arXiv:1302.2676v3 [] 25 Feb 2014

CONVEX BODIES AND MULTIPLICITIES OF IDEALS
KIUMARS KAVEH AND A. G. KHOVANSKII
Dedicated to Viktor Matveyevich Buchstaber for the occasion of his 70th birthday

Abstract. We associate convex regions in Rn to m-primary graded sequences
of subspaces, in particular m-primary graded sequences of ideals, in a large
class of local algebras (including analytically irreducible local domains). These
convex regions encode information about Samuel multiplicities. This is in the
spirit of the theory of Gröbner bases and Newton polyhedra on one hand,
and the theory of Newton-Okounkov bodies for linear systems on the other
hand. We use this to give a new proof, as well as a generalization of a BrunnMinkowski inequality for multiplicities due to Teissier and Rees-Sharp.

Introduction
The purpose of this note is to employ, in the local case, techniques from the
theory of semigroups of integral points and Newton-Okounkov bodies (for the global
case) and to obtain new results as well as new proofs of some previously known
results about multiplicities of ideals in local rings.
Let R = OX,p be the local ring of a point p on an n-dimensional irreducible
algebraic variety X over an algebraically closed field k. Let m denote the maximal
ideal of R and let a be an m-primary ideal, i.e. a is an ideal containing a power
of the maximal ideal m. Geometrically speaking, a is m-primary if its zero set
(around p) is the single point p itself. Let f1 , . . . , fn be n generic elements in a.
The multiplicity e(a) of the ideal a is the intersection multiplicity, at the origin,
of the hypersurfaces Hi = {x | f (x) = 0}, i = 1, . . . , n (it can be shown that
this number is independent of the choice of the fi ). According to Hilbert-Samuel’s
theorem, the multiplicity e(a) is equal to:
dimk (R/ak )
.
k→∞
kn
(This result is analogous to Hilbert’s theorem on the Hilbert function and degree
of a projective variety.) More generally, let R be an n-dimensional Noetherian
local domain over k (where k is isomorphic to the residue field R/m and m is the
maximal ideal). Let a be an m-primary ideal of R. Since a contains a power of the
maximal ideal m, R/a is finite dimensional regarded as a vector space over k. The
Hilbert-Samuel function of the m-primary ideal a is defined by:
n! lim

Ha (k) = dimk (R/ak ).
Date: February 26, 2014.
2010 Mathematics Subject Classification. Primary: 13H15, 11H06; Secondary: 13P10.
Key words and phrases. multiplicity, local ring, Hilbert-Samuel function, convex body.
The first author is partially supported by a Simons Foundation Collaboration Grants for Mathematicians (Grant ID: 210099) and a National Science Foundation (Grant ID: 1200581).
The second author is partially supported by the Canadian Grant N 156833-12.
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For large values of k, Ha (k) coincides with a polynomial of degree n called the
Hilbert-Samuel polynomial of a. The Samuel multiplicity, e(a) of a is defined to be
the leading coefficient of Ha (k) multiplied by n!.
It is well-known that the Samuel multiplicity satisfies a Brunn-Minkowski inequality [Teissier77, Rees-Sharp78]. That is, for any two m-primary ideals a, b ∈ R
we have:
(1)

e(a)1/n + e(b)1/n ≥ e(ab)1/n .

More generally we define multiplicity for m-primary graded sequences of subspaces. That is, a sequence a1 , a2 , . . . of k-subspaces in R such that for all k, m we
have ak am ⊂ ak+m , and a1 contains a power of the maximal ideal m (Definition
5.2). We recall that if a, b are two k-subspaces of R, ab denotes the k-span of all
the xy where x ∈ a and y ∈ b. In particular, a graded sequence a• where each ak
is an m-primary ideal, is an m-primary graded sequence of subspaces. We call such
a• an m-primary graded sequence of ideals.
For an m-primary graded sequence of subspaces we define multiplicity e(a• ) to
be:
dimk (R/ak )
(2)
e(a• ) = n! lim
.
k→∞
kn
(It is not a priori clear that the limit exists.)
We will use convex geometric arguments to prove the existence of the limit in
(2) and a generalization of (1) to m-primary graded sequences of subspaces, for a
large class of local domains R.
Let us briefly discuss the convex geometry part of the story. Let C be a closed
strongly convex cone with apex at the origin (i.e. C is a convex cone and does not
contain any line). We call a closed convex set Γ ⊂ C, a C-convex region if for any
x ∈ Γ we have x + C ⊂ Γ. We say that Γ is cobounded if C \ Γ is bounded. It is
easy to verify that the set of cobounded C-convex regions is closed under addition
(Minkowski sum of convex sets) and multiplication with a positive real number. For
a cobounded C-convex region Γ we call the volume of the bounded region C \ Γ the
covolume of Γ and denote it by covol(Γ). Also we refer to C \Γ as a C-coconvex body.
(Instead of working with convex regions one can alternatively work with coconvex
bodies.) In [Khovanskii-Timorin-a, Khovanskii-Timorin-b], similar to convex bodies and their volumes (and mixed volumes), the authors develop a theory of convex regions and their covolumes (and mixed covolumes). Moreover they prove an
analogue of the Alexandrov-Fenchel inequality for mixed covolumes (see Theorem
2.3). The usual Alexandrov-Fenchel inequality is an important inequality about
mixed volumes of convex bodies in Rn and generalizes the classical isoperimetric
inequality and the Brunn-Minkowski inequality. In a similar way, the result in
[Khovanskii-Timorin-a] implies a Brunn-Minkowski inequality for covolumes, that
is, for any two cobounded C-convex regions Γ1 , Γ2 where C is an n-dimensional
cone, we have:
(3)

covol(Γ1 )1/n + covol(Γ2 )1/n ≥ covol(Γ1 + Γ2 )1/n .

We associate convex regions to m-primary graded sequences of subspaces (in particular m-primary ideals) and use the inequality (3) to prove the Brunn-Minkowski
inequality for multiplicities. To associate a convex region to a graded sequence
of subspaces we need a valuation on the ring R. We will assume that there is a
valuation v on R with values in Zn (with respect to a total order on Zn respecting
2

