ON THE MAXIMAL POLAR QUOTIENT OF AN ANALYTIC PLANE CURVE

by Arkadiusz Płoski^{∗}

IMUJ PREPRINT 2000/12 KRAKÓW

Abstract

We give an explicit formula for the maximal polar quotient of a plane curve singularity with some applications to the Łojasiewicz exponent and the C^{0}-degree of sufficiency.

Introduction

Let C{X, Y } be the ring of convergent power series in two variables X, Y. If f, ϕ ∈ C{X, Y }, we denote by (f, ϕ)_{0} the intersection multiplicity of f and ϕ, equal to the C-codimension of the ideal (f, ϕ) generated by f and ϕ in C{X, Y }. We use ord f to denote the order of the series f. If f, ϕ are without constant terms, then (f, ϕ)_{0} ≥ (ord f)(ord ϕ), the equality holding if and only if f, ϕ are transverse. Let t = t(X, Y) ∈ C{X, Y } be a regular parameter i.e. a series of order 1. Assume that f ∈ C{X, Y } is a reduced (i.e. without multiple factors) power series and t does not divide f.

1991 Mathematics Subject Classification: 32S05

Key words and phrases: plane curve singularity, polar quotients, Łojasiewicz exponent, degree of sufficiency.

∗ Supported in part by the KBN grant No 2 P03 A 00115.

The rational numbers \frac{(f,ϕ)_{0}}{(t,ϕ)_{0}} , where ϕ runs over irreducible factors of the Jacobian J = \frac{∂(t,f)}{∂(X,Y)}, are called polar quotients of f with respect to t. If t = bX − aY then we speak about polar quotients with respect to the direction (a : b) ∈ P^{1}(C). Clearly the set of polar quotients is finite. It is empty if and only if \frac{∂(t,f)}{∂(X,Y)}(0, 0) ≠ 0. If t and f are transverse, then the polar quotients are of the form \frac{(f,ϕ)_{0}}{ord ϕ} and are also called polar invariants. They are topological invariants of the singularity f = 0 (see [T], [LMW]). If t and f are not transverse the notion of polar quotient is also interesting, especially in the case of the singularities at infinity of plane algebraic curves (see [Eph] and [Lê]). Casas-Alvero ([C-A], Chapter 6, Theorem 6.11.5) calculated polar quotients in the general case of reduced power series using the infinitely near points. However, the formulas he got are not so explicit as that given by Merle ([M], Théorème 3.1) in the case of one branch. In this note we study the maximal polar quotient q_{0}(f, t) of f with respect to t:

q_{0}(f, t) = sup{q ∈ R : q is a polar quotient of f with respect to t} .

If \frac{∂(t,f)}{∂(X,Y)}(0, 0) ≠ 0 then q_{0}(f, t) = −∞ by convention.

Our aim is to give an explicit formula for q_{0}(f, t) (Theorem 1.3 of this paper) by means of the maximal polar quotients of the branches and some intersection multiplicities. Our result is inspired by Kuo and Lu’s paper [KL], to prove it we use the Kuo-Lu lemma that enables us to locate the Puiseux roots of derivative \frac{∂f}{∂Y} relatively to the roots of f ([KL], lemma 3.3). As an application of our formula we show that for a given f the sup{q_{0}(f, t) : t does not divide f} is attained if f and t are transverse (Corollary 1.4). The maximal polar quotient are of particular interest. Tessier in his fundamental paper [T] proved (in the case of isolated hypersurface singularities) that the Łojasiewicz exponent L_{0}(f) and the C^{0}-degree of sufficiency are determined by the maximal polar quotient with respect to the generic direction. Using Tessier’s result we give formulas for L_{0}(f) and Suff_{0}(f) (Corollary 1.5) and a correct version (Corollary 1.8) of a formula for Suff_{0}(f) proposed by Lichtin in [Li]. In the appendix we reprove the main result of [KL]. We give also an example showing that the geometric interpretation of this result given in [KL] is not exact.

1 Main result

Let f ∈ C{X, Y } be an irreducible power series. Recall that the semigroup Γ_{0}(f) of f is the set of all intersection numbers (f, ϕ)_{0} where ϕ runs over all power series ϕ ∈ C{X, Y } such that ϕ ∉ (f)C{X, Y }. Let β̄_{0} … β̄_{g} be the minimal system of generators of Γ_{0}(f) : β̄_{0} = min (Γ_{0}(f) \ {0}) = ord f, β̄_{i} = min (Γ_{0}(f) \ (Nβ̄_{0} + … + Nβ̄_{i−1})) for i = 1, …, g, Nβ̄_{0} + … + Nβ̄_{g} = Γ_{0}(f). Let GCD (β̄_{0}, …, β̄_{i}) stand for the greatest common divisor of β̄_{0}, …, β̄_{i}.

