Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

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by  Carles Bivià-Ausina

L OJASIEWICZ EXPONENT OF FAMILIES OF IDEALS, REES MIXED
MULTIPLICITIES AND NEWTON FILTRATIONS
CARLES BIVIA`-AUSINA AND SANTIAGO ENCINAS
Abstract. WegiveanexpressionfortheL ojasiewiczexponentofawideclassofn-tuplesof
ideals (I1,...,In) in On using the information given by a fixed Newton filtration. In order
2
to obtain this expression we consider a reformulation of L ojasiewicz exponents in terms
1
0 of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted
2 homogeneous functions (Cn,0) → (C,0) for which the L ojasiewicz of its gradient map ∇f

n attains the maximum possible value.
a
J

9
2
1. Introduction

]
G
Let O be the ring of complex analytic function germs f : (Cn,0) → C. S. L ojasiewicz
A n
proved in [19] (as a consequence of a more general result of functional analysis) that if I is
.
h
an ideal of O of finite colength and g ,...,g is a generating system of I, then there exists
t n 1 s
a
m a real number α > 0 for which there exist a constant C > 0 and an open neighbourhood U
[ of 0 in Cn such that

2
(1) kxkα 6 Csup|g (x)|,
v i
1 i
3 forallx ∈ U. Theinfimum ofsuch αiscalledtheL ojasiewicz exponent of I andisdenotedby
7
1 L (I). If g : (Cn,0) → (Cm,0)denotes acomplex analytic mapgermsuch that g−1(0) = {0},
0
3. then the L ojasiewicz exponent of g is defined as L (g) = L (I), where I denotes the ideal
0 0
0
of O generated by the component functions of g. If f ∈ O has an isolated singularity
1 n n
1 at the origin, then the L ojasiewicz exponent of the gradient map ∇f : (Cn,0) → (Cn,0) is
:
v
particularly known in singularity theory, by virtue of the result of Teissier [30, p. 280] stating
i
X that the degree of C0-determinacy of f is equal to [L (∇f)] + 1, where [a] stands for the
0
ar integer part of a given a ∈ R. It is known that L (∇f) is an analytical invariant of f but it
0
is still unknown if L (∇f) is a topological invariant of f (see for instance [16]).
0
Let ν : O → R ∪ {+∞} be the asymptotic Samuel function of I (see [17] or [14, p.
I n >0
139]). By a result of Nagata [21] the range of ν is a subset of Q ∪ {+∞}. If J is any
I >0
ideal of O , let us define ν (J) = min{ν (h ),...,ν (h )}, where h ,...,h denotes any
n I I 1 I r 1 r
2010 Mathematics Subject Classification. Primary 32S05; Secondary 13H15.
Key words and phrases. L ojasiewicz exponents, integral closure of ideals, mixed multiplicities of ideals,
monomial ideals.
The first author was partially supported by DGICYT Grant MTM2009–08933. The second author was
partially supported by DGICYT Grant MTM2009–07291and CCG08-UAM/ESP-3928.
1
2 CARLES BIVIA`-AUSINAAND SANTIAGOENCINAS
generating system of J. We will denote by m , or simply by m, the maximal ideal of O .
n n
Lejeune and Teissier proved in [17] the following fundamental facts: L (I) = 1 (therefore
0 νI(m)
L (I) is a rational number), relation (1) holds for α = L (I), for some constant C > 0 and
0 0
some open neighbourhood U of 0 ∈ Cn, and L (I) is expressed as
0
p
(2) L (I) = min : p,q ∈ Z , mp ⊆ Iq ,
0 >1
q
(cid:26) (cid:27)
where J denotes the integral closure of a given ideal J of O . The above expression was
n
one of the motivations that lead the first author to introduce in [5] the notion of L ojasiewicz
exponent of a set of ideals (see Definition 2.6). By substituting m by a proper ideal J of O
n
in (2) we obtain what is known as the L ojasiewicz exponent of I with respect to J (see (10),
(11) and [17]).