addition) such that the residue field of v is k, and moreover the following conditions
(i)-(ii) hold 1. We call such v a good valuation on R (Definition 7.3):
(i) Let S = v(R \ {0}) ∪ {0} be the value semigroup of (R, v). Let C = C(S) be the
closure of the convex hull of S. It is a closed convex cone with apex at the origin.
We assume that C is a strongly convex cone.
Let ℓ : Rn → R be a linear function. For any a ∈ R let ℓ≥a denote the half-space
{x | ℓ(x) ≥ a}. Since the cone C = C(S) associated to the semigroup S is assumed
to be strongly convex we can find a linear function ℓ such that the cone C lies in
ℓ≥0 and it intersects the hyperplane ℓ−1 (0) only at the origin.
(ii) We assume there exists r0 > 0 and a linear function ℓ as above such that for
any f ∈ R, if ℓ(v(f )) ≥ kr0 for some k > 0 then f ∈ mk .
Let Mk = v(mk \ {0}) denote the image of mk under the valuation v. The
condition (ii) in particular implies that for any k > 0 we have Mk ∩ ℓ≥kr0 =
S ∩ ℓ≥kr0 .
As an example, let R = k[x1 , . . . , xn ](0) be the algebra of polynomials localized at
the maximal ideal (x1 , . . . , xn ). Then the map v which associates to a polynomial
its lowest exponent (with respect to some term order) defines a good valuation
on R and the value semigroup S coincides with the semigroup Zn≥0 , that is, the
semigroup of all the integral points in the positive orthant C = Rn≥0 . In the same
fashion any regular local ring has a good valuation, as well as the local ring of a
toroidal singularity (Example 7.5 and Theorem 7.6). More generally, in Section
7 we see that an analytically irreducible local domain R has a good valuation
(Theorem 7.7 and Theorem 7.8; see also [Cutkosky-a, Theorem 4.2 and Lemma
4.3]). A local ring R is said to be analytically irreducible if its completion is an
integral domain. Regular local rings and local rings of toroidal singularities are
analytically irreducible. (We should point out that in the first version of the paper
we had addressed only the case where R is a regular local ring or the local ring of
a toroidal singularity.)
Given a good Zn -valued valuation v on the domain R, we associate the (strongly)
convex cone C = C(R) ⊂ Rn to the domain R which is the closure of convex hull of
the value semigroup S. Then to each m-primary graded sequence of subspaces a•
in R we associate a convex region Γ(a• ) ⊂ C, such that the set C \ Γ(a• ) is bounded
(Definition 7.11). The main result of the manuscript (Theorem 7.12) is that the
limit in (2) exists and:
(4)

e(a• ) = n! covol(Γ(a• )).

The equality (4) and the Brunn-Minkowski inequality for covolumes (see (3) or
Corollary 2.4) are the main ingredients in proving a generalization of the inequality
(1) to m-primary graded sequences of subspaces (Corollary 7.14).
We would like to point out that the construction of Γ(a• ) is an analogue of the
construction of the Newton-Okounkov body of a linear system on an algebraic variety (see [Okounkov03], [Okounkov96], [Kaveh-Khovanskii12], [Lazarsfeld-Mustaţă09]).
In fact, the approach and results in the present paper are analogous to the approach and results in [Kaveh-Khovanskii12] regarding the asymptotic behavior of
Hilbert functions of a general class of graded algebras. In the present manuscript
we also deal with certain graded algebras (i.e. m-primary graded sequences of
1In [Kaveh-Khovanskii12] a valuation v with values in Zn and residue field k is called a valuation
with one-dimensional leaves (see Definition 7.1).
3

subspaces) but instead of dimension of graded pieces we are interested in the codimension (i.e. dimension of R/ak ), that is why in our main theorem (Theorem 7.12)
the covolume of a convex region appears as opposed to the volume of a convex
body ([Kaveh-Khovanskii12, Theorem 2.31]). Also our Theorem 7.12 generalizes
[Kaveh-Khovanskii12, Corollary 3.2] which gives a formula for the degree of a projective variety X in terms of the volume of its corresponding Newton-Okounkov
body, because the Hilbert function of a projective variety X can be regarded as the
difference derivative of the Hilbert-Samuel function of the affine cone over X at the
origin and hence has the same leading coefficient.
On the other hand, the construction of Γ(a) generalizes the notion of the Newton
diagram of a power series (see [Kushnirenko76] and [Arnold-Varchenko-Guseinzade85,
Section 12.7]). To a monomial ideal in a polynomial ring (or a power series ring),
i.e. an ideal generated by monomials, one can associate its (unbounded) Newton
polyhedron. It is the convex hull of the exponents of the monomials appearing in
the ideal. The Newton diagram of a monomial ideal is the union of the bounded
faces of the Newton polyhedron. One can see that for a monomial ideal a, the
convex region Γ(a) coincides with its Newton polyhedron (Theorem 6.5). The main
theorem in this manuscript (Theorem 7.12) for the case of monomial ideals recovers the local case of the well-known theorem of Bernstein-Kushnirenko, about
computing the multiplicity at the origin of a system f1 (x) = · · · = fn (x) = 0 where
the fi are generic functions from m-primary monomial ideals (see Section 6 and
[Arnold-Varchenko-Guseinzade85, Section 12.7]).
Another immediate corollary of (4) is the following: let a be an m-primary ideal
in R = k[x1 , . . . , xn ](0) . Fix a term order on Zn and for each k > 0 let in(ak )
denote the initial ideal of the ideal ak (generated by the lowest terms of elements
of ak ). Then the sequence of numbers
e(in(ak ))
kn
is decreasing and converges to e(a) as k → ∞ (Corollary 7.16).
The Brunn-Minkowski inequality proved in this paper is closely related to
the more general Alexandrov-Fenchel inequality for mixed multiplicities. Take mprimary ideals a1 , . . . , an in a local ring R = OX,p of a point p on an n-dimensional
algebraic variety X. The mixed multiplicity e(a1 , . . . , an ) is equal to the intersection
multiplicity, at the origin, of the hypersurfaces Hi = {x | fi (x) = 0}, i = 1, . . . , n,
where each fi is a generic function from ai . Alternatively one can define the mixed
multiplicity as the polarization of the Hilbert-Samuel multiplicity e(a), i.e. it is the
unique function e(a1 , . . . , an ) which is invariant under permuting the arguments, is
multi-additive with respect to product of ideals, and for any m-primary ideal a the
mixed multiplicity e(a, . . . , a) coincides with e(a). In fact, in the above the ai need
not be ideals and it suffices for them to be m-primary subspaces.
The Alexandrov-Fenchel inequality is the following inequality among the mixed
multiplicities of the ai :
(5)

e(a1 , a1 , a3 , . . . , an )e(a2 , a2 , a3 , . . . , an ) ≥ e(a1 , a2 , a3 , . . . , an )2

When n = dim R = 2 it is easy to see that the Brunn-Minkowski inequality (1)
and the Alexandrov-Fenchel inequality (5) are equivalent. By a reduction of dimension theorem for mixed multiplicities one can get a proof of the Alexandrov-Fenchel
4

