Rev Mat Complut (2015) 28:487–504
DOI 10.1007/s13163-014-0165-3

The Łojasiewicz exponent over a field of arbitrary
characteristic

Szymon Brzostowski

· Tomasz Rodak

Received: 11 March 2014 / Accepted: 5 December 2014 / Published online: 13 January 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Let

K be an algebraically closed field and let K((X

)) denote the field

of generalized series with coefficients in

K. We propose definitions of the local

Łojasiewicz exponent of F

= ( f

, . . . , f

) ∈ K[[X, Y ]]

as well as of the

Łojasiewicz exponent at infinity of F

= ( f

, . . . , f

) ∈ K[X, Y ]

, which gener-

alize the familiar case of

K = C and F ∈ C{X, Y }

(resp. F

∈ C[X, Y ]

), see

Cha˛dzy´nski and Krasi´nski (In: Singularities,

1988

; In: Singularities,

1988

; Ann Polon

Math 67(3):297–301,

1997

; Ann Polon Math 67(2):191–197,

1997

), and prove some

basic properties of such numbers. Namely, we show that in both cases the exponent
is attained on a parametrization of a component of F (Theorems

and

), thus being

a rational number. To this end, we define the notion of the Łojasiewicz pseudoexpo-
nent of F

∈ (K((X

))[Y ])

for which we give a description of all the generalized

series that extract the pseudoexponent, in terms of their jets. In particular, we show
that there exist only finitely many jets of generalized series giving the pseudoexponent
of F (Theorem

). The main tool in the proofs is the algebraic version of Newton’s

Polygon Method. The results are illustrated with some explicit examples.

Keywords

Generalized power series

· Łojasiewicz exponent · Parametrization ·

Newton polygon method

The paper was partially supported by the Polish National Science Centre (NCN) Grants No.
2012/07/B/ST1/03293 and 2013/09/D/ST1/03701.

S. Brzostowski

· T. Rodak (

)

Faculty of Mathematics and Computer Science,
University of Łód´z, ul. Banacha 22, 90-238 Łód´z, Poland
e-mail: rodakt@math.uni.lodz.pl

S. Brzostowski
e-mail: brzosts@math.uni.lodz.pl

123

488

S. Brzostowski, T. Rodak

Mathematics Subject Classification

13F25

· 14B05 · 32S10

1 Introduction

Let f

: (R

, 0) → (R, 0) be a real analytic function. The Łojasiewicz Inequality

asserts that there exist

ν, C > 0 such that

| f (x)| Cdist(x, V ( f ))

, x near 0,

(1)

where V

( f ) stands for the zero set of f . The problem is to determine the smallest

possible exponent

ν in (

). It is known that this exponent is rational [

] and equal to

the rate of growth of f on some analytic path centered near the origin [

]. In the

particular two-dimensional case the optimal exponent

ν can be expressed in terms of

the Puiseux roots of f [

Now, let F

: (C

, 0) → (C

, 0) be an analytic map with an isolated zero at the

origin. In this case a counterpart of the problem described above is to find an optimal
exponent in the inequality

|F(z)| C|z|

(2)

where C is a positive constant and z is in a sufficiently small neighbourhood of 0.
This exponent is called the local Łojasiewicz exponent of F and is denoted by

(F).

Again it is known that it is a rational number and

(F) = sup

ordF

◦

ord

(3)

where

runs through the set of all analytic paths centered in 0 ∈ C

. Moreover, if F

is a regular sequence (i.e. n

= m), then for generic direction ∈ P

−1

the exponent

(F) is attained on the curve F

−1

() (see [

] or [

] for a different proof of this

result). Another observation of this kind is the following

Theorem 1 ([

]) Let F

:= ( f

, . . . , f

), S := { f

× · · · × f

= 0}. Then

(F) = inf{ν ∈ R : ∃

ε,C>0

∀

∈S

|z| < ε ⇒ |F(z)| C|z|

In particular, if F

= 0 and n = 2 then the local Łojasiewicz exponent of F is attained

on one of the curves

{ f

= 0}.

In other words,

(F) =

ord

((t))

ord

(t)

where

(t) ∈ C{t}

\{0}, (0) = 0 and f

((t)) = 0 for some non-zero f

123

The Łojasiewicz exponent

489

If F

: C

→ C

is a polynomial map with finite number of zeroes, then it is

also possible to define so-called Łojasiewicz exponent of F at infinity (or global
Łojasiewicz exponent of F ). Namely, we are looking for the greatest exponent

ν in

the inequality (

), where C is a positive constant and z is outside a sufficiently big

ball. This optimal exponent is called the Łojasiewicz exponent of F at infinity and
is denoted by

∞

(F). Similarly as in the local case, this exponent is rational and is

attained on a curve centered at infinity. More precisely, there exists a meromorphic
map

: E\{0} → C

, where

E is the unit ball in C, such that lim

→0

(t) = ∞ and

∞

(F) is equal to the rate of growth of F on the image of . Thus, we may write

∞

(F) = inf

ord

((t))

ord

(t)

(4)

where

is as above and, in fact, the infimum is just the minimum. Moreover, the

following theorem holds:

Theorem 2 ([

]) Let F

:= ( f

, . . . , f

), S := { f

× · · · × f

= 0}. Then

∞

(F) = sup{ν ∈ R : ∃

,C>0

∀

∈S

|z| > R ⇒ |F(z)| C|z|

In particular, if F

= 0 and n = 2 then there exists a meromorphic map : E\{0} →

such that lim

→0

(t) = ∞, f

◦ = 0 for some non-zero f

and

∞

(F) =

ord

((t))

ord

(t)

