98) k (cid:99) a.a.s., and can therefore dispense with the conditional probability that appears in the fullstatement of Theorem 3.3.When I is positive-dimensional, deg( I ) is no longer equal the number of lattice points under themonomial staircase, but is still determined by staircase combinatorics. The standard pair decompo-sition of a monomial ideal I is a partition of its standard monomials that simultaneously describesits degree and arithmetic degree. Standard pairs were first introduced in [29], and are useful both intheory and in computational applications (e.g., [19]).An admissible pair of I is a pair ( x α , S ), for x α a monomial of K [ x , . . . , x n ] and S ⊆ { x , . . . , x n } ,such that supp( x α ) ∩ S = ∅ and every monomial in x α · K [ S ] is a standard monomial of I . Anadmissible pair is called a standard pair if it is minimal with respect to the partial order givenby ( x α , S ) ≤ ( x β , T ) if x α divides x β and supp( x β − α ) ∪ T ⊆ S . As an abuse of notation, we willalso consider pairs of the form ( x α , S ) for S ⊂ [ n ] to be a standard/admissible pair when the pair( x α , { x i | i ∈ S } ) is a standard/admissible pair. The arithmetic degree of I equals the number ofstandard pairs of I , while its degree equals the number of standard pairs ( x α , S ) with | S | = dim I .For an monomial ideal I , we denote it’s unique minimal generating set as G ( I ).In Section 4, we probabilistically bound the number of standard pairs ( x α , S ) of a random monomialideal for each S ⊆ { x , . . . , x n } as follows: Theorem 1.3.
Fix S a subset of the variables { x , . . . , x n } , and let I ∼ I ( n, D, p ) where p = D − k , k ∈ (0 , n ) . Then there exists a constant C > such that asymptotically almost surely as D → ∞ , CZ ( t, D k − s − (cid:15) ) < { standard pairs ( x α , S ) of I} < Z ( t, D k − s − (cid:15) ) . Proof.
This is Theorem 4.4 for the special case p = D − k , using the specific choices f s = D k − s − (cid:15) and h s +1 = D k − s − (cid:15) . Remark 1.4.
For all the results of this paper, it will be convenient to fix the following set of functionswhich will serve as lower and upper bounds throughout. Let
I ∼ I ( n, D, p ) with D → ∞ and fix Asymptotic Degree f = Q i =1 ( α i + 1) g = Q i =1 ( α i + 1) h = Q i =1 ( α i + 1) Fig 1: This figure illustrates the role of the functions f s , g s , and h s , as defined in Remark1.4, in relation to a random monomial ideal I ∼ I ( n, D, p ). For this figure, s = 0 and n = 2. n ≥ s ≥
0. Then, for C sufficiently small depending only on n and s , fix functions f s ( D ), g s ( D ), h s ( D ) satisfying pD s f s (log f s ) t − → , pD t exp( − CpD s g s ) → , and D t exp( − CpD s h s ) → . In particular, as seen in Theorems 1.1 and 1.3, the functions f s = D − s − (cid:15) /p and g s = h s = D − s + (cid:15) /p satisfy the conditions of the above definition.These functions describe upper and lower bounds on the multidegrees of generators and syzygiesof I in the theorems in this paper as illustrated in Figure 1 for the case s = 0, n = 2. As describedin Proposition 2.1 and Theorem 3.1, f and g provide asymptotic lower and upper bounds on thegenerators of a random ideal in that asymptotically almost surely, all minimal generators x α x α are bounded by f < ( α + 1)( α + 1) < g . Visually, this corresponds to the bottom corners of themonomial staircase (in black) lying between two hyperbolic curves.The function h gives an upper bound to the full staircase in that a.a.s. every monomial satisfying( α + 1)( α + 1) > h belongs to I . Visually, this corresponds to every lattice point above the upperhyperbola being in the shaded region above the monomial staircase.