inequality (5) from the Brunn-Minkowski inequality (1) for dim(R) = 2. The
Brunn-Minkowski inequality (1) was originally proved in [Teissier77, Rees-Sharp78].
In a recent paper [Kaveh-Khovanskii] we give a simple proof of the AlexandrovFenchel inequality for mixed multiplicities of ideals using arguments similar to but
different from those of this paper. This then implies an Alexandrov-Fenchel inequality for covolumes of convex regions (in a similar way that in [Kaveh-Khovanskii12]
and in [Khovanskii88] the authors obtain an alternative proof of the usual AlexandrovFenchel inequality for volumes of convex bodies from similar inequalities for intersection numbers of divisors on algebraic varieties).
We would like to point out that the Alexandrov-Fenchel inequality in
[Khovanskii-Timorin-a] for covolumes of coconvex bodies is related to an analogue of
this inequality for convex bodies in higher dimensional hyperbolic space (or higher
dimensional Minkowski space-time). From this point of view, the AlexandrovFenchel inequality has been proved for certain coconvex bodies in [Fillastre13].
After the first version of this note was completed we learned about the recent
papers [Cutkosky-a, Cutkosky-b] and [Fulger] which establish the existence of limit
(2) in more general settings. We would also like to mention the paper of Teissier
[Teissier78] which discusses Newton polyhedron of a power series, and notes the relationship/analogy between notions from local commutative algebra and convex geometry. Also we were notified that, for ideals in a polynomial ring, ideas similar to
construction of Γ(a• ) (see Definition 7.11) appears in [Mustaţă02] were the highest
term of polynomials is used instead of a valuation. Moreover, in [Mustaţă02, Corollary 1.9] the Brunn-Minkowski-inequality for multiplicities of graded sequences of
m-primary ideals is proved for regular local rings using Teissier’s Brunn-Minkowski
(1).
Finally as the final version of this manuscript was being prepared for publication,
the preprint of D. Cutksoky [Cutkosky-c] appeared in arXiv.org in which the author
uses similar methods to prove Brunn-Minkowski inequality for graded sequences of
m-primary ideals in local domains.
And few words about the organization of the paper: Section 1 recalls basic background material about volumes/mixed volumes of convex bodies. Section 2 is about
convex regions and their covolumes/mixed covolumes, which we can think of as a
local version of the theory of mixed volumes of convex bodies. In Sections 3 and 4
we associate a convex region to a primary sequence of subsets in a semigroup and
prove the main combinatorial result required later (Definition 4.7 and Theorem
4.10). In Section 5 we recall some basic definitions and facts from commutative
algebra about multiplicities of m-primary ideals (and subspaces) in local rings. The
next section (Section 6) discusses the case of monomial ideals and the BernsteinKushnirenko theorem. Finally in Section 7, using a valuation on the ring R, we
associate a convex region Γ(a• ) to an m-primary graded sequence of subspaces a•
and prove the main results of this note (Theorem 7.12 and Corollary 7.14).
Acknowledgement. The first author would like to thank Dale Cutkosky,
Vladlen Timorin and Javid Validashti for helpful discussions. We are also thankful to Bernard Teissier, Dale Cutkosky, Francois Fillastre and Mircea Mustaţă for
informing us about their interesting papers [Teissier78], [Cutkosky-a, Cutkosky-b],
[Fillastre13] and [Mustaţă02].

5

1. Mixed volume of convex bodies
The collection of all convex bodies in Rn is a cone, that is, we can add convex
bodies and multiply a convex body with a positive number. For two convex bodies
∆1 , ∆2 ⊂ Rn , their (Minkowski) sum ∆1 + ∆2 is {x + y | x ∈ ∆1 , y ∈ ∆2 }. Let
vol denote the n-dimensional volume in Rn with respect to the standard Euclidean
metric. The function vol is a homogeneous polynomial of degree n on the cone of
convex bodies, i.e. its restriction to each finite dimensional section of the cone is a
homogeneous polynomial of degree n. In other words, for any collection of convex
bodies ∆1 , . . . , ∆r , the function:
P∆1 ,...,∆r (λ1 , . . . , λr ) = vol(λ1 ∆1 + · · · + λr ∆r ),
is a homogeneous polynomial of degree n in λ1 , . . . , λr . By definition the mixed volume V (∆1 , . . . , ∆n ) of an n-tuple (∆1 , . . . , ∆n ) of convex bodies is the coefficient of
the monomial λ1 · · · λn in the polynomial P∆1 ,...,∆n (λ1 , . . . , λn ) divided by n!. This
definition implies that mixed volume is the polarization of the volume polynomial,
that is, it is the unique function on the n-tuples of convex bodies satisfying the
following:
(i) (Symmetry) V is symmetric with respect to permuting the bodies ∆1 , . . . , ∆n .
(ii) (Multi-linearity) It is linear in each argument with respect to the Minkowski
sum. The linearity in first argument means that for convex bodies ∆′1 ,
∆′′1 , ∆2 , . . . , ∆n , and real numbers λ′ , λ′′ ≥ 0 we have:
V (λ′ ∆′1 + λ′′ ∆′′1 , ∆2 , . . . , ∆n ) = λ′ V (∆′1 , ∆2 , . . . , ∆n ) + λ′′ V (∆′′1 , ∆2 , . . . , ∆n ).
(iii) (Relation with volume) On the diagonal it coincides with the volume, i.e.
if ∆1 = · · · = ∆n = ∆, then V (∆1 , . . . , ∆n ) = vol(∆).
The following inequality attributed to Alexandrov and Fenchel is important and
very useful in convex geometry (see [Burago-Zalgaller88]):
Theorem 1.1 (Alexandrov-Fenchel). Let ∆1 , . . . , ∆n be convex bodies in Rn . Then
V (∆1 , ∆1 , ∆3 , . . . , ∆n )V (∆2 , ∆2 , ∆3 , . . . , ∆n ) ≤ V (∆1 , ∆2 , . . . , ∆n )2 .
In dimension 2, this inequality is elementary. We call it the generalized isoperimetric inequality, because when ∆2 is the unit ball it coincides with the classical
isoperimetric inequality. The celebrated Brunn-Minkowski inequality concerns volume of convex bodies in Rn . It is an easy corollary of the Alexandrov-Fenchel
inequality. (For n = 2 it is equivalent to the Alexandrov-Fenchel inequality.)
Theorem 1.2 (Brunn-Minkowski). Let ∆1 , ∆2 be convex bodies in Rn . Then
vol(∆1 )1/n + vol(∆2 )1/n ≤ vol(∆1 + ∆2 )1/n .
2. Mixed covolume of convex regions
Let C be a strongly convex closed n-dimensional cone in Rn with apex at the
origin. (A convex cone is strongly convex if it does not contain any lines through
the origin.) We are interested in closed convex subsets of C which have bounded
complement.
Definition 2.1. We call a closed convex subset Γ ⊂ C a C-convex region (or simply
a convex region when the cone C is understood from the context) if for any x ∈ Γ
and y ∈ C we have x + y ∈ Γ. Moreover, we say that a convex region Γ is cobounded
6