The effective computation of the L ojasiewicz exponent L (I) of a given ideal I of O is
0 n
a non-trivial problem, since it is intimately related with the determination of the integral
closure of I. The authors applied in [6] the explicit construction of a log-resolution of I to
show an effective method to compute L (I). Newton polyhedra have proven to be a powerful
0
tool in the estimation, and determination in some cases, of L ojasiewicz exponents, as can be
seen in [2], [10], [18] and [22].
Let w = (w ,...,w ) ∈ Zn , we say that a monomial xk1 ···xkn has w-degree d when
1 n >1 1 n
w k +···+w k = d. Apolynomial functionf : Cn → C is said to beweighted homogeneous
1 1 n n
of degree d with respect to w when f is written as a sum of monomials of w-degree d. A
function h ∈ O is termed semi-weighted homogeneous of degree d with respect to w when
n
h is expressed as a sum h = h + h , where h is weighted homogeneous of degree d with
1 2 1
respect to w, h has an isolated singularity at the origin and h is a sum of monomials of
1 2
w-degree greater that d.
The motivation of our work is the article [16] of Krasin´ski-Oleksik-Pl oski, whose main
result is a formula for the L ojasiewicz exponent L (∇f) of any weighted homogeneous func-
0
tion f : C3 → C in terms of the weights and the degree of f. More precisely, if f ∈ O is
3
weighted homogeneous with respect to (w ,w ,w ) of degree d and w = min{w ,w ,w }
1 2 3 0 1 2 3
then it is proven in [16] that
3
d−w d
(3) L (∇f) = min 0, −1 .
0
w w
( 0 i=1(cid:18) i (cid:19))
Y
We remark that when d > 2w , for all i = 1,2,3, then L (∇f) = d−w0. As a consequence
i 0 w0
of (3) we have that if f : C3 → C is a weighted homogeneous function with respect to
(w ,w ,w ), then L (∇f) is a topological invariant of f, by the results of Saeki [27] and Yau
1 2 3 0
[32].
Let us fix a vector of weights w = (w ,...,w ) ∈ Zn and let w = min w . Let f ∈ O
1 n >1 0 i i n
be a semi-weighted homogeneous function of degree d with respect to w. It is well-known
L OJASIEWICZ EXPONENT OF FAMILIES OF IDEALS AND NEWTON FILTRATIONS 3
that
d−w
(4) L (∇f) 6 0.
0
w
0
If d < 2w , for some i ∈ {1,...,n}, then it is easy to find examples where inequality (4)
i
is strict. Assuming d > 2w , for all i = 1,...,n, then it is reasonable to conjecture that
i
equality holds in (4).
In [7] we considered the problem of finding a sufficient condition on f for equality in (4).
We addressed this problem in the framework of L ojasiewicz exponents of sets of n ideals in
O (in the sense of [5]) and weighted homogeneous filtrations. Thus, we introduced in [7] the
n
concept of sets of ideals admitting a w-matching (see Definition 4.1). The application of this
notion to gradient maps lead to determine a wide class of functions for which equality holds
in(4). Inparticular, we foundthat thisequality is trueforevery semi-weighted homogeneous
function f ∈ O of degree d with respect to w such that w divides d, for all i = 1,...,n
n i
(see [7, Corollary 4.16]).
Inthisarticleweshowanextensionofthemainresult of[7]toNewtonfiltrationsingeneral
(see Theorem 3.11). This extension projects to new results about the L ojasiewicz exponent
of the gradient of semi-weighted homogeneous functions. In this direction, we emphasize
that Corollary 4.13 shows a quite wide class of functions f ∈ O for which L (∇f) attains
n 0
the maximum possible value, that is, such that equality holds in (4). The techniques that we
will apply in this article come from multiplicity theory in local rings. More precisely, we use
the notion of mixed multiplicities of a family of ideals of finite colength and its generalization
to suitable families of ideals called Rees mixed multiplicities (see [4]).
Let us consider a Newton polyhedron Γ in Rn. The key ingredient in our approach to
+ +
L ojasiewicz exponents in this article is the notion of Γ -linked pairs (I;J ,...,J ), where
+ 1 n
I,J ,...,J are ideals of O (see Definition 3.10). This notion is expressed via the non-
1 n n
degeneracy condition explored in [8].
