SEMIGROUPS OF VALUATIONS ON LOCAL RINGS, II

STEVEN DALE CUTKOSKY AND BERNARD TEISSIER

Abstract. Given a noetherian local domain R and a valuation ν of its field of fractions which is non negative on R, we derive some very general bounds on the growth of the number of distinct valuation ideals of R corresponding to values lying in certain parts of the value group Γ of ν. We show that this growth condition imposes restrictions on the semigroups ν(R \ {0}) for noetherian R which are stronger than those resulting from the previous paper [4] of the first author. Given an ordered embedding Γ ⊂ (R^{h})lex, where h is the rank of ν, we also study the shape in R^{h} of the parts of Γ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup ν(R \ {0}).

Let (R, m_{R}) be a local domain, with fraction field K. Suppose that ν is a valuation of K with valuation ring (V, m_{V}) which dominates R; that is, R ⊂ V and m_{V} ∩ R = m_{R}. The value groups Γ of ν which can appear when K is an algebraic function field have been extensively studied and classified, including in the papers MacLane [10], MacLane and Schilling [11], Zariski and Samuel [16], Kuhlmann [9] and Moghaddam [12]. These groups are well understood. The most basic fact is that there is an order preserving embedding of Γ into R^{h} with the lex order, where h is the rank of the valuation, which is less than or equal to the dimension of R. The semigroups

S^{R}(ν) = {ν(f) | f ∈ m_{R} − {0}},

which can appear when R is a noetherian domain with fraction field K dominated by ν, are not well understood, although they are known to encode important information about the ideal theory of R and the geometry and resolution of singularities of Spec R. In particular, after [15], the toric resolutions of singularities of the affine toric varieties associated to certain finitely generated subsemigroups of S^{R}(ν) are closely related to the local uniformizations of ν on R.

In Zariski and Samuel’s classic book on Commutative Algebra [16], two general facts about semigroups S^{R}(ν) of valuations on noetherian local domains are proven (in Appendix 3 to Volume II).

1. For any valuation ν of K which is non negative on R, the semigroup S^{R}(ν) is a well ordered subset of the positive part of the value group Γ of ν, of ordinal type at most ω^{h}, where ω is the ordinal type of the well ordered set N, and h is the rank of the valuation.

2. If ν dominates R, the rational rank of ν plus the transcendence degree of V/m_{V}

over R/m_{R} is less than or equal to the dimension of R. The second condition is the Abhyankar inequality [1].

The first author was partially supported by NSF and by the University Paris 7-Denis Diderot.

AMS classification: Primary: 13A18, 14 E15, 16W50. Secondary: 06F05.

In [6], the authors give some examples showing that some surprising semigroups of rank > 1 can occur as semigroups of valuations on noetherian domains, and raise the general question of finding new constraints on value semigroups and classifying semigroups which occur as value semigroups.

The only semigroups which are realized by a valuation on a one dimensional regular local ring are isomorphic to the natural numbers. The semigroups which are realized by a valuation on a regular local ring of dimension 2 with algebraically closed residue field are much more complicated, but are completely classified by Spivakovsky in [14]. A different proof is given by Favre and Jonsson in [7], and we reformulated the theorem in the context of semigroups in [6]. However, very little is known in higher dimensions. The classification of semigroups of valuations on regular local rings of dimension two does suggest that there may be constraints on the rate of growth of the number of new generators on semigroups of valuations dominating a noetherian domain. In [4], such a constraint is found for rank 1 valuations. We prove in this paper that there is such a constraint for valuations of arbitrary rank.

In [4], a very simple polynomial bound is found on the growth of S^{R}(ν) for a rank 1 valuation ν. This bound allowed the construction in [4] of a well ordered subsemigroup of Q_{+} of ordinal type ω, which is not a value semigroup of a noetherian local domain. Thus the above conditions 1 and 2 do not characterize value semigroups on local domains.

•Unless otherwise stated, in this text all local rings are assumed to be noetherian. A valuation of a local domain is a valuation ν of its field of fractions whose ring R_{ν} contains R in such a way that m_{ν} ∩ R ⊆ m_{R}.

In Section 1 of this paper, we describe a polynomial behavior of valuation ideals P_{φ}(R) = {x ∈ R|ν(x) ≥ φ} and P_{φ}^{+}(R) = {x ∈ R|ν(x) > φ}. Given a valuation ν with center p on a local domain R we find very general polynomial bounds on the growth of the sums of the multiplicities of the finitely generated R/p-modules P_{φ}(R)/P_{φ}^{+}(R) when φ runs through growing regions of the value group Γ of ν viewed as a subgroup of R^{h}. These modules are nonzero precisely when φ is 0 or belongs to the semigroup S^{R}(ν). Our results therefore also bound the number of elements of S^{R}(ν) in those regions. This last result generalizes to all ranks the bound given for rank 1 valuations in [4] (restated as Theorem 1.1 in this paper). The statement and proof for higher rank valuations is significantly more complex.

We give an example (Example 1.2) of a rank 2 semigroup T which satisfies all restrictions on the semigroup of a valuation on an s dimensional local domain imposed by our polynomial bounds for modules over the rank 1 convex subgroup Φ_{1} of the group Γ generated by T, but is not a valuation semigroup on an s dimensional local domain. The proof uses our most general bound, Theorem 1.7, in the case of rank 2 valuations.

Our polynomial bounds are estimates of sums over the intersection of S^{R}(ν) with certain regions of Γ. These regions are defined by their intersections with the convex subgroups of Γ and depend on a certain function φ̃ whose precise definition is given in Definition 1.1 of Section 1. Given a valuation ν_{i+1} composed with ν and an element φ in ν_{i+1}(R \{0}) ⊂ Γ/Φ_{i}, the value φ̃ is the smallest element in the semigroup S^{R}(ν_{i}) which projects to φ; it is an element of Γ/Φ_{i−1}. A lower limit in the sum at level i is determined by the values traced out by φ̃ as φ varies in the semigroup ν_{i+1}(R \ {0}), while the upper limit is of the form φ̃ + y_{i}t_{i}, y_{i} ∈ N. It is interesting to consider how close these regions are to being polydiscs. The most desirable situation is when the value group can be embedded by an order preserving homomorphism into (R^{h})lex in such a way that all of these regions are polydiscs. In the first examples that one is likely to consider, this is in fact the case. However, the general situation is not so simple. In Section 2 we give examples showing that the tilde function can exhibit a rather wild behavior. We show that we can make φ̃ decrease arbitrarily fast, and that this is independent of the embedding. We also show that φ̃ can increase arbitrarily fast, and finally that φ̃ can jump back and forth from negative numbers which decrease arbitrarily fast to positive numbers which increase arbitrarily fast. All these examples are independent of the embedding of Γ into R^{h}. In view of the results of Section 1, the behavior of φ̃ is an interesting measure of the complexity of the valuation.

1. Polynomial bounds on valuation ideals

In this section, we derive some very general bounds on the growth of the number of distinct valuation ideals corresponding to values lying in certain parts of the group Γ.

If G is a totally ordered abelian group, then G_{+} will denote the positive elements of G, and G_{≥0} will denote the nonnegative elements. If R is a local ring, m_{R} will denote its maximal ideal, and length_{R}(N) will denote the length of an R-module N.

Suppose that R is a domain and ν is a valuation of R. Let Γ be the value group of ν. We will denote the value semigroup of ν on R by

S^{R}(ν) = {ν(f) | f ∈ m_{R} − {0}}.

S^{R}(ν) is a subsemigroup of the nonnegative part, Γ_{≥0}, of Γ, and if ν dominates R, so that all elements of the maximal ideal m_{R} of R have positive value, then S^{R}(ν) is a subsemigroup of the semigroup Γ_{+} of positive elements of Γ.

Suppose that I ⊂ R is an ideal. We will write

ν(I) = min{ν(f) | f ∈ I − {0}}.

Note that ν(I) ∈ Γ_{≥0} exists since R is noetherian.

Suppose that φ is an element of the value group Γ. We will denote by P_{φ}(R) the ideal {x ∈ R | ν(x) ≥ φ} and by P_{φ}^{+}(R) the ideal {x ∈ R | ν(x) > φ}. When no confusion on the ring is possible we will write P_{φ}, P_{φ}^{+}. We note that P_{φ}(R)/P_{φ}^{+}(R) = 0 if and only if φ ∉ S^{R}(ν) ∪ {0}. The associated graded ring of ν on R is

gr_{ν}(R) = ⨁_{φ∈Γ} P_{φ}(R)/P_{φ}^{+}(R).