if the complement C \ Γ is bounded. In this case the volume of C \ Γ is finite which
we call the covolume of Γ and denote it by covol(Γ). One also refers to C \ Γ as a
C-coconvex body.
The collection of C-convex regions (respectively cobounded regions) is closed under the Minkowski sum and multiplication by positive scalars. Similar to the volume
of convex bodies, one proves that the covolume of convex regions is a homogeneous
polynomial [Khovanskii-Timorin-a]. More precisely:
Theorem 2.2. Let Γ1 , . . . , Γr be cobounded C-convex regions in the cone C. Then
the function
PΓ1 ,...,Γr (λ1 , . . . , λr ) = covol(λ1 Γ1 + · · · + λr Γr ),
is a homogeneous polynomial of degree n in the λi .
As in the case of convex bodies, one uses the above theorem to define mixed
covolume of cobounded regions. By definition the mixed covolume CV (Γ1 , . . . , Γn )
of an n-tuple (Γ1 , . . . , Γn ) of cobounded convex regions is the coefficient of the
monomial λ1 · · · λn in the polynomial PΓ1 ,...,Γn (λ1 , . . . , λn ) divided by n!. That
is, mixed covolume is the unique function on the n-tuples of cobounded regions
satisfying the following:
(i) (Symmetry) CV is symmetric with respect to permuting the regions Γ1 , . . . , Γn .
(ii) (Multi-linearity) It is linear in each argument with respect to the Minkowski
sum.
(iii) (Relation with covolume) For any cobounded region Γ ⊂ C:
CV (Γ, . . . , Γ) = covol(Γ).
The mixed covolume satisfies an Alexandrov-Fenchel inequality [Khovanskii-Timorin-a].
Note that the inequality is reversed compared to the Alexandrov-Fenchel for mixed
volumes of convex bodies.
Theorem 2.3 (Alexandrov-Fenchel for mixed covolume). Let Γ1 , . . . , Γn be cobounded
C-convex regions. Then:
CV (Γ1 , Γ1 , Γ3 , . . . , Γn )CV (Γ2 , Γ2 , Γ3 , . . . , Γn ) ≥ CV (Γ1 , Γ2 , Γ3 , . . . , Γn )2 .
The (reversed) Alexandrov-Fenchel inequality implies a (reversed) Brunn-Minkowski
inequality. (For n = 2 it is equivalent to the Alexandrov-Fenchel inequality.)
Corollary 2.4 (Brunn-Minkowski for covolume). Let Γ1 , Γ2 be cobounded C-convex
regions. Then:
covol(Γ1 )1/n + covol(Γ2 )1/n ≥ covol(Γ1 + Γ2 )1/n .
3. Semigroups of integral points
In this section we recall some general facts from [Kaveh-Khovanskii12] about the
asymptotic behavior of semigroups of integral points. Let S ⊂ Zn × Z≥0 be an
additive semigroup. Let π : Rn × R → R denote the projection onto the second
factor, and let Sk = S ∩ π −1 (k) be the set of points in S at level k. For simplicity,
assume S1 6= ∅ and that S generates the whole lattice Zn+1 . Define the function
HS by:
HS (k) = #Sk .
We call HS the Hilbert function of the semigroup S. We wish to describe the
asymptotic behavior of HS as k → ∞.
7

Let C(S) be the closure of the convex hull of S ∪ {0}, that is, the smallest
closed convex cone (with apex at the origin) containing S. We call the projection
of the convex set C(S) ∩ π −1 (1) to Rn (under the projection onto the first factor
(x, 1) 7→ x), the Newton-Okounkov convex set of the semigroup S and denote it by
∆(S). In other words,
[
{x/k | (x, k) ∈ Sk }.
∆(S) =
k>0

(From the fact that S is a semigroup one can show that ∆(S) is a convex set.) If
C(S) ∩ π −1 (0) = {0} then ∆(S) is compact and hence a convex body.
The Newton-Okounkov convex set ∆(S) is responsible for the asymptotic behavior of the Hilbert function of S (see [Kaveh-Khovanskii12, Corollary 1.16]):
Theorem 3.1. The limit
lim

k→∞

HS (k)
,
kn

exists and is equal to vol(∆(S)).
4. Primary sequences in a semigroup and convex regions
In this section we discuss the notion of a primary graded sequence of subsets in
a semigroup and describe its asymptotic behavior using Theorem 3.1. In section
7 we will employ this to describe the asymptotic behavior of the Hilbert-Samuel
function of a graded sequence of m-primary ideals in a local domain.
Let S ⊂ Zn be an additive semigroup containing the origin. Without loss of
generality we assume that S generates the whole Zn . Let as above C = C(S)
denote the cone of S i.e. the closure of convex hull of S = S ∪ {0}. We also assume
that C is a strongly convex cone, i.e. it does not contain any lines through the
origin.
For two subsets I, J ⊂ S, the sum I + J is the set {x + y | x ∈ I, y ∈ J }. For
any integer k > 0, by the product k ∗ I we mean I + · · · + I (k times).
Definition 4.1. A graded sequence of subsets in S is a sequence I• = (I1 , I2 , . . .)
of subsets such that for any k, m > 0 we have Ik + Im ⊂ Ik+m .
Example 4.2. Let I ⊂ S. Then the sequence I• defined by Ik = k ∗ I is clearly
a graded sequence of subsets.
Let I•′ , I•′′ be graded sequences of subsets. Then the sequence I• = I•′ + I•′′
defined by
Ik = Ik′ + Ik′′ ,
is also a graded sequence of subsets which we call the sum of sequences I•′ and I•′′ .
Let ℓ : Rn → R be a linear function. For any a ∈ R let ℓ≥a (respectively ℓ>a )
denote the half-space {x | ℓ(x) ≥ a} (respectively {x | ℓ(x) > a}), and similarly for
ℓ≤a and ℓ<a . By assumption the cone C = C(S) associated to the semigroup S is
strongly convex. Thus we can find a linear function ℓ such that the cone C = C(S)
lies in ℓ≥0 and it intersects the hyperplane ℓ−1 (0) only at the origin. Let us fix such
a linear function ℓ.
We will be interested in graded sequences of subsets I• satisfying the following
condition:
8

Definition 4.3. We say that a graded sequence of subsets I• is primary if there
exists an integer t0 > 0 such that for any integer k > 0 we have:
(6)

Ik ∩ ℓ≥kt0 = S ∩ ℓ≥kt0 .

Remark 4.4. One verifies that if ℓ′ is another linear function such that C lies in
ℓ′≥0 and it intersects the hyperplane ℓ′−1 (0) only at the origin then it automatically
satisfies (6) with perhaps a different constant t′0 > 0. Hence the condition of being
a primary graded sequence does not depend on the linear function ℓ. Nevertheless
when we refer to a primary graded sequence I• , choices of a linear function ℓ and
an integer t0 > 0 are implied.
Proposition 4.5. Let I• be a primary graded sequence. Then for all k > 0, the
set S \ Ik is finite.
Proof. Since C intersects ℓ−1 (0) only at the origin it follows that for any k > 0
the set C ∩ ℓ<kt0 is bounded which implies that S ∩ ℓ<kt0 is finite. But by (6),

S \ Ik ⊂ S ∩ ℓ<kt0 and hence is finite.
Definition 4.6. Let I• be a primary graded sequence. Define the function HI•
by:
HI• (k) = #(S \ Ik ).
(Note that by Proposition 4.5 this number is finite for all k > 0.) We call it the
Hilbert-Samuel function of I• .
To a primary graded sequence of subsets I• we can associate a C-convex region Γ(I• ) (see Definition 2.1). This convex set encodes information about the
asymptotic behavior of the Hilbert-Samuel function of I• .
Definition 4.7. Let I• be a primary graded sequence of subsets. Define the convex
set Γ(I• ) by
[
Γ(I• ) =
{x/k | x ∈ Ik }.
k>0

(One can show that Γ(I• ) is an unbounded convex set in C.)
Proposition 4.8. Let I• be a primary graded sequence. Then Γ = Γ(I• ) is a
C-convex region in the cone C, i.e. for any x ∈ Γ, x + C ⊂ Γ. Moreover, the region
Γ is cobounded i.e. C \ Γ is bounded.
Proof. Let ℓ and t0 be as in Definition 4.3. From the definitions it follows that the
region Γ contains C ∩ ℓ≥t0 . Thus (C \ Γ) ⊂ (C ∩ ℓ<t0 ) and hence is bounded. Next
let x ∈ Γ. Since x + C ⊂ C, Γ contains the set (x + C) ∩ ℓ≥t0 . But the convex hull

of x and (x + C) ∩ ℓ≥t0 is x + C. Thus x + C ⊂ Γ because Γ is convex.
The following is an important example of a primary graded sequence in a semigroup S.
Proposition 4.9. Let C be an n-dimensional strongly convex rational polyhedral
cone in Rn and let S = C ∩ Zn . Also let I ⊂ S be a subset such that S \ I is
finite. Then the sequence I• defined by Ik := k ∗ I is a primary graded sequence
and Γ(I• ) = conv(I).
9

