ASYMPTOTIC GROWTH OF POWERS OF IDEALS

CĂTĂLIN CIUPERCĂ, FLORIAN ENESCU, AND SANDRA SPIROFF

Abstract. Let A be a locally analytically unramified local ring and J_{1}, …, J_{k}, I ideals such that J_{i} ⊆ \sqrt{I} for all i, the ideal I is not nilpotent, and ⋂_{k} I^{k} = (0). Let C = C(J_{1}, …, J_{k}; I) ⊆ ℝ^{k+1} be the cone generated by {(m_{1}, …, m_{k}, n) ∈ ℕ^{k+1} | J_{1}^{m_{1}} … J_{k}^{m_{k}} ⊆ I^{n}}. We prove that the topological closure of C is a rational polyhedral cone. This generalizes results by Samuel, Nagata, and Rees.

Introduction

In this note we continue the study of the asymptotic properties of powers of ideals initiated by Samuel in [8]. Let A be a commutative noetherian ring with identity and I, J ideals in A with J ⊆ \sqrt{I}. Also, assume that the ideal I is not nilpotent and ⋂_{k} I^{k} = (0). Then for each positive integer m one can define v_{I}(J, m) to be the largest integer n such that J^{m} ⊆ I^{n}. Similarly, w_{J}(I, n) is defined to be the smallest integer m such that J^{m} ⊆ I^{n}. Under the above assumptions, Samuel proved that the sequences {v_{I}(J, m)/m}_{m} and {w_{J}(I, n)/n}_{n} have limits l_{I}(J) and L_{J}(I), respectively, and l_{I}(J)L_{J}(I) = 1 [8, Theorem 1]. It is also observed that these limits are actually the supremum and infimum of the respective sequences. One of the questions raised in Samuel’s paper is whether l_{I}(J) is always rational. This has been positively answered by Nagata [4] and Rees [5]. The approach used by Rees is described in the next section of this paper.

We consider the following generalization of the problem described above. Let J_{1}, …, J_{k}, I be ideals in a locally analytically unramified ring A such that J_{i} ⊆ \sqrt{I} for all i, I is not nilpotent, and ⋂_{k} I^{k} = (0), and let C = C(J_{1}, …, J_{k}; I) ⊆ ℝ^{k+1} be the cone generated by {(m_{1}, …, m_{k}, n) ∈ ℕ^{k+1} | J_{1}^{m_{1}} … J_{k}^{m_{k}} ⊆ I^{n}}. We prove that the topological closure of C is a rational polyhedral cone; i.e., a polyhedral cone bounded by hyperplanes whose equations have rational coefficients. Note that the case k = 1 follows from the results proved by Samuel, Nagata, and Rees; the cone C is the intersection of the half-planes given by n ≥ 0 and n ≤ l_{I}(J)m_{1}. In Section 3 we look at the periodicity of the rate of change of the sequence {v_{I}(J, m)}_{m}, more precisely, the periodicity of the sequence {v_{I}(J, m + 1) − v_{I}(J, m)}_{m}. The last part of the paper describes a method of computing the limits studied by Samuel in the case of monomial ideals.

1. The Rees valuations of an ideal

In this section we give a brief description of the Rees valuations associated to an ideal. For a noetherian ring A which is not necessarily an integral domain, a discrete valuation

on A is defined as follows.

Date: 22 June 2006.

The second author gratefully acknowledges partial financial support from the National Science Foundation, CCF-0515010 and Georgia State University, Research Initiation Grant.

Definition 1.1. Let A be a noetherian ring. We say that v : A → ℤ ∪ {∞} is a discrete valuation on A if {x ∈ A | v(x) = ∞} is a prime ideal P, v factors through A → A/P → ℤ ∪ {∞}, and the induced function on A/P is a rank one discrete valuation on A/P. If I is an ideal in A, then we denote v(I) := min{v(x) | x ∈ I}.

If R is a noetherian ring, we denote by \overline{R} the integral closure of R in its total quotient ring Q(R).