The main goal of the paper is to show that in the above theorems, at least in the two
dimensional case, one may replace the field

C with an algebraically closed field of

arbitrary characteristic. Namely, let

K be an algebraically closed field. The formu-

las (

) and (

) provide the definitions of local and global Łojasiewicz exponents in

K[[x

, . . . , x

]] and in K[x

, . . . , x

], respectively. Now, let n = 2. In this setting,

our main results are Theorems

and

. They give direct two dimensional counterparts

of the above-mentioned Theorems

and

Proofs of these theorems over

C use metric properties of the field or, in the two

dimensional case, the Newton-Puiseux theorem, which is false in positive character-
istic. Thus, we cannot apply these methods. Our idea is to introduce, with the help of
the field of generalized series

K((X

)), some auxiliary notion (called the Łojasiewicz

pseudoexponent), which is, roughly speaking, the greatest vanishing order of the map

= ( f

, . . . , f

) ∈ K((X

))[Y ]

on all paths of the form

(t, y(t)), y(t) ∈ K((t

)).

It turns out (see Theorem

) that this number is rational (if finite) and, what is more

important for us, it is the vanishing order of F on a path

(t, y

(t)), where y

(t) is

a root of some f

. Moreover, in this theorem we prove that all the paths on which

the pseudoexponent is attained are similar to such

(t, y

(t))’s in the sense of jets (see

Definition

). Once Theorem

is proved, the only non-standard information needed

to deduce Theorems

and

is Proposition

. This proposition explains the relation

between the valuations defined by two different types of parametrizations (namely the
standard Hamburger-Noether and the generalized ones).

123

490

S. Brzostowski, T. Rodak

2 The abstract case of an arbitrary field

In the case of a field

K of characteristic 0 one can apply the Newton-Puiseux theorem

to find the roots of an arbitrary f

∈ K((X))[Y ] (a polynomial with coefficients in the

Laurent series field) of positive degree. In short,

K((t)) = K((t

∗

)), where K((t

∗

))

denotes the field of Puiseux series over the field

K. The same is true for fields K of

positive characteristic p but only under the additional assumption that p

deg

f (see

[

, Theorem 5.14]). Thus, in general, one needs to extend the field

K((t

∗

)) even more

to find the algebraic closure of

K((t)). We recall the following notion.

Definition 1 Let

K be a field. By K((t

)) we will denote the field of all generalized

series with coefficients in

K, that is formal sums of the form u(t) =

∈Q

, where

∈ K and the support of u(t), Supp

(t) :=

∈ Q : u

= 0

, is a well-ordered

set.

The fact that the support of every element of

K((t

)) forms a well-ordered set

implies that

K((t

)) is indeed a field (with the natural definitions of addition and

multiplication), an overfield of

K((t

∗

)). But even more is true.

Theorem 3 ([

, Theorem 5.2]) The algebraic closure of the field

K((t

)) is equal

K((t

)).

Although the field

K((t

)) is algebraically closed, it is much bigger than the actual

algebraic closure of

K((t)). The precise description of K((t)) was given by K. Kedlaya

in [

], but we will make no use of this description, working entirely in the larger field

K((t

)).

An alternative way of parametrizing the “zero set” of an f

∈ K[[X, Y ]] of positive

order is by utilizing so-called Hamburger-Noether expansions. More precisely, the
following holds.

Theorem 4 (cf. [

]) Let

K = K, f ∈ K[[X, Y ]], f (0) = 0. Then there exists a

pair

(ϕ(t), ψ(t)) ∈ K[[t]]

\{0} with ϕ(0) = ψ(0) = 0, such that

(ϕ(t), ψ(t)) = 0.

Conversely, for any pair

(ϕ(t), ψ(t)) as above there exists an f ∈ K[[X]][Y ], irre-

ducible as an element of

K[[X, Y ]], with f (0) = 0 and such that

(ϕ(t), ψ(t)) = 0.

The above theorem will also be extended to the case of a pair

(ϕ(t), ψ(t)) ∈

K((t))

(Proposition

below). Anyway, the discussion above motivates the following

definition (cf. also Definition

Definition 2 Let

K be a field. Any pair of the form (t, y(t)) with y ∈ K((t

))

(resp.

(ϕ(t), ψ(t)) ∈ K[[t]]

\{0}, ϕ(0) = ψ(0) = 0) will be called a generalized

(resp. formal) parametrization. We will say that such a pair is a generalized (resp.

123

The Łojasiewicz exponent

491

formal) parametrization of f iff f

(t, y(t)) = 0 (resp. f (ϕ(t), ψ(t)) = 0), where

∈ K((X

))[Y ] (resp. f ∈ K[[X, Y ]]).

We state now the natural generalizations of the classical definitions of the (local

and at infinity) Łojasiewicz exponents. Namely, we adapt (

) and (

) as the defining

conditions allowing

to run through the set of all formal parametrizations (resp.

parametrizations at infinity—cf. Definition

Definition 3 Let

K be a field. For an F ∈ K[[X, Y ]]

with F

(0) = 0 we define the

local Łojasiewicz exponent of F as the number (or

+∞) given by

(F) :=

sup

∈K[[t]]

ord

<∞

ord

◦

ord

Similarly, for an F

∈ K[X, Y ]

we define the Łojasiewicz exponent of F at infinity

or the global Łojasiewicz exponent of F as the number (or

−∞) given by

∞

(F) :=

inf

∈K((t))

ord

◦

ord

The main tool in the paper is the following notion of the Łojasiewicz pseudoexpo-

nent.