Proof. Since S \ I is finite there exists t1 > 0 such that S ∩ ℓ≥t1 ⊂ I. Put
M1 = S ∩ ℓ≥t1 . Because C is a rational polyhedral cone, M1 is a finitely generated
semigroup. Let v1 , . . . , vs be semigroup generators for M1 . Let t0 > 0 P
be bigger
s
than all the ℓ(vi ). For k > 0 take x P
∈ S ∩ ℓ≥kt0 ⊂PM1 . Then x =
i=1 ci vi
for cP
kt0 ≤ ℓ(x) =
c
ℓ(v
)
≤
(
c
)t
.
This
implies
that
i ∈ Z≥0 . Thus P
i
i
i
0
i
i
k ≤ i ci and hence ( i ci ) ∗ M1 ⊂ k ∗ M1 . It follows that x ∈ k ∗ M1 . That is,
S ∩ ℓ≥kt0 = (k ∗ M1 ) ∩ ℓ≥kt0 ⊂ (k ∗ I) ∩ ℓ≥kt0 and hence S ∩ ℓ≥kt0 = (k ∗ I) ∩ ℓ≥kt0
as required. The assertion Γ(I• ) = conv(I• ) follows from the observation that
conv(k ∗ I) = k conv(I).

The following is our main result about the asymptotic behavior of a primary
graded sequence.
Theorem 4.10. Let I• be a primary graded sequence. Then
HI• (k)
lim
k→∞
kn
exists and is equal to covol(Γ(I• )).
Proof. Let t0 > 0 be as in Definition 4.3. Then for all k > 0 we have S ∩ ℓ≥kt0 =
Ik ∩ ℓ≥kt0 . Moreover take t0 to be large enough so that the finite set I1 ∩ ℓ<t0
generates the lattice Zn (this is possible because S and hence I1 generate Zn ).
Consider
S̃ = {(x, k) | x ∈ Ik ∩ ℓ<kt0 }.
T̃ = {(x, k) | x ∈ S ∩ ℓ<kt0 }.
S̃ and T̃ are semigroups in Zn × Z≥0 and we have S̃ ⊂ T̃ . From the definition it
follows that both of the groups generated by S̃ and T̃ are Zn+1 . Also the NewtonOkounkov bodies of S̃ and T̃ are:
∆(S̃) = Γ(I• ) ∩ ∆(t0 ),
∆(T̃ ) = ∆(t0 ),
where ∆(t0 ) = C ∩ ℓ≤t0 . Since S ∩ ℓ≥kt0 = Ik ∩ ℓ≥kt0 , we have:
S \ Ik = T̃k \ S̃k ,
Here as usual S̃k = {(x, k) | (x, k) ∈ S̃} (respectively T̃k ) denotes the set of points
in S̃ (respectively T̃ ) at level k. Hence
HI• (k) = #T̃k − #S̃k .
By Theorem 3.1 we have:
lim

#S̃k
= vol(∆(S̃)),
kn

lim

#T̃k
= vol(∆(t0 )).
kn

k→∞

k→∞

Thus

#(S \ Ik )
= vol(∆(t0 )) − vol(∆(S̃)).
kn
On the other hand, we have:
lim

k→∞

∆(t0 ) \ ∆(S̃) = C \ Γ(I• ),
and hence vol(∆(t0 )) − vol(∆(S̃)) = covol(Γ(I• )). This finishes the proof.
10



Definition 4.11. For a primary graded sequence I• we denote n! limk→∞ HI• (k)/k n
by e(I• ). Motivated by commutative algebra, we call it the multiplicity of I• . We
have just proved that e(I• ) = n! covol(Γ(I• )). Note that it is possible for a primary
graded sequence to have multiplicity equal to zero.
The following additivity property is straightforward from definition.
Proposition 4.12. Let I•′ , I•′′ be primary graded sequences. We have:
Γ(I•′ ) + Γ(I•′′ ) = Γ(I•′ + I•′′ ).
Proof. From definition it is clear that Γ(I•′ + I•′′ ) ⊂ Γ(I•′ ) + Γ(I•′′ ). We need to
show the reverse inclusion. Let I• denote I•′ + I•′′ . For k, m > 0 take x ∈ Ik′ and
′′
y ∈ Im
. Then (x/k) + (y/m) = (mx + ky)/km ∈ (1/km)Ikm . This shows that
(x/k) + (y/m) ∈ Γ(I• ) which finishes the proof.

Let I1,• , . . . , In,• be n primary graded sequences. Define the function PI1,• ,...,In,• :
Nn → N by:
PI1,• ,...,In,• (k1 , . . . , kn ) = e(k1 ∗ I1,• + · · · + kn ∗ In,• ).
Theorem 4.13. The function PI1,• ,...,In,• is a homogeneous polynomial of degree
n in k1 , . . . , kn .
Proof. Follows immediately from Proposition 4.12, Theorem 4.10 and Theorem
2.2.

Definition 4.14. Let I1,• , . . . , In,• be primary graded sequences. Define the mixed
multiplicity e(I1,• , . . . , In,• ) to be the coefficient of k1 · · · kn in the polynomial
PI1,• ,...,In,• divided by n!.
From Theorem 4.10 and Proposition 4.12 we have the following corollary:
Corollary 4.15.
e(I1,• , . . . , In,• ) = n! CV (Γ(I1,• ), . . . , Γ(In,• )),
where as before CV denotes the mixed covolume of cobounded regions.
From Theorem 2.3 and Corollary 2.4 we then obtain:
Corollary 4.16 (Alexandrov-Fenchel inequality for mixed multiplicity in semigroups). For primary graded sequences I1,• , . . . , In,• in the semigroup S we have:
e(I1,• , I1,• , I3,• , . . . , In,• )e(I2,• , I2,• , I3,• , . . . , In,• ) ≥ e(I1,• , I2,• , I3,• , . . . , In,• )2 .
Corollary 4.17 (Brunn-Minkowski inequality for multiplicities in semigroups). Let
I• , J• be primary graded sequences in the semigroup S. We have:
e(I• )1/n + e(J• )1/n ≥ e(I• + J• )1/n .
5. Multiplicities of ideals and subspaces in local rings
Let R be a Noetherian local domain of Krull dimension n over a field k, and
with maximal ideal m. We also assume that the residue field R/m is k.
Example 5.1. Let X be an irreducible variety of dimension n over k, and let p
be a point in X. Then the local ring R = OX,p (consisting of rational functions on
X which are regular in a neighborhood of p) is a Noetherian local domain of Krull
dimension n over k. The ideal m consists of functions which vanish at p.
11