Proposition 1.1 Suppose that f is an irreducible power series with ord f > 1. Then for every regular parameter t we have

q_{0}(f, t) = \frac{GCD (β̄_{0}, …, β̄_{g−1})β̄_{g}}{(t, f)_{0}} .

The proposition above can be easily deduced from the generalization of Merle’s result ([M], Théorème 3.1) given by Ephraim ([Eph], Lemma 1.6). We give a direct proof of (1.1) in section 4 of this paper.

Remark 1.2 If ord f = ord t = 1 and \frac{∂(t,f)}{∂(X,Y)}(0, 0) ≠ 0 then q_{0}(f, t) = −∞, because the set of polar quotients is empty (by convention sup ∅ = −∞). If ord f = ord t = 1 and \frac{∂(t,f)}{∂(X,Y)}(0, 0) = 0 then there is only one polar quotient and it is equal to 1. Hence q_{0}(f, t) = 1.

In the sequel we put max {−∞, a} = a for every a ∈ R. The main result of this note is

Theorem 1.3 Let f = f_{1} … f_{r} (r ≥ 2), with f_{i} ∈ C{X, Y } irreducible, be a reduced power series. Let t be a regular parameter that does not divide f. Then

q_{0}(f, t) = \begin{matrix}r \\ max \\ i = 1 \end{matrix} \{max \{q_{0}(f_{i}, t), \begin{matrix}max \\ j ≠ i \end{matrix} \frac{(f_{i}, f_{j})_{0}}{(t, f_{j})_{0}} \} + \frac{1}{(t, f_{i})_{0}} ∑_{j≠i}(f_{i}, f_{j})_{0}\}

We give the proof of (1.3) in Section 4. If f and t are transverse then our theorem is an intersection theoretical counterpart of the main theorem of Kuo and Lu (see [KL], Theorem A). Eggers in [E] developed Kuo and Lu’s ideas to calculate the polar quotients with respect to the generic direction in terms of characteristics of branches and their intersection multiplicities coded by means of tree-models (see [Ga] for further developments). In this case our result could be also proved by using Eggers’ theorem ([E], Satz 2.1). Let us define for any reduced power series f the invariant q_{0}(f) by putting

(i) if ord f = 1, then q_{0}(f) = −∞, (ii) if f is irreducible, ord f > 1 and β̄_{0}, β̄_{1}, …, β̄_{g} is the minimal

system of generators of the semigroup Γ_{0}(f) then

q_{0}(f) = \frac{GCD (β̄_{0}, …, β̄_{g−1})β̄_{g}}{β̄_{0}}

(iii) if f = f_{1} … f_{r}, r ≥ 2 with irreducible f_{i} (i = 1, …, r) then

q_{0}(f) = \begin{matrix}r \\ max \\ i = 1 \end{matrix} \{max \{q_{0}(f_{i}), \begin{matrix}max \\ j ≠ i \end{matrix} \frac{(f_{i}, f_{j})_{0}}{ord f_{j}} \} + \frac{1}{ord f_{i}} ∑_{j≠i}(f_{i}, f_{j})_{0}\}\}

Using (1.1), (1.2) and (1.3) we get

Corollary 1.4 Suppose ord f > 1. Then for every regular parameter t that does not divide f we have q_{0}(f, t) ≤ q_{0}(f). If t and f are transverse then q_{0}(f, t) = q_{0}(f).

Let S be an analytic set near 0 ∈ C^{2}. We put L_{0}(f, S) = inf{θ > 0 : |grad f(z)| ≥ C|z|^{θ} for z ∈ S near 0} and call L_{0}(f) = L_{0}(f, C^{2}) the Łojasiewicz exponent of f. We say that L_{0}(f) is attained along S if L_{0}(f, S) = L_{0}(f). Let Suff_{0}(f) be the C^{0}-degree of sufficiency i.e. the smallest integer r such that f is topologically equivalent to f + g for all g with ord g ≥ r + 1. A historical note about these two notions is given in [LW].

Corollary 1.5 If f is reduced and ord f > 1 then L_{0}(f) = q_{0}(f) − 1 and Suff_{0}(f) = [q_{0}(f)].