Definition 4 Let

K be a field and let F ∈ K((X

))[Y ]

. The (Łojasiewicz) pseudo-

exponent of F is the number (or

+∞)

¯L

(F) :=

sup

(t)∈K((t

))

ord

(t, y(t)).

Note that the value of ¯

(F) depends on the roles played by the variables X and

Y , however, as long as no confusion is likely, we will simply write ¯

L(F) instead of

¯L

(F).

Remark 1 The above definitions can also be stated more generally – one can consider
the exponents with respect to an intermediate field

L such that K ⊂ L ⊂ K. For

example,

L could be a real closed field. We will not dive into this topic here.

3 Auxiliary results

3.1 Newton’s polygon method

We recall that for a non-zero series z

∈ K((t

)) of the form z(t) =

∈Q

there

are defined:

Throughout this paper, ord of a tuple means the minimal ord of its components.

123

492

S. Brzostowski, T. Rodak

– Its order ord

:= min(Supp(z)),

– Its initial coefficient inco

:= zord

– Its initial form info

:= inco

· tord

Moreover, ord

:= ∞, inco

:= 0, info

:= 0.

Following Abhyankar [

] we will use the symbol

to denote an unspecified (anony-

mous) non-zero element of a field under consideration.

Definition 5 Let z

(t) ∈ K((t

)), z(t) =

∈Q

, let U be an indeterminate over

K((t

)) and let L be an overfield of K(U). We say that a series v(t) ∈ L((t

)) is a

(Q, U)-deformation of z(t), if Q ∈ Q and

info

(v(t) − z(t)) = (U − z

In other words, any series of the form

v(t) =

∈Q

where

–

= z

for q

< Q,

–

= U,

–

∈ L for q > Q,

is a

(Q, U)-deformation of z(t).

We begin with an algebraic restatement of Newton’s Polygon Method. It is a sim-

plified but generalized version of [

, Theorem 14.2], see also [

Proposition 1 (Newton’s polygon method) Let

K be a field and let g ∈ K((X

))

[Y ]\{0}. Write

(X, Y ) = e(X)

(Y − z

(X)) with e(X) ∈ K((X

)),

(5)

where z

(X) ∈ K((X

)) for 1 j k (g(X, Y ) = e(X) = 0 allowed).

Let u

(t) :=

∈Q

∈ K((t

)), where Q ∈ Q, and let v(t) be any (Q, U)-

deformation of 0

∈ K((t

)). Then

→ info

(t, u(t)+v(t)) is independent of the particular choice of the deformation

v(t)

→ For h := inco

(t, u(t) + v(t)) it is h ∈ K[U]\{0} (and even h ∈ K(u

: q

)[U])

→ The following two conditions are equivalent:

(i) There exists 1

k such that ord

(u(t) − z

(t)) > Q

(ii) The polynomial h vanishes for U

= 0

The following two conditions are equivalent:

(iii) For every 1

j k it is ord

(u(t) − z

(t)) < Q

(iv) The polynomial h is constant

123

The Łojasiewicz exponent

493

What is more,

– If U

= 0 is of multiplicity l > 0 as a root of h, then there exist exactly l different

indices j

, . . . , j

∈ {1, . . . , k} for which

ord

(u(t) − z

(t)) > Q, i = 1, . . . , l

– If deg

= m > 0, then there exist exactly m different indices j

, . . . , j

∈

{1, . . . , k} for which

ord

(u(t) − z

(t)) Q, i = 1, . . . , m.

Proof It is easy to see that all the assertions of the theorem can be obtained from the
particular case g

(X, Y ) = e(X)(Y − z(X)), where 0 = e(X) ∈ K((X

)), z(X) ∈

K((X

)) (the theorem being obvious for g(X, Y ) = e(X)). However, for such a g

and any

(Q, U)-deformation v(t) of 0 ∈ K((t

)) we can take r := ord

(u(t) − z(t))

and s

:= ord

(t) to obtain

info

(t, u(t) + v(t))

= info

(e(t)(u(t) + v(t) − z(t))) =

⎧

⎪

⎨
⎪

⎩

(U + ) t

U t

if r

< Q

if r

= Q

ifr

> Q

= (δ

min

(r,Q)

+ δ

max

(r,Q)

) · t

min

(r,Q)+s

where

δ is the Kronecker delta, and the ∈ K are independent of the choice of

v(t) as they are determined by the coefficients of u(t) − z(t) of order Q. Hence,
h

= inco

(t, u(t) + v(t)) ∈ K[U] and h(0) = 0 iff r > Q, which gives “(i)⇔(ii)”.

Similarly, h is constant iff r

< Q so “(iii)⇔(iv)”. The last two assertions are obvious.

We also remark that for a general g

∈ K((X

))[Y ] the fact that h ∈ K(u

: q

)[U] follows immediately, since in particular h = inco

(t, u(t) + Ut

Example 1 Let g

(X, Y ) := Y

− X

−1

− X

−1

∈ K[X, Y ], where K is a field of

positive characteristic p. Then, following [

], we may write

(X, Y ) =

−1

− j X −

∞

−p

−k

Put u

(t) :=

∞

−p

−k

. Then g

(t, u(t)) = 0. Let v(t) := Ut + · · · be a (1, U)-

deformation of 0

∈ K((t

)). We have g(t, u(t) + v(t)) = U(U

−1

− 1)t

+ · · · and

consequently h

(U) = inco

(t, u(t) + v(t)) = U(U

−1

− 1).