If a, b ⊂ R are two k-subspaces then by ab we denote the k-span of all the xy
where x ∈ a and y ∈ b. Note that if a, b are ideals in R then ab coincides with the
product of a and b as ideals.
Definition 5.2.
(i) A k-subspace a in R is called m-primary if it contains a
power of the maximal ideal m.
(ii) A graded sequence of subspaces is a sequence a• = (a1 , a2 , . . .) of k-subspaces
in R such that for all k, m > 0 we have ak am ⊂ ak+m . We call a• an mprimary sequence if moreover a1 is m-primary. It then follows that every ak
is m-primary and hence dimk (R/ak ) is finite. (If each ak is an m-primary
ideal in R we call a• an m-primary graded sequence of ideals.)
When k is algebraically closed, an ideal a in R = OX,p is m-primary if the
subvariety it defines around p coincides with the single point p itself.
Example 5.3. Let a be an m-primary subspace. Then the sequence a• defined by
ak = ak is an m-primary graded sequence of subspaces.
Let a• , b• be m-primary graded sequences of subspaces. Then the sequence
c• = a• b• defined by
ck = ak bk ,
is also an m-primary graded sequence of subspaces which we call the product of a•
and b• .
Definition 5.4. Let a• be an m-primary graded sequence of subspaces. Define the
function Ha• by:
Ha• (k) = dimk (R/ak ).
We call it the Hilbert-Samuel function of a• . The Hilbert-Samuel function Ha (k)
of an m-primary subspace a is the Hilbert-Samuel function of the sequence a• =
(a, a2 , . . .). That is, Ha (k) = dimk (R/ak ).
Remark 5.5. For an m-primary ideal a it is well-known that, for sufficiently large
values of k, the Hilbert-Samuel function Ha coincides with a polynomial of degree
n called the Hilbert-Samuel polynomial of a ([Samuel-Zariski60]).
Definition 5.6. Let a• be an m-primary graded sequence of subspaces. We define
the multiplicity e(a• ) to be:
Ha• (k)
.
kn
(It is not a priori clear that the limit exists.) The multiplicity e(a) of an m-primary
ideal a is the multiplicity of its associated sequence (a, a2 , . . .). That is:
e(a• ) = n! lim

k→∞

Ha (k)
.
kn
(Note that by Remark 5.5 the limit exists in this case.)
e(a) = n! lim

k→∞

The notion of multiplicity comes from the following basic example:
Example 5.7. Let a be an m-primary subspace in the local ring R = OX,p of a
point p in an irreducible variety X over an algebraically closed filed k. Let f1 , . . . , fn
be generic elements in a. Then the multiplicity e(a) is equal to the intersection
multiplicity at p of the hypersurfaces Hi = {x | fi (x) = 0}, i = 1, . . . , n.
12

In Section 7 we use the material in Section 4 to give a formula for e(a• ) in terms
of covolume of a convex region.
One can also define the notion of mixed multiplicity for m-primary ideals as the
polarization of the Hilbert-Samuel multiplicity e(a), i.e. it is the unique function
e(a1 , . . . , an ) which is invariant under permuting the arguments, is multi-additive
with respect to product, and for any m-primary ideal a the mixed multiplicity
e(a, . . . , a) coincides with e(a). In fact one can show that in the above definition of
mixed multiplicity the ai need not be ideals and it suffices for them to be m-primary
subspaces.
Similar to multiplicity we have the following geometric meaning for the notion of
mixed multiplicity when R = OX,p is the local ring of a point p on an n-dimensional
algebraic variety X. Take m-primary subspaces a1 , . . . , an in R. The mixed multiplicity e(a1 , . . . , an ) is equal to the intersection multiplicity, at the origin, of the
hypersurfaces Hi = {x | fi (x) = 0}, i = 1, . . . , n, where each fi is a generic function
from the ai .
6. Case of monomial ideals and Newton polyhedra
In this section we discuss the case of monomial ideals. It is related to the
classical notion of Newton polyhedron of a power series in n variables. We will
see that our Theorem 4.10 in this case immediately recovers (and generalizes) the
local version of celebrated theorem of Bernstein-Kushnirenko ([Kushnirenko76] and
[Arnold-Varchenko-Guseinzade85, Section 12.7]).
Let R be the local ring of an affine toric variety at its torus fixed point. The
algebra R can be realized as follows: Let C ⊂ Rn be an n-dimensional strongly convex rational polyhedral cone with apex at the origin, that is, C is an n-dimensional
convex cone generated by a finite number of rational vectors and it does not contain any lines through the origin. Consider the semigroup algebra over k of the
semigroup of integral points S = C ∩ Zn . In other words, consider
P the algebra
α
of Laurent polynomials consisting of all the f of the form f =
α∈C∩Zn cα x ,
where we have used the shorthand notation x = (x1 , . . . , xn ), α = (a1 , . . . , an ) and
xα = xa1 1 · · · xann . Let R be the localization of this Laurent polynomial algebra at
the maximal ideal m generated by non-constant monomials. (Similarly instead of
R we can take its completion at the maximal ideal m which is an algebra of power
series .)
Definition 6.1. Let a be an m-primary monomial ideal in R, that is, an m-primary
ideal generated by monomials. To a we can associate a subset I(a) ⊂ C ∩ Zn by
I(a) = {α | xα ∈ a}.
The convex hull Γ(a) of I(a) is usually called the Newton polyhedron of the monomial ideal a. It is a convex unbounded polyhedron in C, moreover it is a C-convex
region. The Newton diagram of a is the union of bounded faces of its Newton
polyhedron.
Remark 6.2. It is easy to see that if a is an ideal in R then I = I(a) is a semigroup
ideal in S = C ∩ Zn , that is, if x ∈ I and y ∈ S then x + y ∈ I.
Let a be an m-primary monomial ideal. Then for any k > 0 we have I(ak ) =
k ∗ I(a). It follows from Proposition 4.9 that I• defined by Ik = k ∗ I(a) is a
13