Definition 1.2. Let I be an ideal in a noetherian ring A. An element x ∈ A is said to be integral over I if x satisfies an equation x^{n} + a_{1}x^{n−1} + … + a_{n} = 0 with a_{i} ∈ I^{i}. The set of all elements in A that are integral over I is an ideal \overline{I}, and the ideal I is called integrally closed if I = \overline{I}. If all the powers I^{n} are integrally closed, then I is said to be normal.

Given an ideal I in a noetherian ring A, for each x ∈ A let v_{I}(x) = sup{n ∈ ℕ | x ∈ I^{n}}. Rees [5] proved that for each x ∈ A one can define

\overline{v}_{I}(x) = lim_{k→∞} \frac{v_{I}(x^{k})}{k},

and for each integer n one has \overline{v}_{I}(x) ≥ n if and only if x ∈ \overline{I^{n}}. Moreover, there exist discrete valuations v_{1}, …, v_{h} on A in the sense defined above, and positive integers e_{1}, …, e_{h} such that, for each x ∈ A,

(1.1)  \overline{v}_{I}(x) = min \{\frac{v_{i}(x)}{e_{i}} | i = 1, …, h\}.

We briefly describe a construction of the Rees valuations v_{1}, …, v_{h}. Let 𝔭_{1}, …, 𝔭_{g} be the minimal prime ideals 𝔭 in A such that 𝔭+I ≠ A, and let R_{i}(I) be the Rees ring (A/𝔭_{i})[It, t^{−1}]. Denote by W_{i1}, …, W_{ih_{i}} the rank one discrete valuation rings obtained by localizing the rings \overline{R_{i}(I)} at the minimal primes over t^{−1}\overline{R_{i}(I)}, let w_{ij} (i = 1, …, g, 1 ≤ j ≤ h_{i}) be the corresponding discrete valuations, and let V_{ij} = W_{ij} ∩ Q(A/𝔭_{i}) (i = 1, …, g). Then define v_{ij}(x) := w_{ij}(x + 𝔭_{i}) and e_{ij} := w_{ij}(t^{−1})(= v_{ij}(I)) for all i, and for simplicity, renumber them as e_{1}, …, e_{h} and v_{1}, …, v_{h}, respectively.

Rees [5] proved that v_{1}, …, v_{h} are valuations satisfying (1.1). We refer the reader to the original article [5] for more details on this construction.

Remark 1.3. With the notation established above, for every positive integer n we have

\overline{I^{n}} = ⋂_{i=1}^{h} I^{n}V_{i} ∩ R.

In particular, we have the following.

Remark 1.4. If K, L are ideals in A, v_{1}, …, v_{h} are the Rees valuations of L, and v_{i}(K) ≥ v_{i}(L) for all i = 1, …, h, then \overline{K} ⊆ \overline{L}.

The rationality of l_{I}(J) can now be obtained as consequence of the results of Rees. Indeed, by [8, Theorem 2], if J = (a_{1}, … a_{s}), then l_{I}(J) = min{l_{I}(a_{i}) | i = 1, … s}, and for each i we have l_{I}(a_{i}) = \overline{v}_{I}(a_{i}), which is rational.

Finally, recall the following definition.

Definition 1.5. A local noetherian ring (A, 𝔪) is analytically unramified if its 𝔪-adic completion Â is reduced.

Rees [6] proved that for every ideal I in an analytically unramified ring there exists an integer k such that for all n ≥ 0, \overline{I^{n+k}} ⊆ I^{n}.

2. The cone structure

Throughout this section A is a locally analytically unramified ring and I and \underline{J} = J_{1}, …, J_{k} are ideals in A such that J_{i} ⊆ \sqrt{I} for all i. Let C = C(J_{1}, …, J_{k}; I) ⊆ ℝ^{k+1} denote the cone generated by {(m_{1}, …, m_{k}, n) ∈ ℕ^{k+1} | J_{1}^{m_{1}} … J_{k}^{m_{k}} ⊆ I^{n}}. Also, for each (m_{1}, …, m_{k}) ∈ ℕ^{k}, let v_{I}(\underline{J}, m_{1}, …, m_{k}) denote the largest nonnegative integer n such that J_{1}^{m_{1}} … J_{k}^{m_{k}} ⊆ I^{n}.