The article is organized as follows. In Section 2 we recall the basic definitions and previous
results (mainly from [4] and [5]) that lead to the definition of L ojasiewicz exponent of a set
of ideals. For the sake of completeness we also introduce in Section 2 some auxiliary results
needed in the proof of the main result. In Section 3 we show the main result of the article
(see Theorem 3.11) and discuss some examples. In Section 4 we particularize the techniques
developed in Section 3 to weighted homogeneous filtrations and, as said before, we derive
new results about the L ojasiewicz exponent of gradient maps.
2. The L ojasiewicz exponent of a set of ideals
Let (R,m) be a Noetherian local ring of dimension n and let I be an ideal of R of finite
colength (also called an m-primary ideal). Then we denote by e(I) the Samuel multiplicity
of I (see [12], [14, §11] or [29] for the definition and basic properties of this notion). We
recall that e(I) = ℓ(R/I) if I admits a generating system formed by n elements.
4 CARLES BIVIA`-AUSINAAND SANTIAGOENCINAS
If I ,...,I are ideals of R of finite colength, then we denote by e(I ,...,I ) the mixed
1 n 1 n
multiplicity of I ,...,I in the sense of Risler and Teissier [29] (see also [14, §17] or [31]).
1 n
Definition 2.1. [4] Let (R,m) be a Noetherian local ring of dimension n. Let I ,...,I be
1 n
ideals of R. Then we define the Rees mixed multiplicity of I ,...,I as
1 n
(5) σ(I ,...,I ) = max e(I +mr,...,I +mr),
1 n 1 n
r∈Z+
whenthenumberontherighthandsideisfinite. Ifthesetofintegers{e(I +mr,...,I +mr) :
1 n
r ∈ Z } is non-bounded then we set σ(I ,...,I ) = ∞.
+ 1 n
We remark that if I is an ideal of R of finite colength, for all i = 1,...,n, then
i
σ(I ,...,I ) = e(I ,...,I ). Moreover, if I = ··· = I = I, for some ideal I of R of
1 n 1 n 1 n
finite colength, then e(I ,...,I ) = e(I).
1 n
Let us suppose that the residue field k = R/m is infinite. Let I ,...,I be ideals of
1 n
R and let {a ,...,a } be a generating system of I , where s > 1, for i = 1,...,n. Let
i1 isi i i
s = s +···+s . We say that a property holds for sufficiently general elements of I ⊕···⊕I
1 n 1 n
if there exists a non-empty Zariski-open set U in ks such that the said property holds for
all elements (g ,...,g ) ∈ I ⊕ ··· ⊕ I for which g = u a , i = 1,...,n, where
1 n 1 n i j ij ij
(u ,...,u ,...,u ,...,u ) ∈ U.
11 1s1 n1 nsn P
The next proposition characterizes the finiteness of σ(I ,...,I ).
1 n
Proposition 2.2. [4, p.393] Let I ,...,I be ideals of a Noetherian local ring (R,m) such
1 n
that the residue field k = R/m is infinite. Then σ(I ,...,I ) < ∞ if and only if there exist
1 n
elements g ∈ I , for i = 1,...,n, such that hg ,...,g i has finite colength. In this case,
i i 1 n
we have that σ(I ,...,I ) = e(g ,...,g ) for sufficiently general elements (g ,...,g ) ∈
1 n 1 n 1 n
I ⊕···⊕I .
1 n
If I and J are ideals of finite colength of R such that J ⊆ I then it is well-known that
e(J) > e(I) (see for instance [29, p. 292]). The following result extends this inequality to
Rees mixed multiplicities.
Lemma 2.3. [5, p. 392] Let (R,m) be a Noetherian local ring of dimension n > 1. Let
J ,...,J be ideals of R such that σ(J ,...,J ) < ∞. Let I ,...,I be ideals of R for which
1 n 1 n 1 n
J ⊆ I , for all i = 1,...,n. Then σ(I ,...,I ) < ∞ and
i i 1 n
σ(J ,...,J ) > σ(I ,...,I ).