primary graded sequence in S = C ∩ Zn and the convex region Γ(I• ) associated to
I• coincides with the Newton polyhedron Γ(a) = conv(I(a)) defined above.
More generally, let a• be an m-primary graded sequence of monomial ideals in
R. Associate a graded sequence I• = I(a• ) in S to a• by:
Ik = I(ak ) = {α | xα ∈ ak }.
Clearly for any k we have k ∗ I(a1 ) = I(ak1 ) ⊂ I(ak ). From the above we then
conclude the following:
Proposition 6.3. The graded sequence I• = I(a• ) is a primary graded sequence
in the sense of Definition 4.3.
Let Γ(a• ) denote the convex region associated to the primary graded sequence
I• = I(a• ) (Definition 4.7). We make the following important observation that
a• 7→ Γ(a• ) is additive with respect to the product of graded sequences of monomial
ideals.
Proposition 6.4. Let a• , b• be m-primary graded sequences of monomial ideals in
R. Then I(a• b• ) = I(a• ) + I(b• ). It follows from Proposition 4.12 that:
Γ(a• b• ) = Γ(a• ) + Γ(b• ).
From Proposition 6.3, Proposition 6.4 and Theorem 4.10 we readily obtain the
following.
Theorem 6.5. Let a• be an m-primary graded sequence of monomial ideals in R.
Then:
e(a• ) = n! covol(Γ(a• )).
In particular, if a is an m-primary monomial ideal then:
e(a) = n! covol(Γ(a)).
Here Γ(a) is the Newton polyhedron of a i.e. the convex hull of I(a).
Theorem 6.6. Let a1 , . . . , an be m-primary monomial ideals in R. Then the mixed
multiplicity e(a1 , . . . , an ) is given by:
e(a1 , . . . , an ) = n! CV (Γ(a1 ), . . . , Γ(an )),
where as before CV denotes the mixed covolume of cobounded convex regions.
Remark 6.7. Using Theorem 4.13 one can define the mixed multiplicity of mprimary graded sequences of monomial ideals. Then Theorem 6.6 can immediately
be extended to mixed multiplicities of m-primary graded sequences of monomial
ideals.
One knows that the mixed multiplicity of an n-tuple (a1 , . . . , an ) of m-primary
subspaces in R gives the intersection multiplicity, at the origin, of hypersurfacs
Hi = {x | fi (x) = 0}, i = 1, . . . , n, where each fi is a generic element from the
subspace ai . Theorem 6.6 then gives the following corollary.
Corollary 6.8 (Local Bernstein-Kushnirenko theorem). Let a1 , . . . , an be m-primary
monomial ideals in R. Consider a system of equations f1 (x) = · · · = fn (x) = 0
where each fi is a generic element from ai . Then the intersection multiplicity at
the origin of this system is equal to n! covol(Γ(a1 ), . . . , Γ(an )).
14

Remark 6.9.
(i) When R is the algebra of polynomials k[x1 , . . . , xn ](0) localized at the origin (or the algebra of power series localized at the origin),
i.e. the case corresponding to the local ring of a smooth affine toric variety, Corollary 6.8 is the local version of the classical Bernstein-Kushnirenko
theorem ([Arnold-Varchenko-Guseinzade85, Section 12.7]).
(ii) As opposed to the proof above, the original proof of the Kushnirenko theorem is quite involved.
(iii) Corollary 6.8 has been known to the second author since the early 90’s
(cf. [Khovanskii92]), although as far as the authors know it has not been
published.
7. Main results
Let R be a domain over a field k. Equip the group Zn with a total order
respecting addition.
Definition 7.1 (Valuation). A valuation v : R \ {0} → Zn is a function satisfying:
(1) For all 0 6= f, g ∈ R, v(f g) = v(f ) + v(g).
(2) For all 0 6= f, g ∈ R with f + g 6= 0 we have v(f + g) ≥ min(v(f ), v(g)).
(One then shows that when v(f ) 6= v(g), v(f + g) = min(v(f ), v(g)).)
(3) For all 0 6= λ ∈ k, v(λ) = 0.
We say that v has one-dimensional leaves if whenever v(f ) = v(g), there exists
λ ∈ k with v(g + λf ) > v(g).
From definition S = v(R \ {0}) ∪ {0} is an additive subsemigroup of Zn which
we call the value semigroup of (R, v). Any valuation on R extends to the field
of fractions K of R by defining v(f /g) = v(f ) − v(g). The set Rv = {0 6= f ∈
K | v(f ) ≥ 0} ∪ {0} is a local subring of K called the valuation ring of v. Also
mv = {0 6= f ∈ K | v(f ) > 0} ∪ {0} is the maximal ideal in Rv . The field Rv /mv is
called the residue field of v. One can see that v has one-dimensional leaves if and
only if the residue field of v is k.
Definition 7.2. For a subspace a in R define I = I(a) ⊂ S by:
I = {v(f ) | f ∈ a \ {0}}.
Similarly, for a graded sequence of subspaces a• in R, define I• = I(a• ) by:
Ik = I(ak ) = {v(f ) | f ∈ ak \ {0}}.
For the rest of the paper we assume that R is a Noetherian local domain of
dimension n such that R is an algebra over a field k isomorphic to the residue field
R/m, where m is the maximal ideal of R. Moreover, we assume that R has a good
valuation in the following sense:
Definition 7.3. We say that a Zn -valued valuation v on R with one-dimensional
leaves is good if the following hold:
(i) The value semigroup S = v(R \ {0}) ∪ {0} generates the whole lattice Zn , and
its associated cone C(S) is a strongly convex cone (recall that C(S) is the closure
of convex hull of S). It implies that there is a linear function ℓ : Rn → R such that
C(S) lies in ℓ≥0 and intersects ℓ−1 (0) only at the origin.
(ii) We assume there exists r0 > 0 and a linear function ℓ as above such that for
any f ∈ R if ℓ(v(f )) ≥ kr0 for some k > 0 then f ∈ mk .
15

The condition (ii) in particular implies that for any k > 0 we have:
I(mk ) ∩ ℓ≥kr0 = S ∩ ℓ≥kr0 .
In other words, the sequence M• given by Mk = I(mk ) is a primary graded
sequence in the value semigroup S.
The following is a generalization of Proposition 6.3.
Proposition 7.4. Let v be a good valuation on R. Let a• be an m-primary graded
sequence of subspaces in R. Then the associated graded sequence I• = I(a• ) is a
primary graded sequence in the value semigroup S in the sense of Definition 4.3.
Proof. Let m > 0 be such that mm ⊂ a1 . Then for any k > 0 we have mkm ⊂ ak
which then implies that Mkm ⊂ Ik . This proves the claim.

Example 7.5. As in Section 6 let R be the local ring of an affine toric variety
at its torus fixed point: Take C ⊂ Rn to be an n-dimensional strongly convex
rational polyhedral cone with apex at the origin. Consider
of Laurent
P the algebra
α
c
x
.
Then
R is
polynomials consisting of all the f of the form f =
n
α
α∈C∩Z
the localization of this algebra at the maximal ideal generated by non-constant
monomials. Take a total order on Zn which respects addition and such that the
semigroup S = C ∩ Zn is well-ordered. We also require that if ℓ(α) > ℓ(β) then
α > β, for any α, β ∈ Zn . Such a total order can be constructed as follows: pick
linear functions ℓ2 , . . . , ℓn on Rn such that ℓ, ℓ2 , . . . , ℓn are linearly independent, and
for each i the cone C lies in (ℓi )≥0 . Given α, β ∈ Zn define α > β if ℓ(α) > ℓ(β), or
ℓ(α) = ℓ(β) and ℓ2 (α) > ℓ2 (β), and so on.
Now one defines a (lowest term)
P valuation v on the algebra R with values in
S = C ∩ Zn as follows: For f = α∈S cα xα put:
v(f ) = min{α | cα 6= 0}.
Clearly v extends to the field of fractions of Laurent polynomials and in particular
to R. Similarly v can be defined for formal power series and formal Laurent series.
It is easy to see that v is a valuation with one-dimensional leaves on R. Let us show
that it is moreover a good valuation. Take
P 0 6= f ∈ R. Without loss of generality
we can take f to be a Laurent series f = α∈S cα xα . Applying Proposition 4.9 to
the sequence I• , where Ik = {α | xα ∈ mk }, we know that there exists r0 > 0 with
the following property: if for some α ∈ S we have ℓ(α) ≥ kr0 then xα ∈ mk . On
the other hand, if α ≤ β then ℓ(α) ≤ ℓ(β). Thus ℓ(β) ≥ kr0 and hence xβ ∈ mk .
It follows that if ℓ(v(f )) ≥ kr0 then all the nonzero monomials in f lie in mk and
hence f ∈ mk . This proves the claim that v is a good valuation on R.
The arguments in Example 7.5 in particular show the following:
Theorem 7.6. If R is a regular local ring then R has a good valuation.
Proof. The completion R of R is isomorphic to an algebra of formal power series
over the residue field k. The above construction gives a good valuation on R. One
verifies that the restriction of this valuation to R is still a good valuation.