For each Rees valuation v_{j} of I, denote α_{ij} = v_{j}(J_{i})/e_{j} for all i, j, where e_{j} = v_{j}(I). Then we consider

D_{j} = {(m_{1}, …, m_{k}) ∈ ℝ_{≥0}^{k} | ∑_{s=1}^{k} m_{s}α_{sj} ≤ ∑_{s=1}^{k} m_{s}α_{sl} for all l ≠ j},

and we say that a Rees valuation v_{j} is relevant if D_{j} ≠ {0}. After a renumbering, assume that v_{1}, v_{2}, …, v_{r} (r ≤ h) are the relevant Rees valuations.

Note that each D_{j} is an intersection of half-spaces (hence a polyhedral cone), ⋃_{j=1}^{r} D_{j} = ℝ_{≥0}^{k}, and two cones D_{i}, D_{j} (i ≠ j) either intersect along one common face or have only the origin in common. Let

E_{j} = {(m_{1}, …, m_{k}, n) ∈ ℝ_{+}^{k+1} | (m_{1}, …, m_{k}) ∈ D_{j} and n < ∑_{s=1}^{k} m_{s}α_{sj}}

and

\overline{E}_{j} = \{(m_{1}, …, m_{k}, n) ∈ ℝ_{+}^{k+1} | (m_{1}, …, m_{k}) ∈ D_{j} and n ≤ ∑_{s=1}^{k} m_{s}α_{sj}\}.

Theorem 2.1. Let A be a locally analytically unramified ring. Then for each j = 1, …, r we have

E_{j} ∩ ℚ^{k+1} ⊆ C ∩ (D_{j} × ℝ_{≥0}) ⊆ \overline{E}_{j}.

Proof. Let (m_{1}, …, m_{k}, n) ∈ C ∩(D_{j} ×ℝ_{≥0}). Then there exists t ∈ ℝ such that tm_{1}, …, tm_{k} are positive integers and

J_{1}^{tm_{1}} … J_{k}^{tm_{k}} ⊆ I^{tn}.

Hence, for each Rees valuation v_{j} of I we obtain

tm_{1}v_{j}(J_{1}) + ⋯ + tm_{k}v_{j}(J_{k}) ≥ tnv_{j}(I),

or equivalently,

n ≤ ∑_{s=1}^{k} m_{s}α_{sj}.

For the other inclusion, first observe that it is enough to prove that E_{j} ∩ ℤ^{k+1} ⊆ C ∩ (D_{j} × ℝ_{≥0}). Indeed, if E_{j} ∩ ℤ^{k+1} ⊆ C ∩ (D_{j} × ℝ_{≥0}), then for each α ∈ E_{j} ∩ ℚ^{k+1} there exists a positive integer L such that αL ∈ E_{j} ∩ ℤ^{k+1} ⊆ C ∩ (D_{j} × ℝ_{≥0}). This implies that α ∈ (1/L)(C ∩ (D_{j} × ℝ_{≥0})) = C ∩ (D_{j} × ℝ_{≥0})

Let (m_{1}, …, m_{k}, n) ∈ E_{j} ∩ ℤ^{k+1}. Set α = ∑_{s=1}^{k} m_{s}α_{sj}. Since the ring A is analytically unramified, there exists an integer N such that \overline{I^{t}} ⊆ I^{t−N} for all t. (The convention is that I^{n} = A for n ≤ 0.) Let g be the integer part of α. For any Rees valuation v_{i} of A we then get

v_{i}(I^{g}) = ge_{i} ≤ αe_{i} ≤ (∑_{s=1}^{k} m_{s}α_{si})e_{i} = v_{i}(J_{1}^{m_{1}} … J_{k}^{m_{k}}),

and hence, by Remark 1.4,

J_{1}^{m_{1}} … J_{k}^{m_{k}} ⊆ \overline{I^{g}} ⊆ I^{g−N}.

This implies that

(2.1)  v_{I}(\underline{J}, m_{1}, …, m_{k}) ≥ g − N > α − 1 − N.