This (R/m_{ν} ∩R)-algebra is not in general finitely generated; it is graded by the semigroup S^{R}(ν), which is not finitely generated in general. Our results can be seen as an extension to these algebras of the classical results on N-graded finitely generated algebras.

Suppose that Γ is a totally ordered abelian group, and a, b ∈ Γ. We set

[a, b] = {x ∈ Γ | a ≤ x ≤ b} and [a, b[= {x ∈ Γ | a ≤ x < b}

The concepts of rank of a valuation, the convex (isolated) subgroups of a valuation group and the corresponding composed valuations are discussed in detail in Chapter VI of [16].

Suppose that ν has rank n. Let

0 = Φ_{0} ⊂ Φ_{1} ⊂ ⋯ ⊂ Φ_{n} = Γ

be the sequence of convex subgroups of Γ. Let ν_{i}, for 1 ≤ i ≤ n, be the valuations on the quotient field of R with which ν is composed. We have ν_{1} = ν. Let

p_{n} ⊆ ⋯ ⊆ p_{1}

be the corresponding centers on R of ν_{i}. We define p_{n+1} = (0).

The value group of ν_{i} is Γ_{i} = Γ/Φ_{i−1}. For 1 ≤ i ≤ n, set

t_{i} = ν_{i}(p_{i}) ∈ Φ_{i}/Φ_{i−1} ⊆ Γ/Φ_{i−1}.

Let

λ_{i} : Γ_{i} = Γ/Φ_{i−1} → Γ_{i+1} = Γ/Φ_{i}

be the corresponding maps from the value group of ν_{i} to the value group of ν_{i+1}. If p_{i} ≠ p_{i+1}, then t_{i} is in the kernel of λ_{i}, which is a rank 1 group. When there is no ambiguity, we denote by φ_{i} the image in Γ/Φ_{i−1} of an element φ ∈ Γ.

Definition 1.1. Given φ_{i} ∈ Γ/Φ_{i−1}, denote by φ̃_{i} ∈ Γ/Φ_{i−2} the minimum of ν_{i−1}(f) for f ∈ R such that ν_{i}(f) = φ_{i}.

This minimum exists since the semigroup S^{R}(ν_{i−1}) is well ordered. Note that λ_{i−1}(φ̃_{i}) = φ_{i}.

If p_{i−1} ≠ p_{i}, we remark that for y_{i−1} ∈ N and φ_{i−1} ∈ [φ̃_{i}, φ̃_{i} + y_{i−1}t_{i−1}], we have the inclusions:

p_{i−1}^{y_{i−1}}P_{φ_{i}} ⊂ P_{φ_{i−1}} ⊂ P_{φ_{i}},

P_{φ_{i}} = Pφ̃_{i} and since Φ_{i}/Φ_{i−1} is of rank one, the number of elements of ν_{i−1}(R \ {0}) in the interval [φ̃_{i}, φ̃_{i} + y_{i−1}t_{i−1}] is finite (see [16], loc. cit.).

Lemma 1.2. Suppose that p_{1} ≠ p_{2}. Then for any function A on R-modules with values in R which is additive on short exact sequences of finitely generated R-modules whose unique minimal prime is p_{1}, we have for all y_{1} ∈ N:

(1)  ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}[} A(P_{φ_{1}}/P_{φ_{1}}^{+}) ≤ A(M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}}),

where M_{φ_{2}} = P_{φ_{2}}/P_{φ_{2}}^{+}, a finitely generated torsion free R/p_{2}-module.

Proof. For y_{1} ∈ N, [φ̃_{2}, φ̃_{2} + t_{1}y_{1}[ intersects S^{R}(ν) ∪ {0} in a finite set {τ_{1}, …, τ_{r}}, with

τ_{1} = φ̃_{2} < τ_{2} < ⋯ < τ_{r} < φ̃_{2} + t_{1}y_{1}.

We have inclusions of R modules whose unique minimal prime is p_{1},

P_{τ_{r}}/P_{φ̃_{2}+t_{1}y_{1}} ⊂ P_{τ_{r−1}}/P_{φ̃_{2}+t_{1}y_{1}} ⋯ ⊂ P_{τ_{1}}/P_{φ̃_{2}+t_{1}y_{1}} = P_{φ_{2}}/P_{φ̃_{2}+t_{1}y_{1}}.

By the additivity of A we have

A(P_{φ_{2}}/P_{φ̃_{2}+t_{1}y_{1}}) = ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}[} A(P_{φ_{1}}/P_{φ_{1}}^{+}).

From the inclusion p_{1}^{y_{1}}P_{φ_{2}} ⊂ P_{φ̃_{2}+y_{1}t_{1}}, we have an exact sequence of R-modules whose unique minimal prime is p_{1}:

0 → P_{φ̃_{2}+t_{1}y_{1}}/(P_{φ_{2}}^{+} + p_{1}^{y_{1}}P_{φ_{2}}) → P_{φ_{2}}/(P_{φ_{2}}^{+} + p_{1}^{y_{1}}P_{φ_{2}}) → P_{φ_{2}}/P_{φ̃_{2}+t_{1}y_{1}} → 0.

Since

M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}} ≅ P_{φ_{2}}/(P_{φ_{2}}^{+} + p_{1}^{y_{1}}P_{φ_{2}}),

we have that

A(P_{φ_{2}}/P_{φ̃_{2}+t_{1}y_{1}}) ≤ A(M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}}),

and the conclusions of the lemma follow. □

Suppose that p_{0} is a prime ideal of R such that p_{2} ⊂ p_{1} ⊂ p_{0}. Let e_{m_{i}}(N) denote the multiplicity of an R_{p_{i}} module N with respect to m_{i} = p_{i}R_{p_{i}}.

From the above lemma, we immediately deduce the following result.

Theorem 1.1. Let R be a local domain and ν a rank 1 valuation of R. Let p_{0} be a prime ideal of R containing the center p_{1} of ν. Then

∑_{φ∈[0,yt_{1}[} e_{m_{0}}((P_{φ}/P_{φ}^{+})_{p_{0}}) ≤ e_{m_{0}}((R/p_{1})_{p_{0}})length_{R_{p_{1}}} (R_{p_{1}}/p_{1}^{y}R_{p_{1}})

for all y ∈ N. Thus we have

∑_{φ∈[0,yt_{1}[} e_{m_{0}}((P_{φ}/P_{φ}^{+})_{p_{0}}) ≤ e_{m_{0}}((R/p_{1})_{p_{0}})P_{R_{p_{1}}} (y)

for y ≫ 0, where P_{R_{p_{1}}} (y) is the Hilbert-Samuel polynomial of the local ring R_{p_{1}}.

Proof. This follows from Lemma 1.2, with A(N) = e_{m_{0}}(N ⊗ R_{p_{0}}) for R-modules N. We take ν = ν_{1}, Γ_{2} = 0, φ_{2} = 0, and p_{1} ⊂ p_{0}, so that φ̃_{2} = 0 and M_{φ_{2}} = R, to get for y ∈ N,

∑_{φ∈[0,yt_{1}[} e_{m_{0}}((P_{φ}/P_{φ}^{+})_{p_{0}}) ≤ e_{m_{0}}(R_{p_{0}}/p_{1}^{y}R_{p_{0}}).

The conclusions of the theorem now follow from the associativity formula for multiplicity, [2], Section 7, no. 1, Proposition 3, which shows that

e_{m_{0}}(R_{p_{0}}/p_{1}^{y}R_{p_{0}}) = e_{m_{0}}((R/p_{1})_{p_{0}})e_{m_{1}}(R_{p_{1}}/p_{1}^{y}R_{p_{1}}) = e_{m_{0}}((R/p_{1})_{p_{0}})length_{R_{p_{1}}} (R_{p_{1}}/p_{1}^{y}R_{p_{1}}).

□

If ν has rank 1, and dominates R, so that p_{1} = p_{0} = m_{R} is the maximal ideal of R, we obtain the inequality of [4],

(2)  #(S^{R}(ν) ∩ [0, yt_{1}[ ) < P_{R}(y)

for y ∈ N sufficiently large. This follows from Theorem 1.1 since P_{φ}/P_{φ}^{+} ≠ 0 if and only if there exists f ∈ R such that ν(f) = φ, and since φ = 0 is not in S^{R}(ν).

As was shown in [4], we may now easily construct a well ordered subsemigroup U of Q_{+} such that U has ordinal type ω and U ≠ S^{R}(ν) for any valuation ν dominating a local domain R. We let T be any subset of Q_{+} which has 1 as its smallest element, and

y^{y} < #([y, y + 1[ ) ∩ T) < ∞

for all y ∈ Z_{+}. Let U = ⋃_{n=1}^{∞} nT be the semigroup generated by T. Then U is well ordered by a result of B. H. Neumann (see [17]), and the function #([0, y[ ∩U) grows faster than y^{d} for any d ∈ N. Since the Hilbert-Samuel polynomial of a noetherian local domain R has degree d = dim R < ∞, it follows from formula (2) that U cannot be the semigroup of a valuation dominating a noetherian local domain.