2. The zero-dimensional case
When I is zero-dimensional, its standard pairs are exactly the pairs ( x α , ∅ ) where x α is a standardmonomial of R/ I . In other words, enumerating standard pairs is equivalent to enumerating standard Asymptotic Degree monomials in the zero-dimensional case; this count is also equivalent to the degree of the zero-dimensional ideal.Proposition 2.1 makes precise the image in Figure 1. In particular, it shows that the staircasediagram of a monomial ideal is bounded below by f and above by h , with g providing a tighterbound on just the generators as opposed to the whole of the staircase diagram. As a remark, Propo-sition 2.1 uses the same conventions and notation defined in Remark 1.4, except that in this sectionwe can make the explicit choice C = 1. Proposition 2.1.
Let
I ∼ I ( n, D, p ) with D → ∞ and p = p ( D ) a function of D . Fix functions f ( D ) , g ( D ) , h ( D ) satisfying pf (log f ) n − → , pD n exp( − pg ) → , and D n exp( − ph ) → . Then,1. P (cid:34) f < n (cid:89) i =1 ( α i + 1) < g for all x α ∈ G ( I ) (cid:35) → , and2. P (cid:34) x α ∈ I for all x α s . t . h < n (cid:89) i =1 ( α i + 1) and | α | ≤ D (cid:35) → . Proof.
For the first statement, observe that P [ x α ∈ G ( I )] = pq − (cid:81) ni =1 ( α i +1) . Therefore E (cid:34) { x α ∈ G ( I ) : n (cid:89) i =1 ( α i + 1) > g } (cid:35) = (cid:88) α ∈ Z n ≥ s.t. | α |≤ D and (cid:81) ( α i +1) >g pq − (cid:81) ni =1 ( α i +1) (2.1) ≤ pq g − (cid:88) α ∈ Z n ≥ s.t. | α |≤ D ≤ pq g − D n = p (1 − p ) g − D n ∼ pD n exp ( − pg ) , which goes to 0 as D → ∞ by hypothesis. Moreover, E (cid:34) { x α ∈ G ( I ) : n (cid:89) i =1 ( α i + 1) < f } (cid:35) ≤ (cid:88) α ∈ Z n ≥ s.t. (cid:81) ( α i +1)
For
I ∼ I ( n, D, p ) , with D → ∞ and p = p ( D ) satisfying /D (cid:28) p ≤ , we have P [ Z ( n, f ) ≤ deg( I ) ≤ Z ( n, h )] → . Proof.
Whenever I is zero dimensional, its degree equals the number of standard monomials of R/ I .And by Proposition 2.1, every monomial x α with (cid:81) ( α i + 1) < f is a standard monomial of R/ I ,so there must be at least Z ( n, f ) standard monomials. On the other hand, by the same propositionwe know that every monomial x α with (cid:81) ( α i + 1) > h is not a standard monomial of R/ I , so thereare at most Z ( n, h ) standard monomials.For 1 /D (cid:28) p ≤
1, it follows from [10, Corollary 1.2] that P [dim( R/ I ) = 0] → I ) is bounded between these two quantities.
3. The positive dimensional case
This next proposition will be necessary to compute the degree of higher dimensional ideals. For any T ⊆ [ n ], Theorem 3.1 describes probabilistic constraints on the x α such that ( x α , T C ) is an admissiblepair of I . Analogously, Proposition 2.1 can be viewed as giving constraints on admissible pairs ofthe form ( x α , ∅ ). However, Proposition 2.1 gives slightly better bounds than simply substituting T = { , . . . , n } into Theorem 3.1. Theorem 3.1.
Let
I ∼ I ( n, D, p ) with D → ∞ . For T ⊆ [ n ] , define I| T to be the ideal in K [ x i : i ∈ T ] given by substituting x i (cid:55)→ for i / ∈ T . Set t := | T | and s := n − t . Then there exists a constant C depending only on n and t such that, for any fixed functions f s ( D ) , g s ( D ) , h s ( D ) as in Remark 1.4,the following statements hold:1. P (cid:34) f s < (cid:89) i ∈ T ( α i + 1) < g s for all x α ∈ G ( I| T ) (cid:35) → , and2. P (cid:34) x α ∈ I| T for all x α s . t . h s < (cid:89) i ∈ T ( α i + 1) and | α | ≤ D (cid:35) → . Remark 3.2.