1 n 1 n
Let us recall some basic definitions. We will denote by R the set of non-negative real
+
numbers. We also set Z = Z ∩ R . Let us fix a coordinate system x ,...,x in Cn. If
+ + 1 n
k = (k ,...,k ) ∈ Zn, we will denote the monomial xk1 ···xkn by xk. Let A ⊆ Zn, then
1 n + 1 n +
the Newton polyhedron determined by A, denoted by Γ (A), is the convex hull of the set
+
{k+v : k ∈ A,v ∈ Rn}. A subset Γ ⊆ Rn is called a Newton polyhedron when Γ = Γ (A),
+ + + + +
for some A ⊆ Zn. A Newton polyhedron Γ ⊆ Rn is termed convenient when Γ meets the
+ + + +
x -axis in a point different from the origin, for all i = 1,...,n.
i
L OJASIEWICZ EXPONENT OF FAMILIES OF IDEALS AND NEWTON FILTRATIONS 5
If h ∈ O and h = a xk denotes the Taylor expansion of h around the origin, then the
n k k
support of h is the set supp(h) = {k ∈ Zn : a 6= 0}. If h 6= 0, then the Newton polyhedron
P + k
of h is defined as Γ (h) = Γ (supp(h)). If h = 0 then we set Γ (h) = ∅. If I is an ideal
+ + +
of O and g ,...,g is a generating system of I, then we define the Newton polyhedron of I
n 1 s
as the convex hull of Γ (g )∪···∪Γ (g ). It is easy to check that the definition of Γ (I)
+ 1 + s +
does not depend on the chosen generating system of I. We denote the Newton boundary of
Γ (I) by Γ(I).
+
We say that a proper ideal I of O is monomial when I admits a generating system
n
formed by monomials. We recall that if I is a monomial ideal of O of finite colength, then
n
e(I) = n!V (Rn rΓ (I)), where V denotes n-dimensional volume (see for instance [31, p.
n + + n
239]).
Definition 2.4. Let I ,...,I be monomial ideals of O with σ(I ,...,I ) < ∞. Then
1 n n 1 n
we denote by S(I ,...,I ) the family of maps g = (g ,...,g ) : (Cn,0) → (Cn,0) such
1 n 1 n
that g−1(0) = {0}, g ∈ I , for all i = 1,...,n, and σ(I ,...,I ) = e(g ,...,g ), where
i i 1 n 1 n
e(g ,...,g ) stands for the multiplicity of the ideal of O generated by g ,...,g . The
1 n n 1 n
elements of S(I ,...,I ) are characterized in [4, Theorem 3.10].
1 n
We denote by S (I ,...,I ) the set formed by the maps (g ,...,g ) ∈ S(I ,...,I ) such
0 1 n 1 n 1 n
that Γ (g ) = Γ (I ), for all i = 1,...,n.
+ i + i
Let I ,...,I be ideals of a local ring (R,m) for which σ(I ,...,I ) < ∞. Then we define
1 n 1 n
(6) r(I ,...,I ) = min r ∈ Z : σ(I ,...,I ) = e(I +mr,...,I +mr) .
1 n + 1 n 1 n
We recall that if g : (Cn,0)(cid:8)→ (Cm,0) is an analytic map germ such that g−(cid:9)1(0) = {0},
then L (g) denotes the L ojasiewicz exponent of the ideal generated by the components of g.
0
Theorem 2.5. [5, p. 398] Let I ,...,I be monomial ideals of O such that σ(I ,...,I ) is
1 n n 1 n
finite. If g ∈ S (I ,...,I ), then L (g) depends only on I ,...,I and it is given by
0 1 n 0 1 n
r(Is,...,Is)
(7) L (g) = min 1 n .
0
s>1 s
The previous result motivated the following definition.
Definition 2.6. [5, p. 399] Let (R,m) be a Noetherian local ring of dimension n. Let
I ,...,I be ideals of R for which σ(I ,...,I ) < ∞. We define the L ojasiewicz exponent of
1 n 1 n
I ,...,I as
1 n
r(Is,...,Is)
L (I ,...,I ) = inf 1 n .
0 1 n
s>1 s
Asaconsequence ofLemma2.8, wehavethatr(Is,...,Is) 6 sr(I ,...,I ), foralls ∈ Z .
1 n 1 n >1
Hence L (I ,...,I ) 6 r(I ,...,I ).