Since n < α, we can find δ > 0 such that n < α − δ. Choose l such that lδ > N + 1 and lm_{1}, …, lm_{k}, ln are integers. By (2.1), we obtain v_{I}(\underline{J}, lm_{1}, …, lm_{k}) > lα − N − 1, and by the choice of l, we also have nl < lα − N − 1. Then nl < v_{I}(\underline{J}, lm_{1}, …, lm_{k}), which implies that J_{1}^{lm_{1}} … J_{k}^{lm_{k}} ⊆ I^{ln}; i.e., (m_{1}, …, m_{k}, n) ∈ C. □

Corollary 2.2. The topological closure of C is a rational polyhedral cone.

Proof. From the previous theorem it follows that the topological closure of C ∩ (D_{j} × ℝ_{≥0}) is \overline{E}_{j}, and hence the topological closure of C is the polyhedral cone bounded by the hyperplanes n = ∑_{s=1}^{k} m_{s}α_{sj} (j = 1, …, r) and the coordinate hyperplanes. □

A detailed example of Corollary 2.2 is given below in Example 2.5.

Corollary 2.3. Let a_{1}, a_{2}, …, a_{k} be real numbers. The limit

(2.2)  lim_{m_{1},...,m_{k}→∞} \frac{v_{I}(\underline{J}, m_{1}, …, m_{k})}{a_{1}m_{1} + … + a_{k}m_{k}}

exists if and only if there exists a rational number l such that la_{s} = α_{s1} = α_{s2} = … = α_{sr} for all s = 1, …, k. In this case the limit is equal to l.

Proof. Since the polyhedral cones D_{j} form a partition of ℝ_{≥0}^{k}, the limit (2.2) exists and is equal to l if and only if for each j we have

(2.3)  lim_{\begin{matrix}m_{1},...,m_{k}→∞ \\ (m_{1},...,m_{k})∈D_{j} \end{matrix}} \frac{v_{I}(\underline{J}, m_{1}, …, m_{k})}{a_{1}m_{1} + … + a_{k}m_{k}} = l.

On the other hand, (2.3) holds if and only if la_{s} = α_{sj} for all s = 1, …, k. Indeed, this limit exists and is equal to l if and only if over D_{j} the topological closure of C is bounded by the hyperplane n = la_{1}m_{1} + … + la_{k}m_{k}, which therefore should coincide with the hyperplane n = ∑_{s=1}^{k} m_{s}α_{sj}.

In conlusion, the limit (2.2) exists and is equal to l if and only if all the hyperplanes n = ∑_{s=1}^{k} m_{s}α_{sj} (j = 1, …, r) coincide with n = la_{1}m_{1} + … + la_{k}m_{k}, or equivalently, la_{s} = α_{s1} = α_{s2} = … = α_{sr} for all s = 1, …, k. □

Corollary 2.4. Assume that the ideal I has only one Rees valuation. Then the limit

lim_{m_{1},...,m_{k}→∞} \frac{v_{I}(\underline{J}, m_{1}, …, m_{k})}{a_{1}m_{1} + … + a_{k}m_{k}}

exists if and only if l_{I}(J_{1})/a_{1} = … = l_{I}(J_{k})/a_{k}.

Proof. This is a particular case of the previous Corollary. □

Example 2.5. Let A = ℝ[[X, Y, Z]]/(XY^{2} − Z^{9}) and I = (x, y, z)R as in [3, Example 3.1]. Then R(I) = A[It, t^{−1}], R(I)/t^{−1}R(I) ≅ Q[xt, yt, zt]/(xt)(yt), and there are two Rees valuations v_{1} and v_{2}, corresponding to the minimal primes 𝔭_{1} = (xt, t^{−1}) and 𝔭_{2} = (yt, t^{−1}), over t^{−1}R(I). As shown in [3, Example 3.1], we have v_{1}(x) = 7, v_{1}(y) = v_{1}(z) = 1 and v_{2}(x) = v_{2}(z) = 1, v_{2}(y) = 4. Thus v_{1}(I) = min{v_{1}(x), v_{1}(y), v_{1}(z)} = 1. Likewise v_{2}(I) = 1. Set J_{1} = (x, z^{2}) and J_{2} = (y^{2}, z^{3}). Then v_{1}(J_{1}) = 2, v_{2}(J_{1}) = 1, and v_{1}(J_{2}) = 2, v_{2}(J_{2}) = 3. Therefore, E_{1} = {(m_{1}, m_{2}, n)|n ≤ 2m_{1} + 2m_{2}} and E_{1} = {(m_{1}, m_{2}, n)|n ≤ m_{1} + 3m_{2}}. The boundary planes of E_{1} and E_{2} in ℝ^{3} are z = 2x + 2y and z = x + 3y, respectively. Thus, according to the results of Corollary 2.2, the topological closure of the cone generated by {(m_{1}, m_{2}, n)|J_{1}^{m_{1}}J_{2}^{m_{2}} ⊆ I^{n}} is as pictured below.