Suppose now that p_{0} is a prime ideal of R such that p_{2} ⊆ p_{1} ⊆ p_{0}. Let e_{m_{i}}(N) denote the multiplicity of an R_{p_{i}} module N with respect to m_{i} = p_{i}R_{p_{i}}.

If p_{2} = p_{1}, so that t_{1} is not in the kernel of λ_{1}, let us define

φ_{2}^{+} = min{ν_{2}(f) | f ∈ R and ν_{2}(f) > φ_{2}}.

By [16] (Appendix 3, Corollary to Lemma 4), the interval [φ̃_{2}, φ̃_{2}^{+}[ contains only finitely many elements of S^{R}(ν).

Theorem 1.3. Let R be a local domain and ν_{1}, ν_{2} two valuations of R such that ν_{1} is composed with ν_{2} and the difference of their ranks is equal to one. Let p_{0} be a prime ideal of R containing the centers p_{2} ⊆ p_{1} of ν_{1} and ν_{2}. Then we have:

a) Suppose that p_{1} ≠ p_{2}. Then there exists a function s(ε, φ_{2}) such that for φ_{2} ∈ Γ_{2}, ε > 0 and y_{1} ∈ N such that y_{1} > s(ε, φ_{2}), we have

\begin{matrix}∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}[} e_{m_{0}}(P_{φ_{1}}(R_{p_{0}})/P_{φ_{1}}^{+}(R_{p_{0}})) \\ ≤ (1 + ε)\frac{e_{m_{0}}((R/p_{1})_{p_{0}})}{(dim (R/p_{2})_{p_{1}})!}e_{m_{1}}(P_{φ_{2}}(R_{p_{1}})/P_{φ_{2}}^{+}(R_{p_{1}}))y_{1}^{dim (R/p_{2})_{p_{1}}} . \end{matrix}

b) If p_{1} = p_{2} we have the equalities

\begin{aligned}∑_{φ_{1}∈[φ̃_{2},φ̃_{2}^{+}[} e_{m_{0}}(P_{φ_{1}}(R_{p_{0}})/P_{φ_{1}}^{+}(R_{p_{0}})) & = e_{m_{0}}(P_{φ_{2}}(R_{p_{0}})/P_{φ_{2}}^{+}(R_{p_{0}})) \\ & = e_{m_{0}}((R/p_{1})_{p_{0}})e_{m_{1}}(P_{φ_{2}}(R_{p_{1}})/P_{φ_{2}}^{+}(R_{p_{1}})). \end{aligned}

Proof. Assume first that p_{1} ≠ p_{2}. Taking A(N) = e_{m_{0}}(N_{p_{0}}) in Lemma 1.2, and using the identities P_{φ_{1}}(R_{p_{0}}) ≅ (P_{φ_{1}})_{p_{0}}, we obtain

∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}[} e_{m_{0}}(P_{φ_{1}}(R_{p_{0}})/P_{φ_{1}}^{+}(R_{p_{0}})) ≤ e_{m_{0}}((M_{φ_{2}})_{p_{0}}/p_{1}^{y_{1}}(M_{φ_{2}})_{p_{0}}).

Since (p_{1})_{p_{0}} is the unique minimal prime of (M_{φ_{2}})_{p_{0}}/p_{1}^{y_{1}}(M_{φ_{2}})_{p_{0}}, by [2], Section 7, no. 1, Proposition 3, we have

e_{m_{0}}((M_{φ_{2}})_{p_{0}}/p_{1}^{y_{1}}(M_{φ_{2}})_{p_{0}}) = length_{R_{p_{1}}} ((M_{φ_{2}})_{p_{1}}/p_{1}^{y_{1}}(M_{φ_{2}})_{p_{1}})e_{m_{0}}((R/p_{1})_{p_{0}}).

Since p_{2} is the unique minimal prime of M_{φ_{2}}, there exists a function \overline{s}(φ_{2}) such that

length_{R_{p_{1}}} ((M_{φ_{2}})_{p_{1}}/p_{1}^{y_{1}}(M_{φ_{2}})_{p_{1}}) = H_{(M_{φ_{2}})_{p_{1}}} (y_{1})
= \frac{e_{m_{1}}((M_{φ_{2}})_{p_{1}})}{(dim(R/p_{2})_{p_{1}})!}y_{1}^{dim(R/p_{2})_{p_{1}}} + lower order terms in y_{1}

for y_{1} ≥ \overline{s}(φ_{2}), where H_{(M_{φ_{2}})_{p_{1}}} (y_{1}) is the Hilbert-Samuel polynomial of (M_{φ_{2}})_{p_{1}}. This polynomial bound implies that there exists a function s(ε, φ_{2}) such that

length_{R_{p_{1}}} ((M_{φ_{2}})_{p_{1}}/p_{1}^{y_{1}}(M_{φ_{2}})_{p_{1}}) ≤ (1 + ε)\frac{e_{m_{1}}((M_{φ_{2}})_{p_{1}})}{(dim(R/p_{2})_{p_{1}})!}y_{1}^{dim(R/p_{2})_{p_{1}}}

for y_{1} ≥ s(ε, φ_{2}).

If p_{1} = p_{2}, We have p_{1}(P_{φ_{2}}/P_{φ_{2}}^{+}) = p_{2}(P_{φ_{2}}/P_{φ_{2}}^{+}) = 0, so the first inequality stated in this case follows directly from the additivity of the multiplicity e_{m_{0}}. The second equality follows from the first and the associativity formula of [2], Section 7, no. 1, Proposition 3. □

Corollary 1.4. Suppose that p_{n} = ⋯ = p_{2} = p_{1}. Then

∑_{φ_{n}∈[0,yt_{n}[} ∑_{φ_{n−1}∈[φ̃_{n},φ̃_{n}^{+}[} ⋯ ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}^{+}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+}) ≤ e_{m_{0}}((R/p_{1})_{p_{0}})length_{R_{p_{1}}} (R_{p_{1}}/p_{1}^{y}R_{p_{2}})

for all y ∈ N. Thus we have

∑_{φ_{n}∈[0,yt_{n}[} ∑_{φ_{n−1}∈[φ̃_{n},φ̃_{n}^{+}[} ⋯ ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}^{+}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+}) ≤ e_{m_{0}}((R/p_{1})_{p_{0}})P_{R_{p_{1}}} (y)

for y ≫ 0, where P_{R_{p_{1}}} (y) is the Hilbert-Samuel polynomial of R_{p_{1}}.

Proof. We will prove the formula by induction on the rank n of the valuation. If n = 1, this is just the statement of Theorem 1.1. We will assume that the formula is true for valuations of rank < n, and derived the formula for a rank n valuation ν. Let ν_{2} be the rank n − 1 valuation which ν is composite with. Consider the chain of ideals

(0) = q_{n} ⊂ q_{n−1} = ⋯ = q_{1} = q_{0},

where q_{n−1} = p_{n}, …, q_{1} = p_{2} are the centers on R of the successive valuations ν_{n}, …, ν_{2} with which ν_{2} is composed, and q_{0} = p_{1}. We obtain

(3) ∑_{φ_{n}∈[0,yt_{n}[} ∑_{φ_{n−1}∈[φ̃_{n},φ̃_{n}^{+}[} ⋯ ∑_{φ_{2}∈[φ̃_{3},φ̃_{3}^{+}[} e_{m_{1}}(P_{φ_{2}}/P_{φ_{2}}^{+}) ≤ e_{m_{1}}((R/p_{2})_{p_{1}})P_{R_{p_{2}}} (y) = P_{R_{p_{1}}} (y)

for y ≫ 0. We apply Theorem 1.3 to the valuations ν = ν_{1} and ν_{2} and p_{2} = p_{1} ⊂ p_{0} to obtain for φ_{2} ∈ [φ̃_{3}, φ̃_{3}^{+}[ (or φ_{2} ∈ [φ̃_{n}, φ̃_{n}^{+} + t_{n}y[ if n = 3),

(4)  ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}^{+}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+})_{p_{0}}) ≤ e_{m_{0}}((R/p_{1})_{p_{0}})e_{m_{1}}((P_{φ_{2}}/P_{φ_{2}}^{+})_{p_{1}}).

Now sum over (3) and (4) to obtain the formula for ν.