0 1 n 1 n
The L ojasiewicz exponent given in Definition 2.6 is coherent with the original definition
of L ojasiewicz exponent for an analytic map (see (1)), as is shown in the following result.
6 CARLES BIVIA`-AUSINAAND SANTIAGOENCINAS
Lemma 2.7. Let g = (g ,...,g ) : (Cn,0) → (Cn,0) be an analytic map germ such that
1 n
g−1(0) = {0}. Then
(8) L (g) = L (hg i,...,hg i).
0 0 1 n
Proof. Let us fix integers r,s > 1. Let I denote the ideal generated by the components of g.
It can be proved using Proposition 2.2 and Lemma 2.3 that
e(hgsi+mr,...,hgsi+mr) = e(hgs,...,gsi+mr).
1 n 1 n
Moreover Is = hgs,...,gsi (see for instance [14, p. 344]). Then, by the Rees’ multiplicity
1 n
theorem (see [14, p. 222]) we obtain that e(gs,...,gs) = e(hgsi+mr,...,hgsi+mr) if and
1 n 1 n
only if mr ⊆ Is. Hence we deduce that
r(hgsi,...,hgsi) = min r : mr ⊆ Is .
1 n
Therefore (cid:8) (cid:9)
min r : mr ⊆ Is
L (hg i,...,hg i) = min
0 1 n
s>1 s
(cid:8) (cid:9)
p
= min : p,q ∈ Z , mp ⊆ Iq = L (I),
>1 0
q
(cid:26) (cid:27)
where the last equality follows from (2). (cid:3)
Under the hypothesis of Definition 2.6, let us denote by J a proper ideal of R. An easy
application of Lemma 2.3 shows that
σ(I ,...,I ) = max σ(I +Jr,...,I +Jr).
1 n 1 n
r∈Z+
Hence, let us define
(9) r (I ,...,I ) = min r ∈ Z : σ(I ,...,I ) = σ(I +Jr,...,I +Jr) .
J 1 n + 1 n 1 n
Let I be an ideal of R of finite(cid:8)colength. Then we denote by r (I) the number(cid:9)r (I,...,I),
J J
where I is repeated n times. We deduce from the Rees’ multiplicity theorem (see [14, p.
222]) that if R is quasi-unmixed then r (I) = min{r > 1 : Jr ⊆ I}.
J
Lemma 2.8. [7, p. 581] Let (R,m) be a Noetherian local ring of dimension n. Let I ,...,I
1 n
be ideals of R such that σ(I ,...,I ) < ∞ and let J be a proper ideal of R. Then
1 n
r (Is,...,Is) 6 sr (I ,...,I )
J 1 n J 1 n
1
rJs(I1,...,In) > rJ(I1,...,In)
s
for any integer s > 1.
We remark that the previous lemma was proven in [7] under the assumption that the ideal
J has finite colength, but the same proof works equally for any proper ideal J of O .
n
L OJASIEWICZ EXPONENT OF FAMILIES OF IDEALS AND NEWTON FILTRATIONS 7
If I is an ideal of O of finite colength and J is a proper ideal of O , then the L ojasiewicz
n n
exponent of I with respect to J, denoted by L (I), is defined as the infimum of those α > 0
J
such that there exist a constant C > 0 and an open neighbourhood U of 0 ∈ Cn for which
(10) sup|h (x)|α 6 Csup|g (x)|,
j i
j i
for all x ∈ U, where {h : j = 1,...,r} and {g : i = 1,...,s} are generating systems of J
j i
and I, respectively. As a consequence of [17, §7] we have that L (I) is a rational number
J
and
p
(11) L (I) = min : p,q ∈ Z , Jp ⊆ Iq .
J >1
q
(cid:26) (cid:27)
If g : (Cn,0) → (Cm,0) is an analytic map germ such that g−1(0) = {0} and J is a proper
ideal of O then we denote by L (g) the L ojasiewicz exponent L (I), where I is the ideal
n J J
generated by the component functions of g.
Now we extend Definition 2.6 by considering r (I ,...,I ) instead of r(I ,...,I ).