3.2 Jets and truncations

Definition 6 Let q

∈ Q and let us treat K((t

)) as a K[[t

]]-module. We define

the q-th order open jet space

= ˚

[K((t

))] of K((t

)) as the module

123

494

S. Brzostowski, T. Rodak

K((t

))/(t

) and q -th order closed jet space J

= J

[K((t

))] of K((t

))

as the module

K((t

))

∈Q

The union of the two types of q-th order jet spaces will be denoted by

[K((t

))].

Note that unlike e.g. the smooth functions case, the jet spaces defined above do not

constitute rings (the multiplication is not associative).

The elements of

are called (q-th order) closed jets. A closed jet determined

by a series

v will be denoted by J

[v]. For any set S of generalized series the set of

closed jets

[S ] is defined in the obvious way. Similarly, the elements of ˚

are

called (q-th order) open jets. An open jet determined by a series

v will be denoted by

[v]. For any set S of generalized series the set of open jets ˚

[S ] is defined in the

obvious way. Similarly, there is defined the set of jets

[S ] := ˚

[S ] ∪ J

[S ].

Notation 1 For a closed jet

ι = J

[ϕ] the symbol ˚ι =

[ϕ]) will denote the

interior of

ι i.e. the jet ˚

[ϕ]. Similarly, for any set A of closed jets we put ˚A := {˚ι :

ι ∈ A}.

The jets have canonical representatives of the form

∈ K((t

)) (or

∈ K((t

)) in the case of open jets), nevertheless we find it useful to

distinguish these objects from one another.

Definition 7 Let

ϕ ∈ K((t

)) and q ∈ Q. If ϕ =

∈Q

then we define the

q-th order closed truncation

ϕ as ϕ

(t) :=

and the q-th order

open truncation

ϕ as ϕ

(t) :=

. Thus

∈ J

[ϕ] and ϕ

∈

[ϕ].

In the following, all the formulas involving truncations are to be understood in

the usual way (i.e. at the series level) while the formulas concerning jets are to be
understood as representative-independent (i.e. valid at the jet level), for example
this is the case with the formulas of the type ord

(t, ι), where g ∈ K((X

))[Y ] and

ι ∈ J

Lemma 1 Let

K be a field and m 2. For any m-tuple F ∈ K((X

))[Y ]

of co-

prime polynomials with deg

> 0 and any ϕ ∈ K((t

)) the set A := {q ∈ Q :

ord

(t, ϕ(t) + Ut

) = ord

(t, ϕ(t))} is non-empty. Moreover, there exists min A

and it is rational.

Proof Replacing F

(X, Y ) with F(X, ϕ(X) + Y ) we may assume that ϕ(t) = 0. Let

= ( f

, . . . , f

), where

(X, Y ) = a

i 0

(X)Y

+ · · · + a

i d

(X),

i 0

(X) = 0, i = 1, . . . , m.

123

The Łojasiewicz exponent

495

Since f

, . . . , f

are co-prime, min

ord

i d

(t) = ord

(t, 0) ∈ Q. On the other

hand, ord

(t, Ut

) = min

, j

(ord

i j

(t) + q(d

− j)). Thus,

A = {q ∈ Q : min

, j

(ord

i j

(t) + q(d

− j)) = min

ord

i d

(t)}.

Clearly,

A = ∅. Moreover, since d

> 0 for some i, we infer that there exists min A .

Lemma 2 Let

K be an infinite field. Let F ∈ K((X

))[Y ]

w ∈ K((t

)) and

∈ Q. Then for every (q, U)-deformation v(t) of w(t) we have

ord

(t, w(t) + Ut

) = ord

(t, v(t)).

Moreover, for any N

∈ Q the following conditions are equivalent:

1. ord

(t, w(t) + Ut

) N,

2. ord

(t, ϕ(t)) N, for every representative ϕ(t) of ˚

[w(t)].

Proof As in the proof of the previous lemma, we may assume that

w(t) = 0. Moreover,

it is sufficient to prove only the case m

= 1 and F = 0. Write

(X, Y ) = a

(X)Y

+ · · · + a

(X).

We have

ord

(t, Ut

) = min

(ord

(t) + q(d − j)) = ord

(t, Ut

+ ξ(t))

for any

ξ ∈ L((t

)), ord

ξ(t) q where L is an overfield of K(U). This gives the

first part of the lemma.

“1

⇒2” Take any representative ϕ(t) ∈

[0]. Then one can write ϕ(t) =

with

∈ K, so v(t) := (U − ϕ

+ ϕ(t) is a (q, U)-deformation

of 0. By assumption and the first part of the proof, ord

(t, v(t)) N and substitut-

ing U

= ϕ

into this relation we obviously get

ord

(t, ϕ(t)) N.

“2

⇒1” Let h(U)t

:= info

(t, Ut

). Since h(U) ∈ K[U]\{0} and the field K is

infinite, there exists x

∈ K such that h(x

) = 0. This implies that ord

(t, x

) = α.

But x

∈ ˚

[0], so by assumption it is α N.

3.3 Parametrizations

In what follows, we will utilize an even broader class of parametrizations than the
formal ones (cf. Definition

). Namely, the following strengthening of Theorem

holds true.

123

496

S. Brzostowski, T. Rodak

Proposition 2 Let

K = K. For any f ∈ K((X))[Y ], deg

> 0, there exists a pair

(ϕ(t), ψ(t)) ∈ K((t))

with

∞ > ord

ϕ(t) > 0 such that

(ϕ(t), ψ(t)) = 0.

Conversely, for any pair

(ϕ(t), ψ(t)) as above there exists an irreducible f ∈

K[[X]][Y ] with deg

> 0 and such that

(ϕ(t), ψ(t)) = 0.