More generally, one has:
Theorem 7.7. Suppose R is an analytically irreducible local domain (i.e. the
completion of R has no zero divisors). Moreover, suppose that there exists a regular
local ring S containing R such that S is essentially of finite type over R, R and
16

S have the same quotient field k and the residue field map R/mR → S/mS is an
isomorphism. Then R has a good valuation.
Proof. By Theorem 7.6, S has a good valuation. By the linear Zariski subspace
theorem in [Hubl01, Theorem 1] or [Cutkosky-a, Lemma 4.3] v|R is a good valuation
for R too.

Using Theorem 7.7 and as in [Cutkosky-a, Theorem 5.2] we have the following:
Theorem 7.8. Let R be an analytically irreducible local domain over k. Then R
has a good valuation.
Proposition 7.9. Let I = I(a) be the subset of integral points associated to an
m-primary subspace a in R. Then we have:
dimk (R/a) = #(S \ I).
Proof. Take m > 0 with mm ⊂ a and let r0 and ℓ be as in Definition 7.3. If
ℓ(v(f )) > mr0 then f ∈ mm ⊂ a. Thus the set of valuations of elements in
R \ a is bounded. In particular S \ I is finite. Let {v1 , . . . , vr } = S \ I. Let
B = {b1 , . . . , br } ⊂ R be such that v(bi ) = vi , i = 1, . . . , r. WePclaim that no
linear combination
of b1 , . . . , br lies in a. By contradiction suppose i ci bi = a ∈ a.
P
Then v( i ci bi ) is equal to v(bj ) for some j. This implies that v(bj ) should lie in
I which contradicts the choice of the vi . Thus the image of B in R/a is a linearly
independent set. Among the set of elements in R that are not in the span of a and
B take f with maximum v(f ). If v(f ) = v(b) for some b ∈ B, then we can subtract
a multiple of b from f getting an element g with v(g) > v(f ) which contradicts
the choice of f . Similarly v(f ) can not lie in I otherwise we can subtract an
element of a from f to arrive at a similar contradiction. This shows that the set of
images of elements of B in R/a is a k-vector space basis for R/a which proves the
proposition.

Corollary 7.10. Let a• be an m-primary graded sequence of subspaces in R and
put I• = I(a• ). We then have:
e(a• ) = e(I• ).
Definition 7.11. To the sequence of subspaces a• we associate a C-convex region
Γ(a• ), which is the convex region Γ(I• ) associated to the primary sequence I• =
I(a• ). The convex region Γ(a• ) depends on the choice of the valuation v. By
definition the convex region Γ(a) associated to an m-primary subspace a is the
convex region associated to the sequence of subspaces (a, a2 , a3 , . . .).
Theorem 7.12. Let a• be an m-primary graded sequence of subspaces in R. Then:
Ha• (k)
= n! covol(Γ(a• )).
k→∞
kn
In particular, if a is an m-primary ideal we have e(a) = n! covol(Γ(a)).
e(a• ) = n! lim

The following superadditivity follows from Proposition 4.12. Note that I(a• ) +
I(b• ) ⊂ I(a• b• ) (cf. Proposition 6.4).
Proposition 7.13. Let a• , b• be two m-primary graded sequences of subspaces in
R. We have:
Γ(a• ) + Γ(b• ) ⊂ Γ(a• b• ).
17

From Theorem 7.12, Proposition 7.13 and Corollary 2.4 we readily obtain:
Corollary 7.14. (Brunn-Minkowski for multiplicities) Let a• , b• be two m-primary
graded sequences of subspaces in R. Then:
e(a• )1/n + e(b• )1/n ≥ e(a• b• )1/n .

(7)

Remark 7.15. By Theorem 7.8 and Corollary 7.14 we obtain the Brunn-Minkowski
inequality (7) for an analytically irreducible local domain R. But in fact the assumption that R is analytically irreducible is not necessary: Suppose R is not necessarily
analytically irreducible. First by a reduction theorem the statement can be reduced
to dim R = n = 2. In dimension 2, the inequality (7) implies that the mixed multiplicity of ideals e(·, ·), regarded as a bilinear form on the (multiplicative) semigroup
of m-primary graded sequences of ideals, is positive semidefinite restricted to each
local analytic irreducible component. But the sum of positive semidefinite forms
is again positive semidefinite which implies that Corollary 7.14 should hold for R
itself.
As another corollary of Theorem 7.12 we can immediately obtain inequalities
between the multiplicity of an m-primary ideal, multiplicity of its associated initial
ideal and its length. Let R be a regular local ring of dimension n with a good
valuation (as in Example 7.5 and Theorem 7.6).
Corollary 7.16 (Multiplicity of an ideal versus multiplicity of its initial ideal).
Let a be an m-primary ideal in R and let in(a) denote the initial ideal of a, that is,
the monomial ideal in the polynomial algebra localized at the origin k[x1 , . . . , xn ](0)
corresponding to the semigroup ideal I(a). We have:
e(a) ≤ e(in(a)) ≤ n! dimk (R/a).
k

More generally, if in(a ) denote the monomial ideal in k[x1 , . . . , xn ](0) corresponding to the semigroup ideal I(ak ) then the sequence of numbers
e(in(ak ))
kn
is decreasing and converges to e(a) as k → ∞.
Proof. From definition one shows that Γ(in(a)) is the convex hull of I(a) (see
Theorem 6.5). It easily follows that
I(a) ⊂ Γ(in(a)) ⊂ Γ(a).
We now notice that dimk (R/a) is the number of integral points in S \ I(a) that in
turn is bigger than or equal to the volume of Rn≥0 \ Γ(in(a)) and hence the volume
of Rn≥0 \ Γ(a). More generally, from the definition of Γ(a) we have an increasing
sequence of convex regions:
Γ(in(a)) ⊂ (1/2)Γ(in(a2 )) ⊂ · · · ⊂ Γ(a) =

∞
[

(1/k)Γ(in(ak )).

k=1

Now from Theorem 7.12 we have e(a) = n! covol(Γ(a)) and for each k, e(in(ak )) =
n! covol(Γ(in(ak ))). This finishes the proof.

The inequality e(a) ≤ n! dimk (R/a) is a special case of an inequality of Lech
[Lech64, Theorem 3]. See also Lemma 1.3 in [Fernex-Ein-Mustaţă04].
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Department of Mathematics, School of Arts and Sciences, University of Pittsburgh,
301 Thackeray Hall, Pittsburgh, PA 15260, U.S.A.
E-mail address: kaveh@pitt.edu
Department of Mathematics, University of Toronto, Toronto, Canada; Moscow Independent University; Institute for Systems Analysis, Russian Academy of Sciences
E-mail address: askold@math.utoronto.ca

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