Figure 1. View from the front and rotated 90^{∘} ctr-clockwise around the z-axis.

Example 2.6. Let A = k[[X, Y]], with k a field, and I = (x^{3}, x^{2}y, y^{2}). As shown in [7], I has only one associated Rees valuation. Let J_{1} = (x^{3}y^{7}), J_{2} = (x^{4}y^{6}), and J_{3} = (x^{5}y^{2}). Using the methods in Section 4, we can compute l_{I}(J_{1}) = 9/2, l_{I}(J_{2}) = 13/3, and l_{I}(J_{3}) = 8/3. Then by Corollary 2.4, the limit

lim_{m_{1},m_{2},m_{3}→∞} \frac{v_{I}(J_{1}, J_{2}, J_{3}, m_{1}, m_{2}, m_{3})}{27m_{1} + 26m_{2} + 16m_{3}}

exists and equals \frac{1}{6} since \frac{l_{I}(J_{1})}{27} = \frac{l_{I}(J_{2})}{26} = \frac{l_{I}(J_{3})}{16} = \frac{1}{6}.

3. Periodic Increase

In this section we take a closer look at the graph of the sequence {v_{I}(J, m)}_{m}. To simplify the notation we will simply write v(m) instead of v_{I}(J, m).

We address the question of whether this sequence increases eventually in a periodic way; that is, whether or not there exists a positive integer t such that v(m + t) − v(m + t − 1) = v(m) − v(m − 1) for m ≫ 0, or equivalently, v(m + t) − v(m) = constant, for m ≫ 0. Our work is partly motivated by [4, Theorem 8], where Nagata proves that the deviation v(m) − l_{I}(J)m is bounded. In particular, this implies that there exists a positive constant C such that 0 ≤ v(m + t) − v(m) − v(t) < C for all m, t.

We begin by defining noetherian filtrations.

Definition 3.1. A family of ideals F = {F_{m}}_{m≥0} in a noetherian ring A is called a filtration if F_{0} = A, F_{m+1} ⊆ F_{m}, and F_{m}F_{n} ⊆ F_{m+n} for all m, n ≥ 0. We say that the filtration {F_{m}}_{m≥0} is noetherian if the associated graded ring ⊕_{m≥0}F_{m} is noetherian. Equivalently, the filtration F is noetherian if and only if there exists t such that F_{m+t} = F_{m}F_{t} for all m ≥ t ([1, 4.5.12]).

Proposition 3.2. Let I, J be ideals in a noetherian local ring A such that J ⊆ \sqrt{I} , the ideals I, J are not nilpotent, and ⋂_{k} I^{k} = (0). Assume that J is principal and the ring B = ⨁_{m,n} J^{m} ∩ I^{n} is noetherian. Then there exists a positive integer t such that v(m + t) = v(m) + v(t) for all m ≥ t.