Proof “

⇐” Let (ϕ(t), ψ(t)) ∈ K((t))

with

∞ > ord

ϕ(t) > 0. If ord

ψ(t) > 0 then

the existence of f is a direct consequence of Theorem

. Similarily, it is easy to treat

the case ord

ψ(t) = 0. Thus, we may assume that ord

ψ(t) < 0 and use Theorem

to find a g

∈ K[[X]][Y ]\{0} such that g (ϕ(t),

ψ(t)

) = 0. Now it is enough to

put f

:= g(X, Y

−1

, for a big enough a

∈ N, to assure that f ∈ K[[X]][Y ] and

(ϕ(t), ψ(t)) = 0. Notice that the latter condition together with ord

ϕ(t) < ∞ imply

that necessarily deg

> 0. Thus, we can factor f in K[[X]][Y ] into irreducible

elements and replace f by an irreducible one that also vanishes at

(ϕ(t), ψ(t)). By

the above remark, it has to be deg

> 0 also for the changed f .

“

⇒” Let f ∈ K((X))[Y ], deg

> 0. Write f = a

(X)Y

+ · · · + a

−1

(X)Y +

(X). Note that by considering f (X, X

), where α ∈ Q

, one can arrange

things so that r

:= min

jk−1

(ord

(X)) < ord

(X) and then taking g :=

−r

(X, X

) we have g∈ K[[X]][Y ], deg

>0, g(0) = 0 and X g in K[[X, Y ]].

Applying Theorem

to g we find a parametrization

(t) = (

(t),

(t)) ∈

K[[t]]

\{0} of g such that ord

(t) > 0. Also, ord

(t) < ∞, since other-

wise g

(0,

(t)) = 0, implying X | g in K[[X, Y ]]. Now it is enough to consider

(ϕ(t), ψ(t)) := (

(t),

(t) ·

(t)) ∈ K((t))

to fulfill the needed conditions.

Thus, it is natural to define what follows.

Definition 8 Let

K be a field. Any pair of the form (ϕ(t), ψ(t)) ∈ K((t))

with

∞ > ord

ϕ(t) > 0 will be called a Laurent parametrization. If f ∈ K((X))[Y ] and

(ϕ(t), ψ(t)) = 0 we will say that such a pair is a Laurent parametrization of f . For

∈ K[X, Y ] a pair (ϕ(t), ψ(t)) ∈ K((t))

with ord

(ϕ(t), ψ(t)) < 0 and such that

(ϕ(t), ψ(t)) = 0 will be called a parametrization of f at infinity.

The following property is immediate.

Corollary 1 Let

K = K. For any f ∈ K[X, Y ], deg f > 0, there exists a parame-

trization of f at infinity.

Proof If deg

> 0, it is enough to consider f (X

−1

, Y ) and use Proposition

find

(ϕ(t), ψ(t)) ∈ K((t))

with

∞ > ord

ϕ(t) > 0 such that f (ϕ(t)

−1

, ψ(t)) = 0.

Similarily for the case deg

> 0.

An important connection between the Laurent and the generalized parametrizations

is given in the proposition below. Note that the proof is mainly for the case of a field
K of positive characteristic, since otherwise a standard application of the Implicit
Function Theorem suffices.

123

The Łojasiewicz exponent

497

Proposition 3 Let

K = K. Let (ϕ(t), ψ(t)) ∈ K((t))

and

(t, y(t)) ∈ K((t

))

a Laurent (resp. a generalized) parametrization of the same irreducible and monic

∈ K((X))[Y ]. Then for every g = g(X, Y ) ∈ K((X))[Y ] it is

ord

(t, y(t)) =

ord

(ϕ(t), ψ(t))

ord

ϕ(t)

(6)

Proof Consider the field

E :=

K((t))[Y ]

( f (t,Y ))

∼

= K((t))(y(t)). It is a finite extension

of the field

F := K((t)). Define v

, v

: E → R ∪ {∞} by the formulas

([g]) :=

ord

(ϕ(t), ψ(t))

ord

ϕ(t)

and

([g]) := ord

(t, y(t)),

(7)

where g

= g(t, Y ) ∈ K((t))[Y ]. Since both (ϕ(t), ψ(t)) and (t, y(t)) are parame-

trizations of f , it is easy to see that

and

are correctly defined (recall also that by

definition ord

ϕ(t) = 0).

We claim that

, v

are valuations on the field

E. Indeed, most of the needed

conditions follow at once from the corresponding properties of the order function.
The only thing worth a closer look is the implication: “

([g]) = ∞ ⇒ [g] = 0”. For

this is immediate, because f

(t, Y ) is the minimal polynomial of y(t) over F, so

(t, y(t)) = 0 implies f (t, Y ) | g in F[Y ]. For v

, let g

(ϕ(t), ψ(t)) = 0 and consider

the set

I := {h ∈ F[Y ] : h(ϕ(t), ψ(t)) = 0}. Obviously, this is an ideal in F[Y ],

which itself is a PID, so it is generated by a single element ˜

∈ F[Y ]. But f (t, Y ) is

irreducible in

F[Y ] and also belongs to I . Hence, f (t, Y ) ∼ ˜f. Since g ∈ I , we

thus conclude that f

(t, Y ) | g and [g] = 0.

Now, observe that for h

= h(t) ∈ F it is

([h]) =

ord

(ϕ(t))

ord

ϕ(t)

= ord

(t) = v

([h]).