□

Corollary 1.5. In the special case where ν_{1} is a valuation of rank one and ν_{2} is the trivial valuation, we have p_{2} = 0, Γ_{2} = 0, φ̃_{2} = 0 ∈ Γ_{1} and the inequality of Theorem 1.3 reduces to:

(5)  \begin{matrix}∑_{φ_{1}∈[0,y_{1}t_{1}[} e_{m_{0}}(P_{φ}(R_{p_{0}})/P_{φ}^{+}(R_{p_{0}})) \\ ≤ (1 + ε)e_{m_{0}}((R/p_{1})_{p_{0}})\frac{e_{m_{1}}(R_{p_{1}})}{(dimR_{p_{1}})!}y_{1}^{dimR_{p_{1}}} \end{matrix}

for y_{1} > s(ε).

Corollary 1.6. Taking p_{0} = p_{1}, we deduce that when p_{1} ≠ p_{2} we have for y_{1} > s(ε, φ_{2}) an inequality

\begin{matrix}#(λ_{1}^{−1}(φ_{2}) ∩ S^{R}(ν) ∩ [φ̃_{2}, φ̃_{2} + y_{1}t_{1}[) \\ ≤ (1 + ε)\frac{e_{m_{1}}(P_{φ_{2}}(R_{p_{1}})/P_{φ_{2}}^{+}(R_{p_{1}}))}{(dim(R/p_{2})_{p_{1}})!}y_{1}^{dim(R/p_{2})_{p_{1}}} . \end{matrix}

When p_{1} = p_{2}, we have

#(λ_{1}^{−1}(φ_{2}) ∩ S^{R}(ν)) ≤ length_{R_{p_{1}}} (P_{φ_{2}}(R_{p_{1}})/P_{φ_{2}}^{+}(R_{p_{1}})) < ∞.

Let

(0) = p_{n+1} ⊂ p_{n} ⊆ ⋯ ⊆ p_{1}

be the centers of the valuations with which ν is composed. Define I = {i ∈ {1, …, n}/p_{i} ≠ p_{i+1}} and note that n ∈ I. By [16] (Appendix 3) we know that if i ∉ I, if we define

φ_{i}^{+} = min{ν_{i}(f)|f ∈ R and ν_{i}(f) > φ_{i}},

the intersection S^{R}(ν) ∩ [φ̃_{i}, φ̃_{i}^{+}[ is finite. Let us agree that in this case, for large y_{i} the interval [φ̃_{i+1}, φ̃_{i+1} + t_{i}y_{i}[ coincides with this intersection. Remember also that if i ∉ I we have dim(R/p_{i+1})_{p_{i}} = 0 and e_{m_{i}}((R/p_{i+1})_{p_{i}}) = (dim(R/p_{i+1})_{p_{i}})! = 1.

Theorem 1.7. Let R be a local domain and ν a valuation of R which is of rank n. There exist functions s_{n}(ε) and s_{i}(ε, y_{i+1}, y_{i+2}, …, y_{n}) for 1 ≤ i ≤ n − 1, such that, using the notations and conventions introduced above, we have

\begin{matrix}∑_{φ_{n}∈[0,t_{n}y_{n}[} ∑_{φ_{n−1}∈[φ̃_{n},φ̃_{n}+t_{n−1}y_{n−1}[} ⋯ ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+t_{1}y_{1}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+})_{p_{0}}) \\ ≤ (1 + ε)\frac{∏_{i=0}^{n} e_{m_{i}}((R/p_{i+1})_{p_{i}})}{∏_{i=1}^{n}(dim(R/p_{i+1})_{p_{i}})!} ∏_{i=1}^{n} y_{i}^{dim(R/p_{i+1})_{p_{i}}} \end{matrix}

for y_{n}, y_{n−1}, …, y_{1} ∈ N satisfying

y_{n} ≥ s_{n}(ε), y_{n−1} ≥ s_{n−1}(ε, y_{n}), …, y_{1} ≥ s_{1}(ε, y_{2}, …, y_{n}).

Proof. The proof of this formula is by induction on the rank n of the valuation ν. We first prove the formula in the case when n = 1. We apply (5) to the ring R_{p_{0}} and observe that for φ_{1} ∈ Γ_{1}, P_{φ_{1}}(R_{p_{0}}) ≅ (P_{φ_{1}})_{p_{0}}, to obtain

∑_{φ_{1}∈[0,t_{1}y_{1}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+})_{p_{0}}) ≤ (1 + ε)\frac{e_{m_{0}}((R/p_{1})_{p_{0}})}{(dim R_{p_{1}})!}e_{m_{1}}(R_{p_{1}})y_{1}^{dim R_{p_{1}}}

for y_{1} ≥ s_{1}(ε), which is the formula for n = 1.

We now assume that the formula is true for valuations of rank < n. We will derive the formula for a rank n valuation ν. We apply the formula to the rank n − 1 valuation ν_{2} which ν is composite with, and the chain of prime ideals

(0) = q_{n} ⊂ q_{n−1} ⊂ ⋯ ⊂ q_{1} ⊂ q_{0}

where q_{n−1} = p_{n}, …, q_{1} = p_{2} are the centers on R of the successive valuations ν_{n}, …, ν_{2} with which ν_{2} is composed, and q_{0} = p_{1} is the new ”mute” prime ideal. We obtain the inequality (6)

∑_{φ_{n}∈[0,t_{n}y_{n}[} ⋯ ∑_{φ_{2}∈[φ̃_{3},φ̃_{3}+t_{2}y_{2}[} e_{m_{1}}((P_{φ_{2}}/P_{φ_{2}}^{+})_{p_{1}}) ≤ (1+ε_{1})\frac{∏_{i=1}^{n} e_{m_{i}}((R/p_{i+1})_{p_{i}})}{∏_{i=2}^{n}(dim (R/p_{i+1})_{p_{i}})!} ∏_{i=2}^{n} y_{i}^{dim (R/p_{i+1})_{p_{i}}}

for

y_{n} ≥ s_{n}(ε_{1}), y_{n−1} ≥ s_{n−1}(ε_{1}, y_{n}), …, y_{2} ≥ s_{2}(ε_{1}, y_{3}, …, y_{n−1}).

We apply Theorem 1.3 to the valuations ν = ν_{1} and ν_{2}, and p_{2} ⊆ p_{1} ⊆ p_{0}, to obtain for φ_{2} ∈ [φ̃_{3} + t_{2}y_{2}[, In the case where p_{2} ≠ p_{1} (7)

∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+})_{p_{0}}) ≤ (1 + ε_{1})\frac{e_{m_{0}}((R/p_{1})_{p_{0}})}{(dim (R/p_{2})_{p_{1}})!}e_{m_{1}}((P_{φ_{2}}/P_{φ_{2}}^{+})_{p_{1}})y_{1}^{dim (R/p_{2})_{p_{1}}}

for y_{1} > s(ε_{1}, φ_{2}). Since #(S^{R}(ν) ∩ [φ̃_{3}, φ̃_{3} + t_{2}y_{2}[ ) < ∞, we may define

s_{1}(ε_{1}, y_{2}, y_{3}, …, y_{n}) = max{s(ε_{1}, φ_{2}) | φ_{2} ∈ [φ̃_{3}, φ̃_{3}+t_{2}y_{2}[, φ_{3} ∈ [φ̃_{4}, φ̃_{4}+t_{3}y_{3}[, …, φ_{n} ∈ [0, t_{n}y_{n}[}.

In the case where p_{2} = p_{1} we have by theorem 1.3, b) the equality

∑_{φ_{1}∈[φ̃_{2},φ̃_{2}^{+}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+})_{p_{0}}) = e_{m_{0}}((P_{φ_{2}}/P_{φ_{2}}^{+})_{p_{0}}) = e_{m_{0}}((R/p_{1})_{p_{0}})e_{m_{1}}((P_{φ_{2}}/P_{φ_{2}}^{+})_{p_{1}}),

and define s_{1}(ε_{1}, y_{2}) = 1.

Finally, we set

ε_{1} = 2^{\frac{1}{2}log_{2}(1+ε)} − 1,

so that (1 + ε_{1})^{2} = 1 + ε, and sum over (7) and (6) after multiplication by the appropriate factor to obtain the desired formula for ν. □

As an immediate corollary, we obtain

Corollary 1.8. The sum

∑_{φ_{n}∈[0,t_{n}y_{n}[} ∑_{φ_{n−1}∈[φ̃_{n},φ̃_{n}+t_{n−1}y_{n−1}[} ⋯ ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+t_{1}y_{1}[} e_{m_{0}}((P_{φ_{1}}/P_{φ_{1}}^{+})_{p_{0}})

is bounded for y_{1} ≫ y_{2} ≫ ⋯ ≫ y_{n} ≫ 0 by a function which behaves asymptotically as

\frac{∏_{i=0}^{n} e_{m_{i}}((R/p_{i+1})_{p_{i}})}{∏_{i=1}^{n}(dim(R/p_{i+1})_{p_{i}})!} ∏_{i=1}^{n} y_{i}^{dim (R/p_{i+1})_{p_{i}}} .