J 1 n 1 n
Definition 2.9. [7, p. 581] Let (R,m) be a Noetherian local ring of dimension n. Let
I ,...,I be ideals of R such that σ(I ,...,I ) < ∞. Let J be a proper ideal of R. We
1 n 1 n
define the L ojasiewicz exponent of I ,...,I with respect to J, denoted by L (I ,...,I ), as
1 n J 1 n
r (Is,...,Is)
(12) L (I ,...,I ) = inf J 1 n .
J 1 n
s>1 s
Under the conditions of the previous definition, we observe that L (I ,...,I ) is expressed
J 1 n
as a limit inferior (see [7, p. 581] for details), that is:
r (Is,...,Is)
L (I ,...,I ) = liminf J 1 n .
J 1 n
s→∞ s
If I is an m-primary ideal of R, then we denote by L (I) the number L (I,...,I), where
J J
I is repeated n times. We remark that when R is quasi-unmixed then L (I) is given by
J
p
(13) L (I) = min : p,q ∈ Z , Jp ⊆ Iq ,
J >1
q
(cid:26) (cid:27)
by a direct application of the Rees’ multiplicity theorem.
Wepoint out that we are denoting L (I ,...,I ) by L (I ,...,I ) and that L (I ,...,I )
m 1 n 0 1 n J 1 n
is not defined when J is the zero ideal. If I is an ideal of O , for all i = 1,...,n, then the
i n
subscript inL (I ,...,I ) correspondstothecommonly usednotationtorefer toL ojasiewicz
0 1 n
exponents in a neighbourhood of 0 ∈ Cn, as defined in (1). If J is a proper ideal of O , we
n
also remark that the result analogous to Lemma 2.7 obtained by writing L instead of L in
0 J
equality (8) also holds; it follows by a straightforward reproduction of the proof of Lemma
2.7 consisting of replacing m by J.
For the sake of completeness, we recall the following two results, which will be applied in
the next section.
8 CARLES BIVIA`-AUSINAAND SANTIAGOENCINAS
Lemma 2.10. [7, p. 582] Let (R,m) be a quasi-unmixed Noetherian local ring of dimension
n. Let I ,...,I be ideals of R for which σ(I ,...,I ) < ∞. If J ,J are proper ideals of R
1 n 1 n 1 2
such that J has finite colength then
2
L (I ,...,I ) 6 L (J )L (I ,...,I ).
J1 1 n J1 2 J2 1 n
Proposition 2.11. Let (R,m) be a Noetherian local ring of dimension n. Let J be a proper
ideal of R. For each i = 1,...,n let us consider ideals I and J of R such that I ⊆ J and
i i i i
σ(I ,...,I ) = σ(J ,...,J ) < ∞. Then
1 n 1 n
(14) L (I ,...,I ) 6 L (J ,...,J ),
J 1 n J 1 n
Proof. It follows by replacing m by I in the proof of [5, Proposition 4.7] (cid:3)
Weremarkthatif(R,m)isaNoetherianquasi-unmixed localringofdimensionnandI ,I
1 2
are ideals of R of finite colength such that I ⊆ I then L (I ) > L (J ), as a consequence
1 2 J 1 J 2
of (13). However, the analogous inequality for L ojasiewicz exponents of sets of ideals does
not hold in general, as the following example shows.
Example 2.12. Let us consider the ideals I ,I of O defined by I = hx3i and I = hy3i.
1 2 2 1 2
Let J ,J be the ideals of O defined by J = hx3,xyi, J = I . Then we have that I ⊆ J ,
1 2 2 1 2 2 i i
i = 1,2, but L (I ,I ) = 3 and L (J ,J ) = 6.
0 1 2 0 1 2
3. Newton filtrations
Let us fix a Newton polyhedron Γ ⊆ Rn. If v ∈ Rn r{0} then we define
+ + +
ℓ(v,Γ ) = min{hv,ki : k ∈ Γ }
+ +
∆(v,Γ ) = k ∈ Γ : hv,ki = ℓ(v,Γ ) ,
+ + +
where h,i stands for the standard sc(cid:8)alar product in Rn. A fa(cid:9)ce of Γ is any set of the
+
form ∆(v,Γ ), for some v ∈ Rn r{0}. Hence we also say that ∆(v,Γ ) is the face of Γ
+ + + +
supported by v. The dimension of a face ∆ of Γ is the minimum of the dimensions of the
+
affine subspaces of Rn containing ∆. If ∆ is a face of Γ of dimension n − 1 then we say
+
that ∆ is a facet of Γ .