(8)

Define

| · |

:= 2

−v

(·)

, i

= 1, 2. Then | · |

, | · |

are two absolute values on the field

E, that by (

) agree on the subfield

F with the absolute value | · | := 2

−

ord

(·)

. Since

the extension

E ⊃ F is finite and (F, | · |) is complete (| · | defines the usual t-adic

topology on

F), we can apply [

, Thm. 9.8] to conclude that

| · |

= | · |

. But this

means that also

= v

. Now (

) follows from (

Remark 2 Let

K = K. It is well-known that if f, g ∈ K[[X, Y ]], f (0) = g(0) = 0

and f is irreducible, then the Hilbert-Samuel multiplicity of

( f, g) is given by the

formula

( f, g) = ord

(ϕ(t), ψ(t)),

where

(ϕ(t), ψ(t)) is a formal parametrization of f (see e.g. [

, Thm. 3.14]). From

Proposition

it follows that in a generic coordinate system we have

123

498

S. Brzostowski, T. Rodak

( f, g) = ord

(X,Y )

(X, Y )ord

(t, y(t)),

where

(t, y(t)) is a generalized parametrization of f .

4 Main results

We start with a general result concerning the pseudoexponent ¯

L. It contains, inter alia,

a description of all the jets extracting the pseudoexponent, a result that is inspired by
[

], where the classical case of germs of functions (in the real analytic setting) has

been considered.

Theorem 5 Let

K = K. Then for every tuple F = ( f

, . . . , f

) ∈ K((X

))[Y ]

with deg

> 0 the Łojasiewicz pseudoexponent of F is attained on a generalized

parametrization of a component of F . Furthermore, if m

2 and f

, . . . , f

are co-

prime (as polynomials), then there exists a set ˚

J ⊂

∈Q

of open jets of

K((t

))

such that:

(a) ˚

J is non-empty and finite,

(b) For every ˚

ι ∈ ˚J there exists a root w ∈ K((t

)) of some f

(t, Y ) = 0 representing

ι, i.e. ˚ι = ˚

[w] for some q ∈ Q, with

¯L(F) = ord

(t, w(t)) = ord

(t, ˚ι),

ϕ ∈ K((t

)) it is

¯L(F) = ord

(t, ϕ(t)) ⇔ ˚

[ϕ] ∈ ˚J for some q ∈ Q.

Proof It is clear that we can assume that f

(t, Y ), . . . , f

(t, Y ) have no common root

K((t

)). Also, it is enough to consider the case of all the f

being non-zero. Let

i j

} ⊂ K((t

)) be the set of all the roots of f

(t, Y ) × · · · × f

(t, Y ); precisely,

let f

(t, u

i j

(t)) = 0 for 1 i m, 1 j l

(here, possibly, some – but not all –

= 0 for a constant f

). Define

˜L := sup

sup

ord

(t, u

i j

(t))

By the assumptions,

−∞ < ˜L < ∞. We claim that ˜L = ¯L(F). Take any w ∈ K((t

))

different from all the u

i j

and let

ρ := max

, j

(ord

(w(t) − u

i j

(t))) ∈ Q.

Let z

(t) be any (ρ, U)-deformation of 0. It is

info

(t, w

(t) + z(t)) = P

(U)t

(9)

where P

∈ K[U], α

∈ Q and i = 1, . . . , m. By Proposition

and the definition of

ρ, the polynomials P

, . . . , P

do not vanish at U

= 0. Since (

) is valid in particular

for z

(t) := Ut

+ (w(t) − w

(t)), by taking U = 0 in these equalities we see that

info

(t, w(t)) = t

, for i = 1, . . . , m,

123

The Łojasiewicz exponent

499

– and consequently that –

ord

(F(t, w(t))) = min(α

, . . . , α

(10)

On the other hand, one can take z

(t) := Ut

+ (u(t) − u

(t)), where u(t) ∈ {u

i j

(t)}

is chosen in such a way that

ρ = ord

(w(t) − u(t)). Let u(t) =

∈Q

and

w(t) =

∈Q

. Then

(t) + z(t) = w

(t) − u

(t) + Ut

+ u(t) = (U + w

− u

+ u(t).

Thus, (

) takes the form

info

(t, u(t) + (U + w

− u

) = P

(U)t

so putting U

= u

− w

we conclude that

ord

( f

(t, u(t))) α

, for i = 1, . . . , m,

and so

ord

(t, u(t)) min(α

, . . . , α

(11)

But since u

(t) ∈ {u

i j

(t)}, using (

) and the definition of the number ˜

L we get

˜L ord

(t, u(t)) ord

(t, w(t)).

(12)

Now,

w(t) was an arbitrary element of K((t

))\{u

i j

(t)}. Since the resulting inequality

clearly holds for

w(t) ∈ {u

i j

(t)} by the very definition of ˜L, it holds for any w(t) ∈

K((t

)). Thus,

˜L ¯L(F).

Since the other inequality is obvious, the first assertion of the theorem is proved.

Notice also that from the above reasoning one can actually deduce more:

Claim For every

w ∈ K((t

))\{u

i j

} such that ord

(t, w(t)) = ¯L(F), if ρ :=

sup

, j

(ord

(w − u

i j

)) then

¯L(F) = ord

(t, ˚

[w(t)]).

(13)

Indeed, by assumption

w ∈ {u

i j

}. Hence, using the notations of (

), by (

) and

(

) we see that in such a case

¯L(F) = ord

(t, u(t)) = ord

(t, w(t)) = min(α

, . . . , α

)

= ord

(t, w

(t) + z(t)),

123

500

S. Brzostowski, T. Rodak

for any

(ρ, U)-deformation z(t) of 0. In particular, one can take z(t) := Ut

(w(t) − w

(t)). By Lemma

, for every representative

ϕ of ˚

[w(t)] we have

ord

(t, ϕ(t)) ¯L(F). Now, the definition of ¯L(F) implies that (

) holds.