Using the notations of Definition 1.1, and the conventions preceding Theorem 1.7, define the pseudo-boxes

B_{Γ}(y_{1}, …, y_{n}) = {φ ∈ Γ/φ ∈ [φ̃_{2}, φ̃_{2} + t_{1}y_{1}[, φ_{2} ∈ [φ̃_{3}, φ̃_{3} + t_{2}y_{2}[, …, φ_{n} ∈ [0, t_{n}y_{n}[}.

Then we have:

Corollary 1.9. For y_{1} ≫ y_{2} ≫ ⋯ ≫ y_{n} ≫ 0 the number #(S^{R}(ν) ⋂ B_{Γ}(y_{1}, …, y_{n})) is bounded by the same function as in Corollary 1.8.

Remarks 1.10. 1) The only centers p_{i} which contribute to the right hand side of the inequalities are those for which the inclusion p_{i+1} ⊆ p_{i} is strict.

2) The total degree of the monomial appearing on the right hand side is dim R−dim R/p_{1}, which is dim R in the case where ν is centered at the maximal ideal m_{R}.

We now give an application of Theorem 1.7. Suppose that ν is a rank 2 valuation dominating a local domain R. Let Γ_{2} be the value group of the composed valuation ν_{2} of the quotient field of R, and let p_{2} be the center of ν_{2} on R, t_{1} = ν(m_{R}). Theorem 1.3 gives us a family of growth conditions for φ_{2} ∈ Γ_{2} on S^{R}(ν) ∩ [φ̃_{2}, nt_{1}[ for n sufficiently large. To be precise, Theorem 1.3 tells us that for each φ_{2} ∈ Γ_{2}, there exist functions d(φ_{2}) and s(φ_{2}) ∈ N such that

(8)  #(S^{R}(ν) ∩ [φ̃_{2}, φ̃_{2} + nt_{1}[ ) < d(φ_{2})n^{dim R/p_{2}}

for n > s(φ_{2}).

Example 1.2. For every natural number s ≥ 3, there exists a rank 2, well ordered subsemigroup T of the positive part of (Z × Q)lex, which is of ordinal type ω^{2} and satisfies the restrictions (8) for all φ_{2} ∈ N, but is not the semigroup of a valuation dominating an s dimensional local domain.

Proof. Let r = s − 2. Define a subsemigroup of Q_{≥0} by

\begin{aligned}S & = & \{(2^{m} + j) + \frac{α}{2^{(m+1)r}} | m, j, α ∈ N, \\ & & 0 ≤ j < 2^{m}, 0 ≤ α < 2^{(m+1)r}\}. \end{aligned}

Suppose that n is a positive integer. Then there exists a unique expression n = 2^{m} + j with 0 ≤ j < 2^{m}. We have

#(S ∩ [n, n + 1[ ) = 2^{(m+1)r},

and since 2^{m} ≤ n < 2^{m+1},

(9)  n^{r} < #(S ∩ [n, n + 1[ ) ≤ 2^{r}n^{r}.

For y a positive integer, let f(y) = ∑_{n=1}^{y−1} n^{r}. The first difference function f(y + 1) − f(y) = y^{r} is a polynomial of degree r in y. Thus f(y) is a polynomial of degree r + 1 in y, with positive leading coefficient. From

#(S ∩ [0, y) = ∑_{n=1}^{y−1} #(S ∩ [n, n + 1[ )

and (9), we deduce that

(10)  f(y) < #(S ∩ [0, y[ ) ≤ 2^{r}f(y).

Suppose that c ∈ N. Then #((\frac{1}{c}S) ∩ [0, y[ ) = #(S ∩ [0, cy[ ). Thus

(11)  f(cy) < #((\frac{1}{c}S) ∩ [0, y[ ) ≤ 2^{r}f(cy).

For i ∈ N, let

(12)  c(i) = \{ \begin{matrix}1 & if i = 0 \\ i & if i ≥ 1. \end{matrix}

Let

T = ⋃_{m∈N}\{m\} × (\frac{1}{c(m)}S) ⊂ (Z × Q)lex.

T is a well ordered subsemigroup of (Z × Q)lex, of ordinal type ω^{2}.

Suppose that T is the semigroup S^{R}(ν) of a valuation ν dominating an s = r + 2 dimensional local domain R. Then ν has rank 2. Let ν_{2} be the composed valuation ν_{2}(f) = π_{1}(ν(f)) for f ∈ R, where π_{1} : Z × Q → Z is the first projection. By assumption, the center of ν on R is the maximal ideal m_{R} of R. Let p_{2} be the center of ν_{2} on R. We see from an inspection of T that t_{1} = ν(m_{R}) = (0, 1) and t_{2} = ν_{2}(p_{2}) = 1. Further, φ̃_{2} = (φ_{2}, \frac{1}{c(φ_{2})}) for all φ_{2} ∈ Z_{+}. Observe that for all φ_{2} ∈ Z_{+}, and y_{1} ∈ N,

#(T ∩ [φ̃_{2}, φ̃_{2} + y_{1}t_{1}[ ) = #(T ∩ \{φ_{2}\} × [0, y_{1}[ ) = #(\frac{1}{c(φ_{2})}S ∩ [0, y_{1}[ ).

From (11), we see that

(13)  f(c(φ_{2})y_{1}) < #(T ∩ {φ_{2}} × [0, y_{1}[ ) ≤ 2^{r}f(c(φ_{2})y_{1}).

Thus T satisfies the growth conditions (8) on a local domain R with dim R/p_{2} ≥ r + 1. Since T has rank 2, we must have that dim R ≥ dim R/p_{2} + 1.

Since we are assuming that R has dimension s = r + 2, we have that dim R/p_{2} = r + 1 and dim R_{p_{2}} = 1. Since ν_{2} is a discrete rank 1 valuation dominating R_{p_{2}}, this is consistent.

Theorem 1.7 tells us that there exists a function s(y_{2}) and d ∈ Z_{+} such that

#([0, y_{2}[ ×[0, y_{1}[ ∩ T) ≤ dy_{1}^{a}y_{2}^{b}

for y_{1} ≥ s(y_{2}), where a = dim R/p_{2} = r + 1 and b = dim R_{p_{2}} = 1. From (13) and (12), we see that

f(y_{1}) + ∑_{i=1}^{y_{2}−1} f(iy_{1}) ≤ #([0, y_{2}[ ×[0, y_{1}[ ∩ T).

There exists a positive constant e_{1} such that f(y_{1}) ≥ e_{1}y_{1}^{r+1} for all y_{1} ∈ N, and thus there exists a positive constant e such that

f(y_{1}) + ∑_{i=1}^{y_{2}−1} f(iy_{1}) ≥ ey_{1}^{r+1}y_{2}^{r+1}

for y_{1}, y_{2} ∈ N. Thus we have

dy_{1}^{a}y_{2}^{b} = dy_{1}^{r+1}y_{2} > ey_{1}^{r+1}y_{2}^{r+1}

for all large y_{2}, which is impossible.

□

2. Wild behavior of the tilde function

The tilde function φ̃, defined in Definition 1.1, gives critical information about the behavior of valuations of rank larger than one. This is illustrated by its role in the statement of Theorem 1.7, which shows that there is some order in the behavior of semigroups of higher rank valuations. However, the sums in this theorem are all defined starting from the functions φ̃. This function can be extremely chaotic, as we will illustrate in this section.

We will give examples of rank two valuations, showing that φ̃ can decrease arbitrarily fast as φ increases (Example 2.2), φ̃ can increase arbitrarily fast as φ increases (Example 2.3), and that φ̃ can jump back and forth from negative numbers which decrease arbitrarily fast to positive numbers which increase arbitrarily fast as φ increases (Example 2.4). These properties are all independent of order preserving isomomorphism of the value group.

To construct our examples, we will make use of the following technical lemma, and some variants of it. This lemma is a generalization of the notion of generating sequences of valuations on regular local rings of dimension 2 ([10], [14], [7]).

Lemma 2.1. Suppose that σ : Z_{+} → N is a function. Let K(x, y, z) be a rational function field in three variables over a field K. Set

P_{0} = x, P_{1} = y

and

P_{i+1} = z^{σ(i)}P_{i}^{2} − P_{0}^{2^{i+1}}P_{i−1}

for i ≥ 1.