Proof. According to [T] (Sec. 1, Corollaire 2 and Sec. 3, Théorème 8) there is a Zariski-open subset U ⊂ P^{1}(C) such that L_{0}(f) = q_{0}(f, bX− aY) − 1, Suff_{0}(f) = [q_{0}(f, bX − aY)] for (a : b) ∈ U. On the other hand if the line bX − aY = 0 is not tangent to the curve f(X, Y) = 0 then q_{0}(f, bX − aY) = q_{0}(f) by Corollary 1.4.

The following result was proved independently by Bogusławska ([B], Theorem 2) and Kuo and Parusiński ([KP], Theorem 3.1).

Corollary 1.6 ([B],[KP]). If the line bX − aY = 0 is not tangent to the curve f(X, Y) = 0 then the Łojasiewicz exponent L_{0}(f) is attained on the polar curve a\frac{∂f}{∂X} + b\frac{∂f}{∂Y} = 0.

Proof. Assume that the line bX − aY = 0 is not tangent to the curve f(X, Y) = 0. Using parametrizations of the branches of the polar curve a\frac{∂f}{∂X} + b\frac{∂f}{∂Y} = 0 we check that L_{0}(f, \{a\frac{∂f}{∂X} + b\frac{∂f}{∂Y} = 0\}) = q_{0}(f, bX − aY) − 1. Then we use (1.4) and (1.5).

Remark 1.7 Let f = f_{1} … f_{r} (r ≥ 2) be a reduced power series such that the irreducible factors f_{i} are pairwise transverse that is (f_{i}, f_{j})_{0} = (ord f_{i})(ord f_{j}) for i ≠ j. Let I = {i ∈ {1, …, r} : ord f_{i} > 1}. Then

q_{0}(f) = max_{i∈I}{q_{0}(f_{i}) + ∑_{j≠i} ord f_{j}} = max_{i∈I}{q_{0}(f_{i}) − ord f_{i}} + ord f

if I ≠ ∅ and q_{0}(f) = ord f if I = ∅.

Corollary 1.8 ([Li]). Suppose that f = f_{1} … f_{r} (r ≥ 2) where f_{i} are irreducible, pairwise transverse and ord f_{i} > 1 for some i ∈ {1, …, r}. Then

Suff_{0}(f) = \begin{matrix}max \\ i ∈ I \end{matrix} \{[q_{0}(f_{i})] + ∑_{j≠i} ord f_{j}\}

Proof. We apply the second part of Corollary 1.5 and Remark 1.7. In [Li] (p.160) the above formula is given without assumptions imposed on f_{i}.

2 Polar quotients and Puiseux series

Let C\{X\}^{∗} = ∪_{n≥1}C\{X^{\frac{1}{n}} \} be the ring of Puiseux series. If f(X, Y) ∈ C{X, Y } is a power series Y -regular of order p > 0 i.e. such that ord f(0, Y) = p, then f(X, Y) = ∏_{i=1}^{p}(Y − y_{i}(X))U(X, Y) where y_{i}(X) ∈ C{X}^{∗} are without constant term and U(X, Y) ∈ C{X, Y } is a unit i.e. U(0, 0) ≠ 0. We denote by Zer f = ⟨y_{1}(X), …, y_{p}(X)⟩ the sequence y_{1}(X), …, y_{p}(X) regarded as unordered. To simplify the notation we write y_{i} for y_{i}(X). Let f(X, Y) ∈ C{X, Y } be a power series Y -regular of order p > 1, then \frac{∂f}{∂Y} (X, Y) ∈ C\{X, Y \} is Y -regular of order p − 1 > 0 and we can consider the roots of both series: Zer f = ⟨y_{1}, …, y_{p}⟩ and Zer \frac{∂f}{∂Y} = ⟨z_{1}, …, z_{p−1}⟩. The following is basic for us Lemma 2.1 (The Kuo-Lu lemma, [KL] Lemma 3.3) Suppose that f ∈ C{X, Y } has no multiple factors. Then for every i, j ∈ {1, …, p}, i ≠ j, there exists a k ∈ {1, …, p − 1} such that

(*) ord (y_{i} − y_{j}) = ord (y_{i} − z_{k}) .

Moreover, for every i ∈ {1, …, p} and k ∈ {1, …, p − 1} there exists a j ∈ {1, …, p} such that (*) holds.

A simple proof of the Kuo-Lu lemma without using perturbations of power series is given in [GP1], Lemma 3.2.