For the rest of the reasoning, let

M := {u

i j

: ord

(t, u

i j

(t)) = ¯L(F)}, and if

w ∈ M let q(w) := min{q ∈ Q : ord

(t, w(t) + Ut

) = ¯L(F)}. Note that by

Lemma

the number q

(w) is properly defined. We define ˚J := { ˚

(w)

[w] : w ∈ M }.

Of course, the set ˚

J is finite and non-empty by the first part of the proof, so a) holds.

Now, let ˚

ι ∈ ˚J and let w ∈ M be such that ˚ι = ˚

(w)

[w]. Since

ord

(t, w(t) + Ut

(w)

) = ¯L(F),

by Lemma

and the definition of ¯

L(F) for every representative ϕ of ˚ι we have

ord

(t, ϕ(t)) = ¯L(F).

This proves b).

Considering item c). The implication “

⇐” follows from b). So, assume that ¯L(F) =

ord

(t, ϕ(t)) for some ϕ ∈ K((t

)). If ϕ is one of the u

i j

’s, it belongs to

M ,

(ϕ)

[ϕ] ∈ ˚J. Now let ϕ ∈ {u

i j

}. It means that, as before, we can put ρ :=

sup

, j

(ord

(ϕ − u

i j

)) ∈ Q. Now take any u ∈ {u

i j

} such that ord

(ϕ − u) = ρ. Since

then

[ϕ] = ˚

[u], by the Claim we must have

ord

(t, ˚

[u]) = ¯L(F),

so also ord

(t, u(t)) = ¯L(F). In particular, u ∈ M . Moreover, by Lemma

it is also

ord

(t, u(t) + Ut

) = ¯L(F). Hence, the definition of q(u) implies that q(u) ρ.

But this means that

(u)

[ϕ] = ˚

(u)

[u] ∈ ˚J.

Corollary 2 Let

K = K. For every tuple F = ( f

, . . . , f

) ∈ K((X

))[Y ]

the

pseudoexponent ¯

L(F) is a rational number (or +∞).

Proof If deg

0 then F ∈ K((X

))

and we have ¯

L(F) = sup

(t)∈K((t

))

(t, y(t)) = ord

(t) ∈ Q∪{+∞}. If deg

> 0 and f

, . . . , f

are co-prime then

by Theorem

there exists

w(t) ∈ K((t

)) such that ¯L(F) = ord

(t, w(t)) ∈ Q.

Lastly, if h

| f

, j

= 1, . . . , m, where h ∈ K((X

))[Y ] and deg

> 0 then by

Theorem

there exists y

(t) ∈ K((t

)) such that h(t, y(t)) = 0. This gives ¯L(F) =

+∞.

Example 2 Let

K be an algebraically closed field. Consider F := (Y −

−

, (Y −

−

)

) ∈ K((X

))[Y ]

. By the theorem, one easily sees that

¯L(F) = max{1, 2} = 2 and the exponent is realized only by the parametrization of

the first component of F , that is by

(t, y(t)) := (t,

−

+ t). It follows that

(y) = 2 and ˚J = { ˚

[y]} (see the proof of Theorem

123

The Łojasiewicz exponent

501

Theorem 6 Let

K = K. Then for any F = ( f

, . . . , f

) ∈ K[[X, Y ]]

, such that

(0) = 0, there exists a formal parametrization (t) of some f

such that

(F) =

ord

((t))

ord

(t)

Proof We may assume that all f

are non-zero. Moreover, using Weierstrass Prepara-

tion Theorem (after possible change of variables) we may assume that for each f

have f

(X, Y ) ∈ K[[X]][Y ], ord

(X,Y )

(X, Y ) = deg

(X, Y ) > 0. Observe that

(t, y(t)) (resp. (ϕ(t), ψ(t))) is a generalized (resp. formal) parametrization of some

then ord

(t) 1 (resp. ord

ψ(t) ord

ϕ(t), ϕ(t) = 0). Thus, by Theorems

and Proposition

we have

¯L(F) =

sup

(t)∈K((t

))

ord

(t, y(t))

= sup

ord

(t, y(t)) : (t, y(t)) is a generalized parametrization of some f

= sup

ord

(ϕ(t),ψ(t))

ord

(ϕ(t),ψ(t))

: (ϕ(t), ψ(t)) is a formal parametrization of some f

(F) ¯L(F).

This ends the proof.

Theorem 7 Let

K = K. Then for any polynomial map F = ( f

, . . . , f

) ∈

K[X, Y ]

there exists a parametrization at infinity

(t) of some f

such that

∞

(F) =

ord

((t))

ord

(t)

Proof We may assume that all f

are non-zero and (after change of variables) that all

of them satisfy

deg

(X,Y )

(X, Y ) = deg

(X, Y ).