Define, by induction on i, η_{0} = 1, and

(14)  η_{i+1} = 2η_{i} + \frac{1}{2^{i+1}}

for i ≥ 0. Define γ_{0} = 0 and

(15)  γ_{i} = −(\frac{σ(1)}{2^{i}} + \frac{σ(2)}{2^{i−1}} + ⋯ + \frac{σ(i)}{2})

for i > 0.

1. Suppose that f(x, y, z) ∈ K(z)[x, y]. Then for l ∈ N such that deg_{y}f < 2^{l}, there is a unique expansion

(16)  f = ∑_{α} a_{α}(z)x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{l}^{α_{l}}

where a_{α}(z) ∈ K(z) and the sum is over α = (α_{0}, α_{1}, …, α_{l}) ∈ N × {0, 1}^{l}.

2. For f ∈ K(z)[x, y], define from the expansion (16),

(17)  ν(f) = min_{α}\{(0, ord_{z}(a_{α}(z)) + ∑_{i=0}^{ℓ} α_{i}(η_{i}, γ_{i})\} ∈ (\frac{1}{2^{∞}} Z × \frac{1}{2^{∞}} Z)lex.

Then ν defines a rank 2 valuation on K(x, y, z), which is composite with a rank 1 valuation ν_{2} of K(x, y, z). The value group of ν_{2} is \frac{1}{2^{∞}} Z = ⋃_{i=1}^{∞} \frac{1}{2^{i}} Z.

The valuation ν dominates the local ring R = K[x, y, z]_{(x,y,z)} and the center of ν_{2} on R is the prime ideal (x, y).

Proof. We have that deg_{y}P_{i} = 2^{i−1} for i ≥ 1. Suppose that f ∈ K(z)[x, y] and ℓ is such that deg_{y}f < 2^{ℓ}. By the euclidean algorithm, we have a unique expansion

f = g_{0}(x, y) + g_{1}(x, y)P_{l}

with g_{0}, g_{1} ∈ K(z)[x, y] and deg_{y}g_{0} < 2^{l−1}, deg_{y}g_{1} < 2^{l−1}. Iterating, we have a unique expansion of f of the form of (16).

We have

(18)  ν(P_{i}) = (η_{i}, γ_{i})

and

(19)  ν(z^{σ(i)}P_{i}^{2}) = ν(P_{0}^{2^{i+1}}P_{i−1}) < ν(P_{i+1})

for all i.

Observe that for

α = (α_{0}, …, α_{ℓ}), β = (β_{0}, …, β_{ℓ}) ∈ N × {0, 1}^{ℓ},

(20)  ∑_{i=0}^{ℓ} α_{i}η_{i} = ∑_{i=0}^{ℓ} β_{i}η_{i}

implies α = β.

The function ν defined by (17) thus has the property that there is a unique term in the expansion (16) for which the minimum (17) is achieved. We will verify that ν defines a valuation on K(x, y, z). Suppose that f, g ∈ K(z)[x, y]. Let

(21)  f = ∑_{α} a_{α}(z)x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{ℓ}^{α_{ℓ}}

and

(22)  g = ∑_{β} a_{β}(z)x^{β_{0}}P_{1}^{β_{1}} ⋯ P_{ℓ}^{β_{ℓ}}

be the expressions of f and g of the form (16).

f + g = ∑_{α}(a_{α}(z) + b_{α}(z))x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{ℓ}^{ℓ}

is the expansion of f+g of the form (16). Since ord_{z}(a_{α}(z)+b_{α}(z)) ≥ min{ord_{z}(a(z)), ord_{z}(b(z))} for all α, we have that

ν(f + g) ≥ min{ν(f), ν(g)}.

We will now show that ν(fg) = ν(f) + ν(g).

Suppose that

(23)  s = ∑_{δ} d_{δ}(z)x^{δ_{0}}P_{1}^{δ_{1}} ⋯ P_{ℓ}^{δ_{ℓ}}

is an expansion, with d_{δ} ∈ K(z), and δ = (δ_{0}, …, δ_{ℓ}) ∈ N^{ℓ+1} for all δ. We define

Λ(s) = min_{δ}{(0, ord_{z}d_{δ}(z)) + ∑_{i=0}^{ℓ} δ_{i}(η_{i}, γ_{i})}.

Observe that if s is an expansion of the form (16); that is, δ ∈ N × {0, 1}^{ℓ} for all δ, then Λ(s) = ν(s).

Let

c_{ε}(z) = ∑_{α+β=ε} a_{α}(z)b_{β}(z).

Let

(24)  s_{0} = ∑_{ε} c_{ε}(z)x^{ε_{0}}P_{1}^{ε_{1}} ⋯ P_{ℓ}^{ε_{ℓ}}.

s_{0} is an expansion of the form (23), and fg = s_{0}. To simplify the indexing later on in the proof, we observe that we can initially take ℓ as large as we like.

Let α′ (in the expansion (21)) be such that

ν(f) = ν(a_{α′}(z)x^{α′_{0}}P_{1}^{α′_{1}} ⋯ P_{ℓ}^{α′_{ℓ}})

and let β′ (in the expansion (22)) be such that

ν(g) = ν(b_{β′}(z)x^{β′_{0}}P_{1}^{β′_{1}} ⋯ P_{ℓ}^{β′_{ℓ}}).

Let ε′ = α′ + β′. Then

c_{ε′}(z)x^{ε′_{0}}P_{1}^{ε′_{1}} ⋯ P_{ℓ}^{ε′_{ℓ}}

is the only term in s_{0} which achieves the minimum Λ(s_{0}). We have that c_{ε′}(z) = a_{α′}(z)b_{β′}(z) and

(25)  Λ(s_{0}) = ν(f) + ν(g).

If ε ∈ N × {0, 1}^{ℓ} whenever c_{ε} ≠ 0, then we can can compute ν(fg) = Λ(s_{0}) and we are done. Otherwise, there exists an i ≥ 1 such that there exists an ε with ε_{i} ≥ 2 and c_{ε}(z) ≠ 0. We then substitute the identity:

(26)  P_{i}^{2} = \frac{1}{z^{σ(i)}} P_{i+1} + \frac{1}{z^{σ(i)}} x^{2^{i+1}}P_{i−1}

into s_{0} to obtain an expansion of the form (23), where all terms

c_{ε}(z)x^{ε_{0}}P_{1}^{ε_{1}} ⋯ P_{ℓ}^{γ_{ℓ}}

with ε_{i} ≥ 2 are modified to the sum of two terms

\frac{c_{ε}(z)}{z^{σ(i)}}x^{ε_{0}+2^{i+1}}P_{1}^{ε_{1}} ⋯ P_{i−1}^{ε_{i−1}+1}P_{i}^{ε_{i}−2}P_{i+1}^{ε_{i+1}} ⋯ P_{ℓ}^{ε_{ℓ}}+\frac{c_{ε}(z)}{z^{σ(i)}}x^{ε_{0}}P_{1}^{ε_{1}} ⋯ P_{i−1}^{ε_{i−1}}P_{i}^{ε_{i}−2}P_{i+1}^{ε_{i+1}+1} ⋯ P_{ℓ}^{ε_{ℓ}}.

Collecting terms with like monomials in x, P_{1}, …, P_{ℓ}, we obtain a new expansion

s_{1} = ∑ d_{δ}(z)x^{δ_{0}}P_{1}^{δ_{1}} ⋯ P_{ℓ}^{δ_{ℓ}}

of fg. From the identities (19), we see that the minimum Λ(s_{1}) is only obtained by the term d_{δ′}(z)x^{δ′_{0}}P_{1}^{δ′_{1}} ⋯ P_{ℓ}^{δ′_{ℓ}}, where

d_{δ′}(z)x^{δ′_{0}}P_{1}^{δ′_{1}} ⋯ P_{ℓ}^{δ′_{ℓ}} = \{ \begin{matrix}c_{ε′}(z)x^{ε′_{0}}P_{1}^{ε′_{1}} ⋯ P_{ℓ}^{ε′_{ℓ}} & if ε′_{i} < 2 \\ \frac{c_{ε′}(z)}{z^{σ(i)}}x^{ε′_{0}+2^{i+1}}P_{1}^{ε′_{1}} ⋯ P_{i−1}^{ε′_{i−1}+1}P_{i}^{ε′_{i}−2}P_{i+1}^{ε′_{i+1}} ⋯ P_{ℓ}^{ε′_{ℓ}} & if ε′_{i} ≥ 2. \end{matrix}

We have Λ(s_{1}) = Λ(s_{0}).