For example, suppose p = D − k and fix any (cid:15) >
0. As in the theorem, fix a subset ofthe variables T and let s := n − | T | . Then f s = D k − s − (cid:15) , g s = h s = D k − s + (cid:15) are examples of functionssatisfying the hypotheses of Theorem 3.1. This implies that the corners of the staircase diagram for I| T are confined to an arbitrarily narrow strip around the curve (cid:81) i ∈ T ( α i + 1) = D k − s . In particular,the admissible pairs of the form ( x α , T C ) have α values bounded by (cid:81) i ∈ T ( α i + 1) ≤ D k − s + (cid:15) .Note that the map from I ⊆ K [ x , . . . , x n ] to I| T ⊆ K [ x i : i ∈ T ] defined in the theorem isequivalent to saturating by the variables in S , followed by intersecting with K [ x i : i ∈ T ]. Asymptotic Degree Proof.
Fix α i for i ∈ T . Then B ( α, T ) := { β : | β | ≤ D and β i ≤ α i for i ∈ T } ≥ C (cid:89) i ∈ T ( α i + 1) D s . To show this inequality, we will use the average value of (cid:80) i/ ∈ T β i , and particularly the fact that it isless than D/ B ( α, T ) = (cid:88) γ ∈ Z T ≥ γ ≤ α (cid:8) β ∈ Z n ≥ : | β | ≤ D s.t. β i = γ i for i ∈ T (cid:9) = (cid:88) γ ∈ Z T ≥ γ ≤ α (cid:18) D − | γ | + ss (cid:19) . Now we restrict the summation to those γ where | γ | ≤ D/ B ( α, T ) ≥ (cid:88) γ ∈ Z T ≥ γ ≤ α, | γ | 1, using x = − p and r = D s we get the followinginequality: E (cid:104) { x α ∈ G ( I| T ) : (cid:89) ( α i + 1) < f s } (cid:105) ≤ pD s (cid:18) f s (log f s ) t − ( t − (cid:0) f s (log f s ) t − (cid:1)(cid:19) , which goes to zero whenever f s (log f s ) t − (cid:28) D − s /p . For the second statement, we have E (cid:104) { x α (cid:54)∈ I| T : (cid:89) ( α i + 1) > h s and | α | ≤ D } (cid:105) = (cid:88) α ∈ Z T ≥ | α |≤ D, (cid:81) ( α i +1) >h s q B ( α,T ) . An estimate similar to the one in Equation 3.1 shows that this goes to zero whenever D t exp( − CpD s h s )does.Since the degree, in any dimension, can be bounded by the number of admissible pairs with supportof a particular size, the previous result leads to bounds on the degree of I ∼ I ( n, D, p ). Theorem 3.3. Fix s , ≤ s ≤ n . For I ∼ I ( n, D, p ) , if p = p ( D ) is any function such that lim D →∞ P [dim( I ) = s ] > , then P (cid:20)(cid:18) ns (cid:19) Z ( n − s, f s ) < deg( I ) < (cid:18) ns (cid:19) Z ( n − s, h s ) (cid:12)(cid:12)(cid:12)(cid:12) dim( I ) = s (cid:21) → as D → ∞ , where f s ( D ) , h s ( D ) are any functions satisfying the hypotheses of Theorem 3.1.Proof. Suppose dim( I ) = s . Thendeg( I ) = (cid:88) T ⊂ [ n ] , | T | = n − s deg( I| T ) . Since lim D →∞ P [dim( I ) = s ] > 0, for the following events, it suffices to know that the non-conditionalprobabilities go to 1. In particular, suppose that A ( I ) is an event with P [ A ( I )] → 1. Then considerthe following sum: P [ A ( I )] = n (cid:88) k =0 P [ A ( I ) | dim( I ) = k ] P [dim( I ) = k ] . (3.2)If lim D →∞ P [dim( I ) = s ] > 0, this forces P [ A ( I ) | dim( I ) = s ] → P [deg( I| T ) < Z ( n − s, h s )] → 1. Summing over all projections onto n − s coordinates, this shows P (cid:2) deg( I ) < (cid:0) ns (cid:1) Z ( n − s, h s ) (cid:3) → P (cid:34) f s < (cid:89) i ∈ T ( α i + 1) for all x α ∈ G ( I| T ) (cid:35) → , Asymptotic Degree and therefore with probability approaching one, all x α below this curve correspond to admissiblepairs. Asymptotically, this implies P [ Z ( n − s, f s ) < deg( I| T )] → 1, and therefore P (cid:20)(cid:18) ns (cid:19) Z ( n − s, f s ) < deg( I ) (cid:21) → . Remark 3.4. A few words about the conditional nature of the result in Theorem 3.3. As seenin Theorem 1.1 in the introduction, particular choices of p can guarantee that only one dimensionis observed a.a.s., and thus the conditional probability that appears in the general statement ofTheorem 3.3 can be dispensed with. In one of the most natural settings, the case where p ( D ) = D − k , I ∼ I ( n, D, p ) is a.a.s. a particular fixed dimension for any choice of k other than an integer (this iswhy we restricted k not to be an integer in Theorem 1.1).But what happens if we allow k to be integral? This “boundary case” was worked out in detail bythe second author by analyzing the case where pD t approaches a constant, where D − t is one of theintegral thresholds for dimensionality. Theorem 3.5 ([31, Theorem 3]) . Let I ∼ I ( n, D, p ) , let ≤ t ≤ n be an integer and let c > be aconstant. If pD t → c as D → ∞ , then lim D →∞ E [dim I ] = t − (1 − e − c/t ! )( nt ) . If we consider a particular regime I ( n, D, p = D − k ) with k integral, then there is not a single di-mension a.a.s., but it is a boundary case involving exactly two dimensions that appear with nonzeroprobability, distributed as in 3.5. Combining that Theorem with Theorem 3.3 allows an explicit eval-uation of Equation 3.2, which will have exactly two nonzero summands, and provides the “boundarycase” version of Theorem 3.3 with no conditional probabilities in the final statement.In the next section, we take an alternative approach to remove conditioning on dimension, byusing standard pair enumerations that are robust across dimension. 4. Standard pairs To demonstrate the well-behaved nature of standard pair invariants (as opposed to the degree in-variant), consider several experimental samples from the I (3 , D, p = D − ) regime, which is chosenso that I sampled from this distribution will have non-negligible positive probability of being either1 or 2 dimensional, a.a.s. A small value of D = 65 is enough to display the relevant behavior. I ∼ I (3 , , p = 1 / I deg I sp sp sp (cid:104) x x x , x x x , x x x , x x x , x x x , x x x (cid:105) (cid:104) x x , x x x , x x x , x x x , x x x , x x x , x x x (cid:105) (cid:104) x x x , x x x , x x x , x x x , x x , x x , x (cid:105) (cid:104) x x , x x , x x x , x x x , x x , x x , x x x , x x (cid:105) (cid:104) x x , x x x , x x x , x x x , x x , x x (cid:105) (cid:104) x x , x x x , x x x , x x x , x x x , x x , x x (cid:105) Table 1 A collection of randomly generated ideals showing the sensitivity of the degree at the dimension boundary as comparedto the number of standard pairs. Here, sp i is the number i -dimensional standard pairs.Asymptotic Degree Because these parameters were chosen to be a boundary case, we see samples containing bothone- and two-dimensional ideals. The degrees of the dimension one versus dimension two idealsare dramatically different. This small example shows that the degree invariant is not well behavedwhen considering multiply-dimensional sets of ideals. This explains why the asymptotic statementsin Section 3 required conditioning on dimensions (see Remark 3.4).On the other hand, this small glimpse at the data illustrates that the enumeration of standard pairs,whether counted collectively or