Proposition 2.2 With the notations and assumptions introduced above

q_{0}(f, X) = \begin{matrix}p \\ max \\ i = 1 \end{matrix} \{∑_{j≠i} ord (y_{j} − y_{i}) + \begin{matrix}max \\ j ≠ i \end{matrix}\{ord (y_{j} − y_{i})\}\} .

Proof. By definition

q_{0}(f, X) = sup \{\frac{(f, g)_{0}}{(X, g)_{0}} : g is an irreducible factor of \frac{∂f}{∂Y} \}

Using Zeuthen’s rule for intersection multiplicity we get

q_{0}(f, X) = \begin{matrix}p − 1 \\ max \\ k = 1 \end{matrix} \{∑_{i=1}^{p} ord (z_{k} − y_{i})\} .  (1)

Put l_{i} = ∑_{j≠i} ord (y_{j} − y_{i}) + max_{j≠i}{ord (y_{j} − y_{i})} and choose i_{0} ∈ {1, …, p} such that max _{i=1}^{p}{l_{i}} = l_{i_{0}}. Let j_{0} ∈ {1, …, p} be such that

\begin{matrix}max \\ j ≠ i_{0} \end{matrix}\{ord (y_{j} − y_{i_{0}})\} = ord (y_{j_{0}} − y_{i_{0}}) .  (2)

By the first part of the Kuo-Lu lemma there is a root z = z_{k} of the derivative \frac{∂f}{∂Y} such that

ord (z − y_{i_{0}}) = ord (y_{j_{0}} − y_{i_{0}})  (3)

We will check that

ord (z − y_{i}) ≥ ord (y_{i_{0}} − y_{i}) for all i ≠ i_{0} .  (4)

Indeed, we have

ord (z − y_{i}) ≥ min {ord (z − y_{i_{0}}), ord (y_{i_{0}} − y_{i})} = ord (y_{i_{0}} − y_{i}) for

ord (z − y_{i_{0}}) = ord (y_{j_{0}} − y_{i_{0}}) = max_{j≠i}{ord (y_{j} − y_{i_{0}})} ≥ ord (y_{i} − y_{i_{0}})

for a given i ≠ i_{0}. We get

\begin{matrix}p \\ max \\ i = 1 \end{matrix}\{l_{i}\}=l_{i_{0}} = ∑_{j≠i_{0}} ord (y_{j} − y_{i_{0}}) + ord (y_{j_{0}} − y_{i_{0}})
≤ ∑_{j≠i_{0}} ord (z − y_{j}) + ord (z − y_{i_{0}})
= ∑_{j=1}^{p} ord (z − y_{j})

by (4) and (3). Consequently max _{i=1}^{p}{l_{i}} ≤ q_{0}(f, X) by (1). To check the inequality q_{0}(f, X) ≤ max _{i=1}^{p}\{l_{i}\} let z = z_{k} be a root of \frac{∂f}{∂Y} such that

q_{0}(f, X) = ∑_{i=1}^{p} ord (z − y_{i}) .  (5)

Let i_{1} ∈ {1, …, p} be such that

\begin{matrix}p \\ max \\ i = 1 \end{matrix}\{ord (z − y_{i})\} = ord (z − y_{i_{1}}) .  (6)

We will check that

ord (y_{i} − y_{i_{1}}) ≥ ord (y_{i} − z) for i ≠ i_{1} .  (7)

We have

ord (y_{i} − y_{i_{1}})≥min {ord (y_{i} − z), ord (y_{i_{1}} − z)}
=min \{ord (y_{i} − z), \begin{matrix}p \\ max \\ j = 1 \end{matrix}\{ord (z − y_{j})\}
=ord (y_{i} − z) .

By the second part of the Kuo-Lu lemma there is a j_{1} ∈ {1, …, p} such that

ord (z − y_{i_{1}}) = ord (y_{j_{1}} − y_{i_{1}}) .  (8)

Therefore we get

q(f, X)=∑_{i=1}^{p} ord (z − y_{i}) = ∑_{i≠i_{1}} ord (z − y_{i}) + ord (z − y_{i_{1}})

≤ ∑_{i≠i_{1}} ord (y_{i} − y_{i_{1}}) + ord (y_{j_{1}} − y_{i_{1}})
≤ ∑_{i≠i_{1}} ord (y_{i} − y_{i_{1}}) + \begin{matrix}max \\ i ≠ i_{1} \end{matrix}\{ord (y_{i} − y_{i_{1}})\}
=l_{i_{1}} ≤ \begin{matrix}p \\ max \\ i = 1 \end{matrix}\{l_{i}\} .