By descending induction on the invariants

n = max{δ_{1} + ⋯ + δ_{ℓ} | some δ_{i} ≥ 2 and d_{δ}(z) ≠ 0}

and

m = #{(δ_{1}, …, δ_{ℓ}) ∈ N^{ℓ} | δ_{1} + ⋯ + δ_{ℓ} = n, some δ_{i} ≥ 2 and d_{δ}(z) ≠ 0},

making substitutions of the form (26), we eventually obtain an expression s of fg of the form (23), with (δ_{0}, …, δ_{ℓ}) ∈ N × {0, 1}^{ℓ} for all ℓ. We then compute ν(fg) = Λ(s) = Λ(s_{0}) = ν(f) + ν(g). We have thus completed the verification that ν is a valuation.

□

Example 2.2. Suppose that f : N → Z is a decreasing function, and K is a field. Then there exists a valuation ν of the three dimensional rational function field K(x, y, z) with value group (\frac{1}{2^{∞}} Z × Z)lex, which dominates the regular local ring R = K[x, y, z]_{(x,y,z)}, such that for any valuation ω equivalent to ν with value group (\frac{1}{2^{∞}} Z × Z)lex, for all sufficiently large n ∈ N, there exists λ ∈ \frac{1}{2^{∞}} Z ∩ [0, n[ such that π_{2}(λ̃) < f(n), where π_{2} : \frac{1}{2^{∞}} Z × Z → Z is the second projection.

Proof. We choose positive integers σ(i) so that

γ_{i} = −(\frac{σ(1)}{2^{i}} + \frac{σ(2)}{2^{i−1}} + ⋯ + \frac{σ(i)}{2}) < f(i2^{i+3})

for all positive integers i, where η_{i} are defined by (14), and so that γ_{i} ∈ Z. Let ν be the valuation dominating R defined by Lemma 2.1, with this choice of σ. The value group of ν is (\frac{1}{2^{∞}} Z × Z)lex.

Let ω be a valuation equivalent to ν with value group (\frac{1}{2^{∞}} Z × Z)lex. We have ω(x) = (a, b) for some a ∈ \frac{1}{2^{∞}} Z_{+} and b ∈ Z, and ω(z) = (0, c) for some c ∈ Z_{+} since convex subgroups have to be preserved under an automorphism of ordered groups. From the relations (19) we see that ω(P_{i}) = η_{i}ω(x) + γ_{i}ω(z) for i ≥ 1, which implies b = 0, since π_{2}(ω(P_{i})) ∈ Z. We also have ω(a(z)) = ord_{z}(a)ω(z) for a(z) ∈ K(z). Let ω_{2} be the valuation on K(x, y, z) defined by ω_{2}(f) = π_{1}(ω(f)), where π_{1} : \frac{1}{2^{∞}} Z × Z → \frac{1}{2^{∞}} Z is the first projection.

Let

e = ⌈a⌉, and n_{0} = ⌈a⌉2^{e+2}.

Suppose that n ≥ n_{0}. We will find λ ∈ \frac{1}{2^{∞}} Z ∩ [0, n[ such that π_{2}(λ̃) < f(n).

There exists i ≥ e such that

⌈a⌉2^{i+2} ≤ n < ⌈a⌉2^{i+3}.

Let λ = aη_{i}. From η_{i} = \frac{1}{3}(2^{i+2} − \frac{1}{2^{i}}), we obtain

λ = aη_{i} < ⌈a⌉2^{i+2} ≤ n.

Since c ∈ Z_{+}, i ≥ e and f is decreasing, we have

cγ_{i} ≤ γ_{i} < f(i2^{i+3}) ≤ f(⌈a⌉2^{i+3}) < f(n).

Thus

π_{2}(λ̃) ≤ π_{2}(ω(P_{i})) = cγ_{i} < f(n).
□

Example 2.3. Suppose that g : N → Z is an increasing function, and K is a field. Then there exists a valuation ν of the three dimensional rational function field K(x, y, z) with value group (\frac{1}{2^{∞}} Z×Z)lex, which dominates the regular local ring R = K[x, y, z]_{(x,y,z)}, such that for any valuation ω equivalent to ν with value group (\frac{1}{2^{∞}} Z × Z)lex, for all sufficiently large n ∈ N, there exists λ ∈ \frac{1}{2^{∞}} Z∩[0, n[ such that π_{2}(λ̃) > g(n), where π_{2} : \frac{1}{2^{∞}} Z×Z → Z is the second projection.

Proof. The proof is a variation of the proof of Example 2.2. We outline it here. We first must establish a modification of Lemma 2.1. Suppose that τ : Z_{+} → N is a function. Set

Q_{0} = x, Q_{1} = y

and

Q_{i+1} = Q_{i}^{2} − z^{τ(i)}Q_{0}^{2^{i+1}}Q_{i−1}

for i ≥ 1.

As in Lemma 2.1, define by induction on i, η_{0} = 1, and

(27)  η_{i+1} = 2η_{i} + \frac{1}{2^{i+1}}

for i ≥ 0. Define δ_{0} = 0 and

(28)  δ_{i} = \frac{τ(1)}{2^{i}} + \frac{τ(2)}{2^{i−1}} + ⋯ + \frac{τ(i)}{2}

for i > 0.

Suppose that f(x, y, z) ∈ K[x, y, z]. Then for l ∈ N such that deg_{y}f < 2^{l}, there is a unique expansion

(29)  f = ∑_{α} a_{α}(z)x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{l}^{α_{l}}

where a_{α}(z) ∈ K[z] and the sum is over α = (α_{0}, α_{1}, …, α_{l}) ∈ N × {0, 1}^{l}. This is established as (16) in the statement of Lemma 2.1. We have a stronger statement which is valid in the polynomial ring K[x, y, z], since the leading coefficients of the Q_{i}, as polynomials in y, have 1 as their leading coefficient.

For f ∈ K[x, y, z], define from the expansion (29),

(30)  ν(f) = min_{α}\{(0, ord_{z}(a_{α}(z)) + ∑_{i=0}^{ℓ} α_{i}(η_{i}, δ_{i})\} ∈ (\frac{1}{2^{∞}} Z × \frac{1}{2^{∞}} Z)lex.

Then ν defines a rank 2 valuation on K(x, y, z), which is composite with a rank 1 valuation ν_{2} of K(x, y, z). The value group of ν_{2} is \frac{1}{2^{∞}} Z = ⋃_{i=1}^{∞} \frac{1}{2^{i}} Z.

ν dominates the local ring R = K[x, y, z]_{(x,y,z)} and the center of ν_{2} on R is the prime ideal (x, y).

We have ν(Q_{i}) = (η_{i}, δ_{i}) and

(31)  ν(Q_{i}^{2}) = ν(z^{τ(i)}Q_{0}^{2^{i+1}}Q_{i−1}) < ν(Q_{i+1})

for all i.

We now construct the example. We choose positive integers τ(i) so that

δ_{i} = \frac{τ(1)}{2^{i}} + \frac{τ(2)}{2^{i−1}} + ⋯ + \frac{τ(i)}{2} > g(i2^{i+3})

for all positive integers i, and δ_{i} ∈ Z for all i. Let ν be the valuation constructed above, which dominates R. The value group of ν is (\frac{1}{2^{∞}} Z × Z)lex.

Let ω be a valuation equivalent to ν with value group (\frac{1}{2^{∞}} Z × Z)lex. We have ω(x) = (a, b) for some a ∈ \frac{1}{2^{∞}} Z_{+} and b ∈ Z, and ω(z) = (0, c) for some c ∈ Z_{+}. From the relations (31) we see that ω(Q_{i}) = η_{i}ω(x) + δ_{i}ω(z) for i ≥ 1, which implies b = 0, since π_{2}(ω(Q_{i})) ∈ Z. We also have ω(a(z)) = ord_{z}(a)ω(z) for a(z) ∈ K(z). Let ω_{2} be the valuation on K(x, y, z) defined by ω_{2}(f) = π_{1}(ω(f)), where π_{1} : \frac{1}{2^{∞}} Z × Z → \frac{1}{2^{∞}} Z is the first projection.

Let

e = ⌈a⌉ and n_{0} = ⌈a⌉2^{e+2}.

Suppose that n ≥ n_{0}. We will find λ ∈ \frac{1}{2^{∞}} Z ∩ [0, n[ such that π_{2}(λ̃) > g(n).

There exists i ≥ e such that

⌈a⌉2^{i+2} ≤ n < ⌈a⌉2^{i+3}.

Let λ = aη_{i}. From η_{i} = \frac{1}{3}(2^{i+2} − \frac{1}{2^{i}}), we obtain

λ = aη_{i} < ⌈a⌉2^{i+2} ≤ n.