even dimension-by-dimension, behaves predictably across dimensionborders. In other words, the count of standard pairs of a particular dimension is uncorrelated withthe true dimension of the ideal. This is obviously false for degree since the definition of degree relieson counting only the standard pairs in a particular dimension. Table 1 also demonstrates a usefulfact we’ll use in Section 4, which is that the standard pair count is always zero for standard pairsof dimension greater than the ideal of the dimension. This observation will be important to thehypotheses on Lemma 4.3 and Theorem 4.4. (See Remark 4.2, preceding those results.)Our next goal in this section is to prove that for any choices of parameters in I ( n, D, p ), andany choice of a subset S of the variables of the ring, there is a region of lattice points guaranteed,asymptotically almost surely, to be standard standard pairs of the form ( x α , S ). Theorem 4.1. Let I ∼ I ( n, D, p ) with D → ∞ . Fix S ⊆ [ n ] , with s := | S | , T := [ n ] \ S , and t := | T | = n − s . Let p = p ( D ) → and let functions f s , h s +1 → ∞ and f s ≤ D be as in Remark 1.4.Then P [ for all α ∈ L ( f s , h s +1 ) , ( x α , S ) is a standard pair for I ] → , where L ( f s , h s ) := (cid:40) α ∈ Z T ≥ : (cid:89) i ∈ T ( α i + 1) < f s and ( α i + 1) t − > h s +1 (cid:41) .Proof. For convenience, we will write f = f s and h = h s +1 . Then the statement P [for all α ∈ L ( f, h ) , ( x α , S ) is a standard pair for I ] → , is equivalent to the finite set of statements P [for all α ∈ L ( f, h ) , ( x α , S ) is an admissible pair for I ] → , and for all i ∈ T P (cid:2) for all α ∈ L ( f, h ) , ( x α | T \{ i } , S ∪ { i } ) is not an admissible pair for I (cid:3) → . (4.1)For the geometric intuition underlying the statements of Equation 4.1, see Figure 2. Part 1: P [for all α ∈ L ( f, h ) , ( x α , S ) is an admissible pair for I ] → f satisfies the same conditions as it would in Theorem 3.1. As a consequence of that theorem,we have that P for all α ∈ Z T with (cid:89) j ∈ T ( α j + 1) < f, ( x α , S ) is an admissible pair for I → . For all α ∈ L ( f, h ), we have (cid:81) j ∈ T ( α j + 1) < f , and thus P [for all α ∈ L ( f, h ) , ( x α , S ) is an admissible pair for I ] → . Part 2 : For fixed i ∈ T , P (cid:2) for all α ∈ L ( f, h ) , ( x α | T \{ i } , S ∪ { i } ) is not an admissible pair I (cid:3) → Asymptotic Degree x α x x U U Fig 2: For n = 2, D = 12, and x α = x x , this figure illustrates the conditions underwhich ( x α , ∅ ) is a standard pair of I ∼ I ( n, D, p ). For x α to be a standard monomial, nomonomial in the dark gray region can be a generator of I . Additionally, to insure x α doesnot belong to a higher-dimensional standard pair, at least one monomial from each of thelight gray regions, U and U , must be a generator of I . In the language of Equation 4.1,at least one generator chosen in region U guarantees that ( x , { x } ) is not an admissiblepair for I (note x = x α | x ). Similarly, a generator chosen in region U guarantees that( x , { x } ) is not an admissible pair for I .Notice that for a fixed product, the sum is maximized in the case where the product is mostasymmetric, and so since (cid:81) i ∈ T ( α i + 1) < f ≤ D and α i ≥ 0, we have that | α | < D . Now let T (cid:48) = T \ { i } and S (cid:48) = S ∪ { i } . Again we apply Theorem 3.1, noting