This ends the proof of (2.2).

3 Characteristic, order of contact and intersection multiplicity

In this section we recall some notions of the theory of branches. Our main reference is [Z] (see also [D],[GP1],[GP2] for nontransverse case). Let f = f(X, Y) ∈ C{X, Y } be an irreducible power series Y -regular of order p > 1. Clearly (f, X)_{0} = p. The Propositions 3.1, 3.2 and 3.3 are well-known. Proposition 3.1 Let Zer f = ⟨y_{1}, …, y_{p}⟩. Put b_{0} = p. Then there exists a sequence of strictly positive integers b_{1} < … < b_{h} such that for every i ∈ {1, …, p}:

(i) {b ∈ R : b = ord (y_{i} − y_{j}) for some j ≠ i} = {b_{1}/b_{0}, …, b_{h}/b_{0}} (ii) #{j ∈ {1, …, p} : ord (y_{i} − y_{j}) = b_{k}/b_{0}} =

GCD (b_{0}, …, b_{k−1}) − GCD (b_{0}, …, b_{k}) . We call (b_{0}, b_{1}, …, b_{h}) the characteristic sequence of f (with respect to coordinates X, Y). We put B_{k} = GCD (b_{0}, …, b_{k}) for k = 0, 1, …, h and b̄_{k} = b_{k} + \frac{1}{B_{k−1}} ∑_{i=1}^{k−1}(B_{i−1} −B_{i})b_{i} for k = 1, …, h. The sum of an empty family is equal to zero. Thus we have b̄_{1} = b_{1}. We put b̄_{0} = b_{0}. Proposition 3.2 The sequence b̄_{0}, b̄_{1}, …, b̄_{h} is a system of generetors of the semigroup Γ_{0}(f) with respect to b̄_{0} = (f, X)_{0} i.e. b̄_{k} = min {Γ_{0}(f)\(Nb̄_{0}+…+Nb̄_{k−1})} for k = 1, …, h and Nb̄_{0}+…+Nb̄_{h} = Γ_{0}(f).

Let us consider two Y -regular power series f, g ∈ C{X, Y }. Let Zer f = ⟨y_{1}(X), …, y_{p}(X)⟩ and Zer g = ⟨z_{1}(X), …, z_{q}(X)⟩. We define the order of contact of f and g (in coordinates X, Y) by putting

cont(f, g) = max {ord (y_{i}(X) − z_{j}(X)) : 1 ≤ i ≤ p, 1 ≤ j ≤ q} .

Clearly cont(f, g) = cont(g, f). Moreover it is easy to check that cont(f, g) = max _{i=1}^{p}{ord (y_{i}(X) − z_{j}(X))} for every j ∈ {1, …, q}. Proposition 3.3 Suppose that ord f(0, Y) > 1 and let (b_{0}, …, b_{h}) be the characteristic sequence of f. Let k > 0 be the smallest integer such that cont(f, g) ≤ b_{k}/b_{0} (we put b_{h+1}/b_{0} = +∞). Then

\frac{(f, g)_{0}}{(X, g)_{0}} = ∑_{i=1}^{k−1}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + B_{k−1}cont(f, g) .

Using (3.1) and (3.3) we prove Proposition 3.4 Let f = f(X, Y) ∈ C{X, Y } be irreducible, Y - regular, ord f(0, Y) > 1 of characteristic (b_{0}, b_{1}, …, b_{h}). Then

(i) q_{0}(f, X) = ∑_{i=1}^{h}(B_{i−1} − B_{i})b_{i}/b_{0} + b_{h}/b_{0} = B_{h−1}b̄_{h}/b̄_{0} , (ii) for every irreducible, Y -regular power series g ∈ C{X, Y }

\frac{(f, g)_{0}}{(X, g)_{0}} ≤ q_{0}(f, X) if and only if cont(f, g) ≤ \frac{b_{h}}{b_{0}} .