More specifically,

(X, Y ) = Y

+ a

(X)Y

−1

+ · · · + a

(X),

where deg

(X) k. Observe that

∞

(F) = inf

ord

(ϕ(t),ψ(t))

ord

ϕ(t)

: ϕ(t), ψ(t) ∈ K((t)),

ord

ϕ(t) ord

ψ(t), ord

ϕ(t) < 0

and

(ϕ(t), ψ(t)) = 0 ∧ ord

(ϕ(t), ψ(t)) < 0 ⇒ ord

ϕ(t) ord

ψ(t). (14)

123

502

S. Brzostowski, T. Rodak

Let ˜

= ( ˜f

, . . . , ˜f

), where ˜f

(X, Y ) = f

−1

, Y ). Using Proposition

we get

∞

(F) − ¯L( ˜F). By Theorem

there exists y

(t) ∈ K((t

)) such that ¯L( ˜F) =

ord

˜F(t, y

(t)) and ˜f

(t, y

(t)) = 0 for some j

. Let (by Propositions

and

)

(ϕ

(t), ψ

(t)) ∈ K((t))

be a Laurent parametrization of ˜

such that for every

= g(X, Y ) ∈ K((X))[Y ] it is

ord

(t, y

(t)) =

ord

(ϕ

(t), ψ

(t))

ord

(t)

Put

(t) := (

(t)

, ψ

(t)). Since ord

(t) > 0, (t) is a parametrization at infinity

of f

. Moreover, by (

) we have ord

= ord

/ϕ

. Consequently,

∞

(F) − ¯L( ˜F) = −ord

˜F(t, y

(t))

= −

ord

˜F(ϕ

(t), ψ

(t))

ord

(t)

ord

((t))

ord

(t)

∞

(F).

The following two examples demonstrate how to use the above theorems to calculate

the Łojasiewicz exponent.

Example 3 A. Płoski in [

] proved that a rational number is equal to the Łojasiewicz

exponent of a holomorphic mapping of

if and only it appears in the sequence

, 2, 3, 4, 4

, 4

, 5, . . .

that is, is a positive integer or of the form N

b
a

, where 0

< b < a < N, a, b, N ∈ Z.

Let

K be an algebraically closed field. We will check that any number from the above

sequence is realized as the local Łojasiewicz exponent of some pair F

∈ K[[X, Y ]]

It is immediate to see from the definition of the local Łojasiewicz exponent that
L

, Y ) = N. Following Płoski, let us consider F := (Y

− X

, X

−b

Observe that all the formal parametrizations of the second component of F are of the
form

(ϕ(t), 0) or (0, ψ(t)). The first component factors as

− X

ε∈K

− εX

1
a

for some

α 1, and so by Proposition

we may assume that its formal parametriza-

tions are of the form

, εt

). Since

ord

(ϕ(t), 0)

ord

(ϕ(t), 0)

= a + 1,

ord

(0, ψ(t))

ord

(0, ψ(t))

= a,

ord

, εt

)

ord

, εt

)

= N +

by Theorem

we get that

(F) = N +

b
a

Example 4 Let

K be an algebraically closed field. Inspired by [

] let us consider

:= ((X + Y

)

−1

, Y

−1

− (X + Y

)

−1

) ∈ K[X, Y ]

, where p

2, q 1,

123

The Łojasiewicz exponent

503

, q ∈ Z. It is clear that if (t) := (ϕ

(t), ϕ

(t)) ∈ K((t))

is a parametrization

at infinity of the first component of F , then ord

(t), ord

(t) < 0 and hence

ord

(t) = ord

(t). Consequently,

ord

((t))

ord

(t)

= (p − 1) ·

ord

(t)

ord

(t)

− 1
q

Now, let

(t) := (ψ

(t), ψ

(t)) ∈ K((t))

be a parametrization at infinity of the

second component of F . If

(t) = (ψ

(t), 0) then we get ord

((t))

ord

(t)

= p − 1,

which is bigger than

−1
q

and hence can be discarded for the computation of

∞

(F)

(cf. Definition

). It follows that 1

− (ψ

(t) + ψ

(t))

−1

−p

(t) = 0 with ψ

= 0.

Using this relation we get ord

(ψ

(t) + ψ

(t)) =

−q

−1

· ord

(t), and this for

ord

(t) < 0 implies that ord

(ψ

(t) + ψ

(t)) ord

(t) ord

(t), the

inequalities being strict if q

> 1. Consequently, we easily see that the only possibilities

are

ord

(t) =

ord

(t), if q > 1

ord

(t), if q = 1

⎧

⎪

⎨
⎪

⎩

−q

−1

· ord

(t), if ord

(t) 0, q > 1 and p < q

· ord

(t),

if ord

(t) < 0 and q > 1

ord

(t),

if q

= 1

Again because of the relation

satisfies, we get

ord

((t))

ord

(t)

ord

−q

(t)

ord

(t)

⎧

⎪

⎨
⎪

⎩

− 1, if ord

(t) 0, q > 1 and p < q

−q

if ord

(t) < 0 and q > 1

− q, if q = 1

− 1.

Now, since it is an easy matter to actually find parametrizations

giving equality in

the above formula, we conclude by Theorem

that

∞

(F) = min

− 1

− 1.

Similarly, one can check that for G

:= (X

−q−1

, X

−q

− 1) ∈ K[X, Y ]

with p

> q > 0, p, q ∈ Z, we have

∞

(G) = −

Finally, we immediately see that

∞

(X, X Y − 1) = −1.

123

504

S. Brzostowski, T. Rodak

Summing up,

∞

takes all rational numbers as its values, over any algebraically

closed field

We end the paper by asking the following:

Question 1 Can Example

be strengthened – are the numbers N

b
a

, where 0

< b <

< N, a, b, N ∈ Z, all the possible (finite) Łojasiewicz exponents L

(F) that can

be realized for F

∈ K[[X, Y ]]

, for every algebraically closed field

Question 2 Is our definition of the local Łojasiewicz exponent equivalent to Lejeune
and Teissier’s “integral closure definition” used in [

], or to Płoski’s “characteristic

polynomial definition” (cf. [

]), for every algebraically closed field

Open Access

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which permits any use, distribution, and reproduction in any medium, provided the original author(s) and
the source are credited.

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The Łojasiewicz exponent over a field of arbitrary characteristic

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