that h = h s +1 . The theoremimplies that P for all α ∈ Z T (cid:48) with (cid:89) j ∈ T (cid:48) ( α j + 1) > h, ( x α , S (cid:48) ) is not an admissible pair for I → . For all α ∈ L ( f, h ), we have (cid:81) j ∈ T (cid:48) ( α j + 1) > h , and thus P (cid:2) for all α ∈ L ( f, h ) , ( x α | T (cid:48) , S (cid:48) ) is not an admissible pair for I (cid:3) → . Remark 4.2. Notice that the case where | S | > dim I the conditions in Remark 1.4 requires that f s must be a decreasing to zero function. This ensures the set L ( f s , h s +1 ) is empty for D sufficiently Asymptotic Degree large, and thus Theorem 4.1 is vacuously true in these cases, which matches the fact that in thesecases, the dimension results in [10] imply that there are no standard pairs with support S .The next results bound the size of L ( f s , h s +1 ) to give probabistic estimates on the number ofstandard pairs in the case where | S | ≤ dim I . Lemma 4.3. Fix T ⊆ [ n ] , with T (cid:54) = ∅ and let t = | T | . Given f, h with f → ∞ and f t − (cid:29) h t . Thenthere exists some constant C ≤ such that | L ( f, h ) | = CZ ( t, f ) + O (cid:0) f (log( f )) t − (cid:1) . Proof. The case of t = 1 causes issues with the remainder of the proof we first prove that case. Noticeif t = 1, the requirement h (cid:28) f t − implies h (cid:28) h is eventually strictly lessthan 1. That means for D sufficiently large, we can expand to give L ( f, h ) = { a ∈ Z ≥ | a + 1 < f } .Thus | L ( f, h ) | = f − 1, thus the statement is true.For the remainder of this proof, we will use a result by Davenport[8], which bounds the differencebetween the number of lattice points in a region and its volume. For this it will be convenient toconsider the following volume: W t ( c, d ) := V ol (cid:16)(cid:110) x ∈ R t ≥ : (cid:89) ( x i + c ) ≤ d (cid:111)(cid:17) . Equivalently, we may define this volume as W t ( c, d ) = V ol (cid:0)(cid:8) x ∈ R t : x i ≥ c − (cid:81) ( x i + 1) ≤ d (cid:9)(cid:1) .We use the convention that W ( c, d ) = 1.Applying the main theorem from Davenport [8], as d → ∞ , we get W t (1 , d ) = Z ( t, d ) ± O (cid:32) t − (cid:88) i =0 W i (1 , d ) (cid:33) . Then since W (1 , d ) = 1 and W (1 , d ) = d , we have W (1 , d ) (cid:29) W (1 , d ). Recall that Z ( t, d ) = O ( d (log d ) t − ), then by inducting on i , we have that W i (1 , d ) (cid:29) W i − (1 , d ). This allows us to simplifythe previous equaiton to W t (1 , d ) = Z ( t, d ) ± O ( W t − (1 , d )) . Again we can apply the theorem by Davenport, this time to | L ( f, h ) | to yield the following: | L ( f, h ) | = W t ( t − √ h, f ) ± O (cid:16) W t − ( t − √ h, f ) (cid:17) . Observe that W t ( c, d ) = c t W t (1 , d/c t ). So as long as d/c t → ∞ , we can use the asymptotic behaviorof W t (1 , d ). In particular, we can expand W t ( c, d ) as follows W t ( c, d ) = c t W t (1 , d/c t )= c t Z ( t, d/c t ) ± O ( c t W t − (1 , d/c t ))= c t (cid:18) d/c t log t − ( d/c t )( t − O ( d/c t log t − d/c t ) (cid:19) ± O ( c t W t − (1 , d/c t ))= d log t − ( d/c t )( t − O ( d log t − d/c t ) ± O ( c t W t − (1 , d/c t ))= d log t − ( d/c t )( t − O ( d log t − d/c t ) . Asymptotic Degree In this case, we use d = f and c = t − √ h , so d/c t = fh t/t − and thus d/c t → ∞ . This allows us toapply the previous formula to yield the following: | L ( f, h ) | = W t ( t − √ h, f ) ± O ( W t − ( t − √ h, f ))= f log t − (cid:16) fh t/t − (cid:17) ( t − O (cid:18) f log t − fh t/t − (cid:19) = f (log( f ) − ( t/t − 1) log( h )) t − ( t − O (cid:0) f (log( f ) − ( t/t − 1) log( h )) t − (cid:1) h (cid:28) f so log( h ) < C log( f )= C f log t − ( f )( t − O (cid:0) f log t − ( f ) (cid:1) = CZ ( t, f ) + O (cid:0) f log t − ( f ) (cid:1) . Theorem 4.4. Fix S a proper subset of the variables { x , . . . , x n } and let I ∼ I ( n, D, p ) . Then for f s , h s as in Remark 1.4, a constant C > and arbitrary (cid:15) > , the following hold.1. If D − s (cid:28) p (cid:28) then P [ { standard pairs ( x α , S ) of I} = 0] → . 2. If D − n + (cid:15) ≤ p (cid:28) D − s then P [ CZ ( t, f s ) < { standard pairs ( x α , S ) of I} < Z ( t, h s )] → . Proof. The first case is clear from the dimension bounds in [10, Theorem 3.4], so we focus on thesecond case. Lower Bound: Notice that if s = n , the constraints in the second case are vacuous and so thestatement is true vacuously. Thus we will restrict to the case where s < n . Now applying Theorem 4.1we find that for any choice of h s +1 we have P [ { standard pairs ( x α , S ) in I } ≥ | L ( f s , h s +1 ) | ] → . Since choosing a smaller f s only weakens the condtion inside the probability, and since p (cid:28) D − s ,WLOG we may assume f s ≥ D − s − δ /p for any choice of δ > 0. Choose h s +1 = D − s − δ /p . Now wecompute f t − s h ts +1 ≥ (cid:0) D − s − δ /p (cid:1) t − ( D − s − δ /p ) t = pD − st + s − ( t − δ D − st − t + tδ = pD n − (2 t − δ Then since p ≥ D − n + (cid:15) , we can simply choose δ < (cid:15)/ (2 t − 1) to allow f t − s h ts +1 → ∞ . Then by applyingLemma 4.3 to L ( f s , h s +1 ) we get P [ { standard pairs ( x α , S ) in I } ≥ CZ ( t, f s )] → . Upper Bound: If ( x α , S ) is a standard pair then for all β ∈ Z S , x α + β / ∈ I . Applying Theorem 3.1implies that if (cid:81) i/ ∈ S α i (cid:29) D k then P (cid:2) x α + β / ∈ I, ∀ β ∈ Z S (cid:3) → 0. Thus as an upper bound, P [ { standard pairs ( x α , S ) in I } ≤ Z ( t, h s )] → . Asymptotic Degree Since the arithmetic degree for a monomial ideal is given by the number of standard pairs, theresults on standard pairs allow us to compute the asymptotic arithmetic degree of a random monomialideal. Corollary 4.5. Let I ∼ I ( n, D, p ) , and suppose p = D − k for < k < n . Then there exist constants C , C > such that asymptotically almost surely as D → ∞ , C Z ( t, D k − (cid:15) ) < arith-deg( I ) < C Z ( t, D k + (cid:15) ) . Proof. This is a consequence of applying Theorem 4.4 with S = ∅ and h = D k + (cid:15) and f = D k + (cid:15) while noting that there are always asymptotically more standard pairs with S = ∅ than any othersubset of the variables.With some refinement, it should be possible to extend the results of Section 4 to give us the“expected Hilbert polynomial” of a random monomial ideal. Since the Hilbert polynomial is a poly-nomial, it is somewhat less clear what is a reasonable notion of “expected”. However, for the choiceof “expected” determined by the simply taking the expected value of the coefficients, we can use thefact that Hilbert polynomial can be expressed as a sum of binomial coefficients indexed over standardpairs of the ideal: HP S/I ( t ) = (cid:88) ( x α ,S ) standard pairs of I (cid:18) t − | α | + | S | − | S | − (cid:19) . 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