Proof. To prove (i) let us observe that by Proposition 3.1:

∑_{j≠i} ord (y_{i}−y_{j})+\begin{matrix}max \\ j ≠ i \end{matrix}\{ord (y_{j}−y_{i})\} = ∑_{i=1}^{h}(B_{i−1}−B_{i})\frac{b_{i}}{b_{0}} +\frac{b_{h}}{b_{0}} = B_{h−1} \frac{b̄_{h}}{b̄_{0}}

for every i ∈ {1, …, p}. Then we use Proposition 2.2. To check (ii) let us suppose first that cont(f, g) ≤ b_{h}/b_{0}. Then the smallest k > 0 such that cont(f, g) ≤ b_{k}/b_{0} is less or equal to h. By Proposition 3.3 we get

\frac{(f, g)_{0}}{(X, g)_{0}} ≤ ∑_{i=1}^{k−1}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + B_{k−1} \frac{b_{k}}{b_{0}} = B_{k−1} \frac{b̄_{k}}{b̄_{0}}
≤ \frac{B_{h−1}b̄_{h}}{b̄_{0}} = q_{0}(f, X)

for the sequence B_{k−1}b̄_{k} is increasing for k > 0 (b̄_{k+1} −(B_{k−1}/B_{k})b̄_{k} = b_{k+1} − b_{k} > 0 for k = 1, …, h − 1).

Now, suppose that cont(f, g) > b_{h}/b_{0}, Again by Proposition 3.3 we get

\frac{(f, g)_{0}}{(X, g)_{0}} =∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + cont(f, g) > ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + \frac{b_{h}}{b_{0}}
= \frac{B_{h−1}b̄_{h}}{b̄_{0}} = q_{0}(f, X) .

This ends the proof.

4 Proof

It is easy to check

Lemma 4.1 If Φ is a local isomorphism i.e. a pair of power series without constant term such that Jac Φ(0, 0) ≠ 0 then

q_{0}(f, t) = q_{0}(f ∘ Φ, t ∘ Φ) .

Therefore to prove (1.1) and (1.3) it suffices to consider the case t = X.

Proof of Proposition 1.1. Let (β̄_{0}, …, β̄_{g}) be the minimal system of generators of Γ_{0}(f). It is known ([D], p. 332-333) that three cases are possible

(a) b̄_{0} = β̄_{0} then h = g and (b̄_{0}, …, b̄_{g}) = (β̄_{0}, …, β̄_{g}),

(b) b̄_{0} > β̄_{0} is a multiple of β̄_{0}, then h = g + 1 and (b̄_{0}, …, b̄_{g+1}) = (b̄_{0}, β̄_{0}, …, β̄_{g})

(c) b̄_{0} > β̄_{0} and b̄_{0} is not a multiple of β̄_{0}, then h = g and (b̄_{0}, b̄_{1}, …, b̄_{g}) = (β̄_{1}, β̄_{0}, β̄_{2}, …, β̄_{g}).

Therefore B_{h−1}b̄_{h} = GCD (b̄_{0}, …, b̄_{h−1})b̄_{h} = GCD (β̄_{0}, …, β̄_{g−1})β̄_{g} and Proposition 1.1 follows from (3.4)(i).

Proof of Theorem 1.3. It suffices to prove the following: if f = f_{1} … f_{r} (r ≥ 2) with f_{i} irreducible, is Y -regular, then

q_{0}(f, X) = \begin{matrix}r \\ max \\ i = 1 \end{matrix} \{max \{q_{0}(f_{i}, X), \begin{matrix}max \\ j ≠ i \end{matrix} \frac{(f_{i}, f_{j})_{0}}{(X, f_{j})_{0}} \} + \frac{1}{(X, f_{i})_{0}} ∑_{j≠i}(f_{i}, f_{j})_{0}\}

According to Proposition 2.2 we have to calculate the quantities

l_{i} = ∑_{j≠i} ord (y_{j} − y_{i}) + \begin{matrix}max \\ j ≠ i \end{matrix}\{ord (y_{j} − y_{i})\} .

Without restriction of the generality we may assume that i = 1 and y_{1} = y_{1}(X) is a root of the equation f_{1}(X, Y) = 0. Let us suppose that ord f_{1}(0, Y) > 1 and let (b_{0}, …, b_{h}) be the characteristic of f_{1}. Set

I_{1} = {i ≠ 1 : y_{i} = y_{i}(X) is a root of f_{1}(X, Y) = 0}

and

I_{2} = {i : y_{i} = y_{i}(X) is a root of f_{2}(X, Y) … f_{r}(X, Y) = 0} .

Therefore we can write

l_{1} = ∑_{i∈I_{1}} ord (y_{i}(X) − y_{1}(X)) + ∑_{i∈I_{2}} ord (y_{i}(X) − y_{1}(X)) +
+max \begin{cases}max \\ i ∈ I_{1} \end{cases}\{ord (y_{1}(X) − y_{i}(X))\}, \begin{matrix}max \\ i ∈ I_{2} \end{matrix}\{ord (y_{i}(X) − y_{1}(X))\}\} .