+
It is easy to observe that a face ∆ of Γ is compact if and only if it is supported by a
+
vector v ∈ (R r{0})n. The union of all compact faces of Γ will be denoted by Γ; this is
+ +
also known as the Newton boundary of Γ . We remark that Γ determines and is determined
+ +
by Γ, since Γ = Γ+Rn.
+ +
We denote by Γ the union of all segments joining the origin and some point of Γ. There-
−
fore Γ− is a compact subset of Rn+. If Γ+ is convenient, then Γ− is equal to the closure of
Rn rΓ .
+ +
If ∆ is a face of Γ , then C(∆) denotes the cone formed by all half-rays emanating from
+
the origin and passing through some point of ∆.
L OJASIEWICZ EXPONENT OF FAMILIES OF IDEALS AND NEWTON FILTRATIONS 9
We say that a vector v ∈ Zn r {0} is primitive when the non-zero coordinates of v are
+
mutually prime integer numbers. Then any facet of Γ is supported by a unique primitive
+
vector of Zn. Let us denote by F(Γ ) the set of primitive vectors of Rn supporting some
+ + +
facet of Γ and by F (Γ ) the set of vectors v ∈ F(Γ ) such that ∆(v,Γ ) is compact. Let
+ c + + +
us remark that if Γ is convenient then F(Γ ) = F (Γ )∪{e ,...,e }, where e ,...,e is
+ + c + 1 n 1 n
the canonical basis of Rn.
LetussupposethatF (Γ ) = {v1,...,vr}. Thereforeℓ(vi,Γ ) 6= 0,foralli = 1,...,r. Let
c + +
us denote by M the least common multiple of the set of integers {ℓ(v1,Γ ),...,ℓ(vr,Γ )}.
Γ + +
Hence we define the filtrating map associated to Γ as the map φ : Rn → R given by
+ + +
M
φ (k) = min Γ hk,vii : i = 1,...,r , for all k ∈ Rn.
Γ ℓ(vi,Γ ) +
(cid:26) + (cid:27)
We observe that φ (Zn) ⊆ Zn, φ (k) = M , for all k ∈ Γ, and the map φ is linear on
Γ + + Γ Γ Γ
each cone C(∆), where ∆ is any compact face of Γ .
+
Let us define the map ν : O → R ∪{+∞} by ν (h) = min{φ (k) : k ∈ supp(h)}, for
Γ n + Γ Γ
all h ∈ O , h 6= 0; we set ν (0) = +∞. We refer to ν as the Newton filtration induced by
n Γ Γ
Γ (see also [8, 15]).
+
From now on, we will assume that Γ is a convenient Newton polyhedron in Rn.
+ +
Let h ∈ O and let h = a xk be the Taylor expansion of h around the origin. If A is
n k k
a compact subset of Rn then we denote by h the sum of all terms a xk such that k ∈ A.
P A k
If supp(h) ∩ A = ∅, then we set h = 0. Let J be an ideal of O and let g ,...,g be a
A n 1 s
generating system of J. We recall that J is said to be Newton non-degenerate (see [3] or
[28]) when
x ∈ Cn : (g ) (x) = ··· = (g ) (x) = 0 ⊆ x ∈ Cn : x ···x = 0 ,
1 ∆ s ∆ 1 n
(cid:8) (cid:9) (cid:8) (cid:9)
as set germs at 0 ∈ Cn, for each compact face ∆ of Γ (J) (see Theorem 3.6). It is immediate
+
to check that this definition does not depend on the chosen generating system of J. In
particular, any monomial ideal is Newton non-degenerate.
The next result compares the asymptotic Samuel function and the Newton filtration.
Proposition 3.1. [3, p. 26] Let J ⊆ O be an ideal of finite colength. Let Γ denote the
n
Newton boundary of Γ (J) and let M = M . Then Mν 6 ν and equality holds if and only
+ Γ J Γ
if J is Newton non-degenerate.