Proof. In the ring ⨁_{n≥0} I^{n} consider the filtration {F_{m}} with F_{m} = ⨁_{n≥0} J^{m} ∩ I^{n}. Since B = ⊕_{m≥0}F_{m} is noetherian, there exists a positive integer t such that F_{m+t} = F_{m}F_{t} for all m ≥ t. We will prove that this implies v(m + t) = v(m) + v(t) for all m ≥ t. First note that the inequality v(m + t) ≥ v(m) + v(t) always holds. By contradiction, assume that v(m + t) > v(m) + v(t) for some m ≥ t. This implies that the component of degree v(m) + v(t) + 1 in F_{m+t} is J^{m+t}, and since F_{m+t} = F_{m}F_{t} we then obtain

J^{m+t} = J^{t}(J^{m} ∩ I^{v(m)+1}) + J^{m}(J^{t} ∩ I^{v(t)+1}).

Let J = (z). Then we have

(z)^{m+t} = z^{m+t}(I^{v(m)+1} : z^{m}) + z^{m+t}(I^{v(t)+1} : z^{t}).

From the definition of v(−), both (I^{v(m)+1} : z^{m}) and (I^{v(t)+1} : z^{t}) are contained in the maximal ideal, and by the Nakayama Lemma, we must have z nilpotent, contradicting our assumptions. □

Remark 3.3. It is not always true that the ring B is noetherian. For such an example see [2, Lemma 5.6].

Note that there are a few other natural conditions that ensure the periodic increase of the sequence {v(m)}_{m}. We comment on these below.

Remark 3.4. If the ring G(I) = ⨁_{n≥0} I^{n}/I^{n+1} is reduced, then we have v(m) = mv(1) for all m. In particular, the sequence v(m + 1) − v(m) is constant. Indeed, let x ∈ J \ I^{v(1)+1}. The image of x in I^{v(1)}/I^{v(1)+1} ⊆ G(I) is nonzero, and since G(I) is reduced, so is the image of x^{m} in I^{mv(1)}/I^{mv(1)+1}. This implies that J^{m} ⊈ I^{mv(1)+1}, and hence v(m) ≤ mv(1).

The point of view formulated in the above remark can be refined to include the case when J is not necessarily principal, but it comes at the expense of strengthening the hypotheses.

Remark 3.5. Assume that I is normal and J = (a_{1}, …, a_{s}). Then for every m we have v_{I}(J, m) = min{v_{I}((a_{j}), m) | j = 1, …, s}. Indeed, if n := min{v_{I}((a_{j}), m) | j = 1, …, s}, then a_{j}^{m} ∈ I^{n} for all j = 1, …, s. This implies that J^{m} ⊆ \overline{J^{m}} = \overline{(a_{1}^{m}, …, a_{s}^{m})} ⊆ \overline{I^{n}} = I^{n}, so v_{I}(J, m) ≥ n. On the other hand, if v_{I}(J, m) > n, we have J^{m} ⊆ I^{v_{I}((a_{j}),m)+1} for some j and hence a_{j}^{m} ∈ I^{v_{I}((a_{j}),m)+1}, a contradiction. If I is normal and all the rings ⨁_{m,n}(a_{j}^{m}) ∩ I^{n} are noetherian (j = 1, …, s), by Proposition 3.2 we obtain that there exists t_{j} such that v_{I}((a_{j}), m + t_{j}) = v_{I}((a_{j}), m) + v_{I}((a_{j}), t_{j}) for m ≥ t_{j}. If we have t_{1} = t_{2} = … = t_{s} = t (the sequences v_{I}((a_{j}), m) increase eventually in a periodic way with the same period), then we have v_{I}(J, m + t) = v_{I}(J, m) + v_{I}(J, t) for m ≥ t. Indeed, by the above observation, v_{I}(J, m + t) = v_{I}((a_{j}), m + t_{j}) for some j, and hence v_{I}(J, m + t) = v_{I}((a_{j}), m) + v_{I}((a_{j}), t) ≤ v_{I}(J, m) + v_{I}(J, t). The other inequality always holds.

Note that in the situation described in Remark 3.4, when G(I) = ⨁_{n≥0} I^{n}/I^{n+1} is reduced (which implies that I is normal) we have t_{1} = t_{2} = … = t_{s} = 1.

Our final observation introduces a bigraded ring associated to the ideals J and I that can be used in examining the periodicity of the rate of change of the sequence {v(m)}_{m}.