We get

∑_{i∈I_{1}} ord (y_{i}(X) − y_{1}(X)) = ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} by Proposition 3.1,

∑_{i∈I_{2}} ord (y_{i}(X)−y_{1}(X)) = \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} by Zeuthen’s rule,

\begin{matrix}max \\ i ∈ I_{1} \end{matrix}\{ord (y_{i}(X) − y_{1}(X))\} = \frac{b_{h}}{b_{0}} by Proposition 3.1,

\begin{matrix}max \\ i ∈ I_{2} \end{matrix}\{ord (y_{i}(X) − y_{1}(X))\} = \begin{matrix}max \\ i ≠ 1 \end{matrix}\{cont(f_{1}, f_{i})\} by definition.

Consequently we get

(*) l_{1} = \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} + ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} +
+max \{\frac{b_{h}}{b_{0}} , \begin{matrix}max \\ i ≠ 1 \end{matrix}cont(f_{1}, f_{i})\} .

Let us consider two cases.

Case 1: max_{i≠1}{cont(f_{1}, f_{i})} ≤ b_{h}/b_{0}. By Proposition 3.4 we have q_{0}(f_{1}, X) = ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + \frac{b_{h}}{b_{0}} and \frac{(f_{1},f_{i})_{0}}{(X,f_{i})_{0}} ≤ q_{0}(f_{1}, X) for all i ≠ 1. Therefore we get by (*):

l_{1} = \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} + ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + \frac{b_{h}}{b_{0}}
= \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i}) + q_{0}(f_{1}, X)
= \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} + max \{q_{0}(f_{1}, X), \begin{matrix}max \\ i ≠ 1 \end{matrix} \frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}} \}

Case 2: max_{i≠1}{cont(f_{1}, f_{i})} > b_{h}/b_{0}. Let us fix i ≠ 1. If cont(f_{1}, f_{i}) > b_{h}/b_{0} then

\frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}} = ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + cont(f_{1}, f_{i})

by (3.3), if cont(f_{1}, f_{i}) ≤ b_{h}/b_{0} then

\frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}} ≤ ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}}

again by (3.3). Therefore

\begin{matrix}max \\ i ≠ 1 \end{matrix} \frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}} = ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + \begin{matrix}max \\ i ≠ 1 \end{matrix}\{cont(f_{1}, f_{i})\} .

Using (*) we get

l_{1} = \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} + ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + \begin{matrix}max \\ i ≠ 1 \end{matrix}\{cont(f_{1}, f_{i})\}
= \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} + \begin{matrix}max \\ i ≠ 1 \end{matrix} \frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}}
= \frac{1}{(X, f_{1})_{0}} ∑_{i≠1}(f_{1}, f_{i})_{0} + max \{q_{0}(f_{1}, X), \begin{matrix}max \\ i ≠ 1 \end{matrix} \frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}} \}

for q_{0}(f_{1}, X) = ∑_{i=1}^{h}(B_{i−1} − B_{i})\frac{b_{i}}{b_{0}} + \frac{b_{h}}{b_{0}} < \begin{matrix}max \\ i ≠ 1 \end{matrix} \frac{(f_{1}, f_{i})_{0}}{(X, f_{i})_{0}} .

It remains to consider the case ord f_{1}(0, Y) = 1. Then (X, f_{1})_{0} = 1, q_{0}(f_{1}, X) = −∞ and we get easily

l_{1} = ∑_{j≠1}(f_{1}, f_{j})_{0} + \begin{matrix}max \\ j ≠ 1 \end{matrix} \frac{(f_{1}, f_{j})_{0}}{(X, f_{j})_{0}} .

This ends the proof.

Appendix

We reprove here the main result of [KL].

Let f = f(X, Y) ∈ C{X, Y } be a power series Y -regular of order p = ord f. Let y_{i} = y_{i}(X) ∈ C{X}^{∗} (i = 1, …, p) be the sequence of all solutions (without constant term) of the equation f(X, Y) = 0. We put

l_{i} = ∑_{j≠i} ord (y_{i} − y_{j}) + \begin{matrix}max \\ j ≠ i \end{matrix}\{ord (y_{i} − y_{j})\} for i = 1, …, p .