Since c ∈ Z_{+}, i ≥ e and g is increasing, we have

cδ_{i} ≥ δ_{i} > g(i2^{i+3}) ≥ g(⌈a⌉2^{i+3}) > g(n).

Thus

π_{2}(λ̃) = π_{2}(ω(Q_{i})) = cδ_{i} > g(n).
□

Example 2.4. Suppose that f : N → Z is a decreasing function, g : N → Z is an increasing function, and K is a field. Then there exists a rank 2 valuation ν of the five dimensional rational function field K(x, y, u, v, z) with value group (H ×Z)lex, where H = (\frac{1}{2^{∞}} Z + \frac{1}{2^{∞}} Z\sqrt{2}) ⊂ R, which dominates the regular local ring R = K[x, y, u, v, z]_{(x,y,u,v,z)}, such that for any valuation ω equivalent to ν with value group (H×Z)lex, for all sufficiently large n ∈ N, there exists λ_{1} ∈ H ∩ [0, n[ such that π_{2}(λ̃_{1}) < f(n)and there exists λ_{2} ∈ H ∩ [0, n[ such that π_{2}(λ̃_{2}) > g(n), where π_{2} : \frac{1}{2^{∞}} Z × Z → \frac{1}{2^{∞}} Z is the second projection.

Proof. We need an extension of the method of Lemma 2.1 for constructing valuations which we first outline. Suppose that σ : Z_{+} → N and τ : Z_{+} → N are functions. Define P_{0} = x, P_{1} = y and P_{i+1} = z^{σ(i)}P_{i}^{2} − P_{0}^{2^{i+1}}P_{i−1} for i ≥ 1. Define Q_{0} = u, Q_{1} = v and Q_{i+1} = Q_{i}^{2} − z^{τ(i)}Q_{0}^{2^{i+1}}Q_{i−1} for i ≥ 1.

Suppose that f ∈ K[z, x, y, u, v] and deg f < 2^{l}. Then there is a unique expansion

f = ∑_{β} g_{β}(z, x, y)u^{β_{0}}Q_{1}^{β_{1}} ⋯ Q_{ℓ}^{β_{ℓ}}

where the sum is over β = (β_{0}, β_{1}, …, β_{ℓ}) ∈ N × {0, 1}^{ℓ} and g_{β} ∈ K[z, x, y] for all β (since the leading coefficient of each Q_{i} with respect to y is 1). Each g_{β}(z, x, y) has a unique expansion

g_{β} = ∑_{α} a_{α,β}(z)x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{ℓ}^{α_{ℓ}}

where the sum is over α = (α_{0}, α_{1}, …, α_{ℓ}) ∈ N × {0, 1}^{ℓ} and a_{α,β}(z) ∈ K(z) for all α, β. Thus f has a unique expansion

(32)  f = ∑_{α,β} a_{α,β}(z)x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{ℓ}^{α_{ℓ}}u^{β_{0}}Q_{1}^{β_{1}} ⋯ Q_{ℓ}^{β_{ℓ}}

with α, β ∈ N × {0, 1}^{ℓ}, and a_{α,β}(z) ∈ K(z).

Set η_{0} = 1 and

η_{i+1} = 2η_{i} + \frac{1}{2^{i+1}}

for i ≥ 0. Set γ_{0} = 0 and

γ_{i} = −(\frac{σ(1)}{2^{i}} + \frac{σ(2)}{2^{i−1}} + ⋯ + \frac{σ(i)}{2})

for i > 0. Set δ_{0} = 0 and

δ_{i} = \frac{τ(1)}{2^{i}} + \frac{τ(2)}{2^{i−1}} + ⋯ + \frac{τ(i)}{2}

for i > 0.

Let H = (\frac{1}{2^{∞}} Z + \frac{1}{2^{∞}} Z\sqrt{2}) ⊂ R.

We define from the expansion (32),

(33) ν(f) = min_{α,β}\{(0, ord_{z}(a_{α,β}(z)) + ∑ α_{i}(η_{i}, γ_{i}) + ∑ β_{i}(η_{i}\sqrt{2}, δ_{i})\} ∈ (H × \frac{1}{2^{∞}} Z)lex.

We have

ν(z^{σ(i)}P_{i}^{2}) = ν(P_{0}^{2^{i+1}}P_{i−1}) < ν(P_{i+1})

for all i, ν(P_{i}) = (η_{i}, γ_{i}) for all i,

ν(Q_{i}^{2}) = ν(z^{τ(i)}Q_{0}^{2^{i+1}}Q_{i−1}) < ν(Q_{i+1})

for all i, ν(Q_{i}) = (η_{i}\sqrt{2}, δ_{i}) for all i. Further, there is a unique term in the expansion (32) which achieves the minimum (33). We have

ν(f) ∈ \frac{1}{2^{∞}} Z × \frac{1}{2^{∞}} Z

if and only if f has the form

f = a_{α,0}(z)x^{α_{0}}P_{1}^{α_{1}} ⋯ P_{ℓ}^{α_{ℓ}} + higher value terms,

and

ν(f) ∈ \frac{1}{2^{∞}} Z\sqrt{2} × \frac{1}{2^{∞}} Z

if and only if f has the form

f = a_{0,β}(z)u^{β_{0}}Q_{1}^{β_{1}} ⋯ Q_{ℓ}^{β_{ℓ}} + higher value terms.

Observe that for all β, we have a_{0,β}(z) ∈ K[z] in the expansion (32).

We now construct the valuation of the example. For i ∈ Z_{+}, choose σ(i) ∈ N such that

γ_{i} < f(i2^{i+3})

and choose τ(i) ∈ Z_{+} so that

g(i2^{i+3}) < δ_{i}

for all i ∈ Z_{+}, and γ_{i}, δ_{i} ∈ Z for all i. Let ν be the valuation constructed above, which dominates R. The value group of ν is (H × Z)lex.

Let ω be a valuation equivalent to ν with value group (H×Z)lex. There exist α, β, γ, δ ∈ \frac{1}{2^{∞}} Z, b_{1}, b_{2} ∈ Z, c ∈ Z_{+}, such that

ω(x) = (a_{1}, b_{1}), ω(u) = (a_{2}, b_{2}), ω(z) = (0, c)

where

a_{1} = α + β\sqrt{2} > 0, a_{2} = γ + δ\sqrt{2} > 0

and αδ − βγ ≠ 0. We have ω(P_{i}) = η_{i}ω(x) + γ_{i}ω(z) and ω(Q_{i}) = η_{i}ω(u) + δ_{i}ω(z) for i ≥ 1, which implies b_{1} = b_{2} = 0, since π_{2}(ω(P_{i})), π_{2}(ω(Q_{i})) ∈ Z.

Set

e = max{⌈a_{1}⌉, ⌈a_{2}⌉}.

Let n_{0} = e2^{e+2}.

Suppose that n ≥ n_{0}. We will show that there exists λ_{1} ∈ H ∩ [0, n[ such that π_{2}(λ̃_{1}) < f(n) and there exists λ_{2} ∈ H ∩ [0, n[ such that π_{2}(λ̃_{2}) > g(n).

There exists i ≥ e such that e2^{i+2} ≤ n < e2^{i+3}. Let λ_{1} = a_{1}η_{i}. From η_{i} = \frac{1}{3}(2^{i+2} − \frac{1}{2^{i}}) we obtain

λ_{1} = a_{1}η_{i} < ⌈a_{1}⌉2^{i+2} ≤ e2^{i+2} ≤ n.

Since c ∈ Z_{+}, i ≥ e and f is decreasing, we have

cγ_{i} ≤ γ_{i} < f(i2^{i+3}) ≤ f(e2^{i+3}) < f(n).

Thus

π_{2}(λ̃_{1}) ≤ π_{2}(ω(P_{i})) = cγ_{i} < f(n).

Let λ_{2} = a_{2}η_{i}. We have

λ_{2} = a_{2}η_{i} < ⌈a_{2}⌉2^{i+2} ≤ e2^{i+2} ≤ n.

Since c ∈ Z_{+}, i ≥ e and g is increasing, we have

cδ_{i} ≥ δ_{i} > g(i2^{i+3}) ≥ g(e2^{i+3}) > g(n).

Thus

π_{2}(λ̃_{2}) = π_{2}(ω(Q_{i})) = cδ_{i} > g(n).
□

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Dale Cutkosky, Bernard Teissier, 202 Mathematical Sciences Bldg, Institut Mathématique de Jussieu

University of Missouri, UMR 7586 du CNRS, Columbia, MO 65211 USA 175 Rue du Chevaleret, 75013 Paris,France cutkoskys@missouri.edu teissier@math.jussieu.fr