Remark 3.6. Let C be the ring ⨁_{m≥0,n≥0} F_{m,n}, where F_{m,n} = J^{m} ∩ I^{n}/J^{m} ∩ I^{n+1}, with the multiplication defined naturally such that F_{m,n}F_{m′,n′} ⊆ F_{m+m′,n+n′}. Let F_{m} = ⨁_{n≥0} F_{m,n}. Note that F_{m} is a filtration on G(I) = ⨁_{n≥0} I^{n}/I^{n+1} and F_{m,n} = 0 for n < v(m), while F_{m,v(m)} ≠ 0 for all m. As in the above remark, one can check that v(m + t) = v(m) + v(t) is equivalent to F_{m,v(m)}F_{t,v(t)} ≠ 0.

So, if there exists t such that F_{t,v(t)} contains a nonzerodivisor on C, then v(m + t) = v(m) + v(t) for all m. However, note that C a domain implies that F_{0} = G(I), the associated graded ring of I, is a domain as well, and then Remark 3.4 applies.

4. Computations

In this section we describe a method of determining L_{J}(I) = inf{m/n | J^{m} ⊆ I^{n}} (and l_{I}(J) = 1/L_{J}(I)) for two monomial ideals I and J in a polynomial ring k[x_{1}, …, x_{r}] over a field k. Whenever J = (a_{1}, …, a_{s}), one has L_{J}(I) = max{L_{(a_{j})}(I) | j = 1, …, s} ( [8, Theorem 2]), so we may assume that J is a principal ideal. Let I = (x_{1}^{b_{i1}}x_{2}^{b_{i2}} … x_{r}^{b_{ir}} | i = 1, …, t) and J = (x_{1}^{c_{1}}x_{2}^{c_{2}} … x_{r}^{c_{r}}).

First observe that J^{m} ⊆ I^{n} if and only if there exist nonnegative integers y_{1}, …, y_{t} with y_{1} + … + y_{t} = n such that

(4.1)  ∑_{i=1}^{t} b_{ij}y_{i} ≤ c_{j}m for all j = 1, …, r.

Set B_{ij} = (1/c_{j})b_{ij}, z_{i} = y_{i}/(y_{1} + … + y_{t}) = y_{i}/n and z = (z_{1}, …, z_{t}) ∈ ℚ^{t}.

So J^{m} ⊆ I^{n} if and only if there exist z_{i} = y_{i}/n with y_{1} + … + y_{t} = n such that

(4.2)  m/n ≥ \frac{1}{nc_{j}} ∑_{i=1}^{t} b_{ij}y_{i} = ∑_{i=1}^{t} B_{ij}z_{i} for all j = 1, …, r.

Consider the function α : ℝ^{t} → ℝ , α(z) = max_{1≤j≤r}{∑_{i=1}^{t} B_{ij}z_{i}} and the subsets of the rationals Λ_{1} = {m/n | J^{m} ⊆ I^{n}} and Λ_{2} = {α(z) | z_{1}, …, z_{t} ∈ ℚ_{≥0}, z_{1} + … + z_{t} = 1}. We will prove that

(4.3)  inf Λ_{1} = inf Λ_{2}

The inequality ≥ follows from (4.2). For the other inequality, we will show that Λ_{2} ⊆ Λ_{1}. Let α(z) ∈ Λ_{2} with z_{i} = p_{i}/q (1 ≤ i ≤ t, p_{1} + … + p_{t} = q, and p_{i}, q nonnegative integers). The coefficients B_{ij} are rationals, so after clearing the denominators we obtain α(z) = h/lq for some nonnegative integers h, l. By (4.2), since z_{i} = lp_{i}/lq for all i, we have h/lq ∈ Λ_{1}, which finishes the proof of (4.3).

Note that inf Λ_{2} = inf{α(z) | z_{1}, …, z_{t} ∈ ℝ_{≥0}, z_{1} + … + z_{t} = 1}, so we need to minimize the function

α(z) = max{∑_{i=1}^{t} B_{ij}z_{i}|j = 1, …, r}

subject to the constraints

z_{1}, …, z_{t} ≥ 0 and z_{1} + … + z_{t} = 1.