Then we have Kuo and Lu’s formula for the Łojasiewicz exponent ([KL], Theorem A).With the above notation assume that f is a reduced power series with ord f > 1. Then, we have

L_{0}(f) = \begin{matrix}p \\ max \\ i = 1 \end{matrix}\{l_{i} − 1\} .

Proof. By Proposition 2.2 we have q_{0}(f, X) = max _{i=1}^{p}{l_{i}}. On the other hand q_{0}(f, X) = q_{0}(f) by the second part of Corollary 1.4. Then we use the first part of Corollary 1.5.

Note here that the original result was proved for distinguished polynomials. The quantities l_{i} was defined in Kuo and Lu’s work by means of some perturbations of the roots y_{i} = y_{i}(X). The simple expressions for l_{i} we use are due to Bogusławska ([B], lemma 3).

Let c_{i} = max_{j≠i}{ord (y_{i} − y_{j})} for i = 1, …, p. According to [KL] a root y_{i} is minimal if for any root y_{j} with ord (y_{i} − y_{j}) = c_{i} we get c_{j} ≤ c_{i}. If is easy to check that if c_{i} < c_{k} then l_{i} < l_{k}. Hence we get Corollary to Kuo and Lu’s formula ([KL], Theorem A’). We have

L_{0}(f) = \begin{matrix}max \\ i ∈ I \end{matrix} \{l_{i} − 1\} .

where I is the set of all i ∈ {1, …, p} such that the root y_{i} is minimal.

For every branch f we denote by K_{0}(f) the knot corresponding to f. Let f be a branch of characteristic (β_{0}, …, β_{g}). We define the self-link σ_{0}(f) of f to be σ_{0}(f) = link (K_{0}(f), K_{0}(f̃)) where f̃ is a branch of characteristic (β_{0}, …, β_{g}) such that cont(f, f̃) = β_{g}/β_{0} (see [KL], p. 300-301). Then σ_{0}(f) = (f, f̃)_{0} = GCD (β̄_{0}, …, β̄_{g−1})β̄_{g} = q_{0}(f)(ord f) by Proposition 3.3.

In [KL] the authors assert ([KL], Corollary to Theorem B) the following: “suppose that y_{1} = y_{1}(X) is a minimal root of f = f_{1} … f_{r} and let f_{i} = f_{i}(X, Y) be such that f_{i}(X, y_{1}(X)) = 0. Assume that knots K_{0}(f_{i}) and K_{0}(f_{j}) are not isotopic for i ≠ j. Then

(ord f_{i})l_{1} = ∑_{j≠i}(f_{1}, f_{j})_{0} + σ_{0}(f_{i}).”

The example given below shows that the above statement (and consequently Theorem B of [KL]) are not true.

Example. Let 1 < β_{0} < … < β_{g}, g > 1 be a sequence of coprime integers such that the divisors GCD (β_{0}, …, β_{i}) form a strictly decreasing sequence. Let f_{1}(X, Y) = 0 (resp. f_{2}(X, Y) = 0) be the minimal equation of the Puiseux series y_{1} = X^{β_{1}/β_{0}} + … + X^{β_{g−1}/β_{0}} (resp. y_{2} = X^{β_{1}/β_{0}} + … + X^{β_{g}/β_{0}}). Let f = f_{1}f_{2}. The series f_{1} and f_{2} have different characteristics and hence the knots K_{0}(f_{1}) and K_{0}(f_{2}) are not isotopic. For every root y_{i} of f we get c_{i} = cont(f_{1}, f_{2}) = β_{g}/β_{0}, therefore all roots of f are minimal. The calculation from the proof of Theorem 1.3 (Section 4 of this paper, Case 2) gives

l_{1} = \frac{(f_{1}, f_{2})_{0}}{ord f_{1}} + \frac{(f_{1}, f_{2})_{0}}{ord f_{2}}

and

q_{0}(f_{1}) < \frac{(f_{1}, f_{2})_{0}}{ord f_{2}} .

Therefore

(ord f_{1})l_{1} =(f_{1}, f_{2})_{0} + ord f_{1} \frac{(f_{1}, f_{2})_{0}}{ord f_{2}}
>(f_{1}, f_{2})_{0} + (ord f_{1})q_{0}(f_{1}) = (f_{1}, f_{2})_{0} + σ_{0}(f_{1}).

This contradicts the statement quoted above.

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Department of Mathematics, Technical University, AL. 1000 L PP 7, 25-314 Kielce, Poland

e-mail: matap@tu.kielce.pl