As a consequence of the previous result, if J is an ideal of finite colength of O and
n
r = min{r : re ∈ Γ (J)}, for all i = 1,...,n, then max{r ,...,r } 6 L (J) and equality
i i + 1 n 0
holds if J is a Newton non-degenerate ideal (see [3, p. 27] for details).
Given an integer r ∈ Z , we denote by A the ideal of O generated by the elements
>0 r n
h ∈ O such that ν (h) = r (we assume that the ideal generated by the empty set is 0). In
n Γ
particular, A = hxk : k ∈ Γ i.
MΓ +
10 CARLES BIVIA`-AUSINAAND SANTIAGOENCINAS
Moreover, we denote by B the ideal of O generated by the elements h ∈ O for which
r n n
ν (h) > r. Then
Γ
B = h ∈ O : φ (supp(h)) ⊆ [r,+∞[ ∪{0},
r n Γ
for all r > 0 and ν (h) = max{r > 0 : h ∈ B }, for all h ∈ O , h 6= 0. We will refer
Γ (cid:8) r (cid:9) n
indistinctly to the map ν and to the family of ideals {B } as the Newton filtration
Γ r r>1
induced by Γ .
+
It is immediate to check that
(a) B is an integrally closed monomial ideal of finite colength, for all r > 1;
r
(b) B B ⊆ B , for all r,s > 1;
r s r+s
(c) B = O .
0 n
If I is an ideal of O , then we denote by ν (I) the maximum of those r such that I ⊆ B .
n Γ r
Then, if g ,...,g denotes any generating system of I, we have
1 s
ν (I) = min{ν (g ),...,ν (g )}.
Γ Γ 1 Γ s
Given an integer r > 0, we observe that A ⊆ B and A 6= B in general. Moreover it
r r r r
follows easily that A = B if and only if A is an ideal of finite colength of O .
r r r n
Let us remark that supp(A ) = Γ ∩ Zn, A has finite colength and e(A ) =
MΓ + + MΓ MΓ
n!V (Γ ), since Γ is convenient and A is a monomial ideal (see the paragraph before
n − MΓ
Definition 2.4).
Proposition 3.2. Let us fix a family of ideals J ,...,J of O such that σ(J ,...,J ) < ∞.
1 n n 1 n
Let ν (J ) = r , for all i = 1,...,n, and let M = M . Then
Γ i i Γ
r ···r
(15) σ(J ,...,J ) > 1 nn!V (Γ ).
1 n Mn n −
Proof. ByProposition2.2wehavethatσ(J ,...,J ) = e(g ,...,g ), forasufficientlygeneral
1 n 1 n
element (g ,...,g ) ∈ J ⊕ ··· ⊕ J . Then the result arises as a direct application of [8,
1 n 1 n
Theorem 3.3]. (cid:3)
As a consequence of [8, Theorem 3.3], equality in (15) is characterized by means of a
condition imposed to any element (g ,...,g ) ∈ J ⊕ ··· ⊕ J such that e(g ,...,g ) =
1 n 1 n 1 n
σ(J ,...,J ) (we refer the reader to [8] for details). By coherence with the nomenclature of
1 n
[8, Theorem 3.3] we introduce the following definition.
Definition 3.3. Let J ,...,J be a family of ideals of O such that σ(J ,...,J ) < ∞.
1 n n 1 n
Let M = M . We say that (J ,...,J ) is non-degenerate on Γ , or that (J ,...,J ) is
Γ 1 n + 1 n
Γ -non-degenerate, when equality holds in (15). That is, when σ(J ,...,J ) = r1···rne(A ).
+ 1 n Mn M
Under the hypothesis of the previous definition, let us suppose that J is principal, that
1
is, J = hhi, for some h ∈ O . Then, in order to simplify the notation, we will write
1 n
(h,J ,...,J ) instead of (hhi,J ,...,J ). We will adopt the same simplification if any other
2 n 2 n
ideal J is principal, for some i ∈ {1,...,n}. Hence, the previous definition applies to germs
i
of complex analytic maps (g ,...,g ) : (Cn,0) → (Cn,0) such that g−1(0) = {0}.
1 n