Let Δ_{k} = {z ∈ ℝ_{≥0}^{t}| ∑_{i=1}^{t} B_{ik}z_{i} ≥ ∑_{i=1}^{t} B_{ij}z_{i} for all j ≠ k}. Clearly Δ_{1} ∪ … ∪ Δ_{r} = ℝ_{≥0}^{t}, so

it is enough to minimize the function α on each Δ_{k}.

In conclusion, for each k = 1, …, r, the problem reduces to minimizing the objective function

α(z) = ∑_{i=1}^{t} B_{ik}z_{i}

subject to the constraints

z_{1}, …, z_{t} ≥ 0, z_{1} + … + z_{t} = 1 and

∑_{i=1}^{t} B_{ik}z_{i} ≥ ∑_{i=1}^{t} B_{ij}z_{i} for all j ≠ k.

This is a classical problem linear programming problem which can be algorithmically solved using the simplex method.

Remark 4.1. In general, the limits l_{I}(J) and L_{j}(I) need not be reached by an element of the sequences {v_{I}(J, m)/m}_{m} and {w_{J}(I, n)/n}_{n}, respectively. However, in the monomial case, as the procedure described above shows, there exists a pair (m, n) with J^{m} ⊆ I^{n} and L_{J}(I) = n/m.

Example 4.2. Let A = k[x, y] and I = (x^{3}, x^{2}y, y^{2}), J = (x^{3}y^{7}). In this case, b_{11} = 3, b_{12} = 0, b_{21} = 2, b_{22} = 1, b_{31} = 0, b_{32} = 2, c_{1} = 3, c_{2} = 7 and B_{11} = 3/3 = 1, B_{12} = 0/7 = 0, B_{21} = 2/3, B_{22} = 1/7, B_{31} = 0, B_{32} = 2/7. Then

Δ_{1} = {(z_{1}, z_{2}, z_{3}) ∈ ℝ_{≥0}^{3} | z_{1} + (2/3)z_{2} ≥ (1/7)z_{2} + (2/7)z_{3}}

and

Δ_{2} = {(z_{1}, z_{2}, z_{3}) ∈ ℝ_{≥0}^{3} | (1/7)z_{2} + (2/7)z_{3} ≥ z_{1} + (2/3)z_{2}}.

By using a computer algebra system that has the simplex method implemented, one can obtain that the minimum on each of the sets Δ_{1} and Δ_{2} is 2/9, and hence L_{J}(I) = 2/9.

In fact, the minimum can occur only at the intersection of various regions Δ_{k} (in our case on Δ_{1} ∩ Δ_{2}), for there are no critical points in the interior of Δ_{k}.

Acknowledgement. The authors would like to thank Robert Lazarsfeld for a talk which inspired them to consider the problem treated in the article. They also thank Mel Hochster for pointing out to them the example mentioned in Remark 3.3.

References

[1] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993.

[2] J. B. Fields, Lengths of Tors determined by killing powers of ideals in a local ring, J. Algebra, 247, (2002), 104–133.

[3] R. Hübl and I. Swanson, Discrete valuations centered on local domain, J. Pure Appl. Algebra 161 (2001), no. 1-2, 145–166.

[4] M. Nagata, Note on a paper of Samuel concerning asymptotic properties of ideals, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 30 (1957), 165–175.

[5] D. Rees, Valuations associated with ideals. II, J. London Math. Soc. 31 (1956), 221–228.

[6] D. Rees, A note on analytically unramified local rings, J. London Math. Soc. 36 (1961), 24–28.

[7] J. Sally, One-fibered ideals, in Commutative Algebra, Math. Sci. Research Inst. Publ. 15, Springer-Verlag, New York, 1989, 437–442.

[8] P. Samuel, Some asymptotic properties of powers of ideals, Ann. of Math. (2) 56, (1952), 11–21.

Department of Mathematics, North Dakota State University, Fargo, ND 58105

  E-mail address: catalin.ciuperca@ndsu.edu

Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303

  E-mail address: fenescu@mathstat.gsu.edu

Department of Mathematics, University of Utah, Salt Lake City, UT 84112

  E-mail address: spiroff@math.utah.edu