ON THE POLAR QUOTIENTS OF

AN ANALYTIC PLANE CURVE

Janusz Gwoździewicz and Arkadiusz Płoski

IMUJ PREPRINT 2001/05

Abstract. We give explicit formulae for the polar quotients of a plane many– branched singularity

Introduction

We use standard notations: ℂ{X, Y } is the ring of convergent power series, ord f is the order of f = f(X, Y) ∈ ℂ{X, Y }, (f, ϕ)_{0} denotes the intersection multiplicity of f with ϕ. Recall that (f, ϕ)_{0} ≥ (ord f)(ord ϕ) with equality if and only if f, ϕ are transverse.

Let f = f(X, Y) ∈ ℂ{X, Y } be a reduced power series (i.e. without multiple factors) and let t = t(X, Y) ∈ ℂ{X, Y } be a regular parameter (i.e. a series of order 1) such that t does not divide f. We consider the set of polar quotients of f with respect to t:

Q(f, t) = \{\frac{(f, ϕ)_{0}}{(t, ϕ)_{0}} : ϕ is an irreducible factor of\frac{∂(t, f)}{∂(X, Y)}\} .

The polar quotients were studied by many authors ([T], [M], [E], [Eph], [Ca], [LMW1], [LMW2], [Ga], [D], [LP]). Recently, an explicit formula for the maximal polar quotient q(f, t) = sup Q(f, t) has been given ([P], Theorem 1.3). In this note

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 14H20, 32S10.

Key words and phrases. plane curve singularity, polar quotients, Puiseux’ date.

0The second author was partially supported by the KBN grant P03A00115.

we give similar formulae for all polar quotients. Like in [P] our main tool is the Kuo–Lu lemma ([KL], Lemma 3.3 and Section 2 of this paper). Instead of using Puiseux’ trees (see [KL], [E]) we make our calculations by means of Puiseux’ date (Section 3). The notion of symmetric power explained in [Wh], Appendix V turns out very useful.

Let A be a nonempty set. Then A_{sym}^{p} is the set of all p–tuples regarded as unordered. If α = ⟨a_{1}, …, a_{p}⟩ ∈ A_{sym}^{p} then |α| = {a_{1}, …, a_{p}} is the set corresponding to α. We put deg α = p. If α = ⟨a_{1}, …, a_{p}⟩ and β = ⟨b_{1}, …, b_{q}⟩ then α ⊕ β = ⟨a_{1}, …, a_{p}, b_{1}, …, b_{q}⟩. Clearly |α ⊕ β| = |α| ∪ |β| and deg α ⊕ β = deg α + deg β.

Instead of ⟨\underbrace{a_{1}, …, a_{1}}_{m_{1} times}, …, \underbrace{a_{p}, …, a_{p}}_{m_{p} times}⟩ we write ⟨a_{1} : m_{1}, …, a_{p} : m_{p}⟩.

1. Main result

The semigroup Γ_{0}(f) of an irreducible power series f ∈ ℂ{X, Y } is the set of all intersection numbers (f, ϕ)_{0} where ϕ runs over all power series ϕ ∈ ℂ{X, Y } such that f does not divide ϕ. Let t = t(X, Y) be a regular parameter. Suppose that ord f > 1 and put p = (t, f)_{0}. Let b̄_{0}, …, b̄_{h} be the p-minimal system of generators of Γ_{0}(f) determined by conditions

(i) b̄_{0} = p, (ii) b̄_{k} = min(Γ_{0}(f) \ (ℕb̄_{0} + ⋯ + ℕb̄_{k−1})) for k = 1, …, h, (iii) Γ_{0}(f) = ℕb̄_{0} + ⋯ + ℕb̄_{h}.

Let B_{k} = GCD(b̄_{0}, …, b̄_{k}) be the greatest common divisor of b̄_{0}, …, b̄_{k}.

Proposition 1.1. ([M], [Eph]). Let f ∈ ℂ{X, Y } be an irreducible power series with ord f > 1. Then

Q(f, t) = \{\frac{B_{k−1}b̄_{k}}{b̄_{0}} : k = 1, …, h \} .

The above formula was given by Merle ([M], Théorème 3.1) in transverse case. Then Ephraim ([Eph], Lemma 1.6) observed that Merle’s result holds also for polar quotients with respect to a regular parameter t. In Section 4 we give a new proof of (1.1). Recall that the sequence B_{k−1}b̄_{k}/b̄_{0} is strictly increasing.

If ord f = 1 then Q(f, t) = ∅ if f, t are transverse and Q(f, t) = {1} if f, t are not.

Remark 1.2. Using the inversion formulae for plane branches (see [P], proof of Proposition 1.1) one can calculate the polar quotients of a branch f = 0 in terms of the minimal system of generators β̄_{0}, …, β̄_{g} of Γ_{0}(f). Let e_{k} = GCD(β̄_{0}, …, β̄_{k}) for k = 0, 1, …, g. Then

Q(f, t) = \{\frac{e_{k−1}β̄_{k}}{β̄_{0}} : k = 1, …, g\} ∪ O(f, t)

where O(f, t) = ∅ if (f, t)_{0} = β̄_{0} or (f, t)_{0} = β̄_{1} and O(f, t) = {ord f} otherwise.

Let ϕ, ψ ∈ ℂ{X, Y } be irreducible power series. The contact coefficient (in the sense of Hironaka) with respect to a regular parameter t ∈ ℂ{X, Y } is the rational number

h(ϕ, ψ; t) = \frac{(ϕ, ψ)_{0}}{(t, ψ)_{0}}

In [LMW1] this notion is introduced in the case where t and ψ are transverse. Then h(ϕ, ψ; t) = (ϕ, ψ)_{0}/ ord ψ and we write h(ϕ, ψ) instead of h(ϕ, ψ; t).

Let us consider a reduced power series f = f_{1} ⋯ f_{r} with r > 1 irreducible factors. Fix i ∈ {1, …, r}. For every real number τ > 0 we put J_{i}(τ) = {j ∈ {1, …, r} : h(f_{i}, f_{j}; t) ≤ τ } and J_{i}(τ)^{c} = {1, …, r} \ J_{i}(τ).

By convention h(f_{i}, f_{i}; t) = +∞, hence i ∈ J_{i}(τ)^{c}. We put

q_{i}(τ) = \frac{1}{(t, f_{i})_{0}} \{( ∑_{j∈J_{i}(τ)^{c}}(t, f_{j})_{0})τ + ∑_{j∈J_{i}(τ)}(f_{i}, f_{j})_{0}\} for τ > 0.

To write the above formula more explicitly suppose i = 1 and assume h(f_{1}, f_{r}; t) ≤ h(f_{1}, f_{r−1}; t) ≤ ⋯ ≤ h(f_{1}, f_{2}; t) < h(f_{1}, f_{1}; t) = +∞. Then

q_{1}(τ) = \{ \begin{matrix}\frac{(t,f_{1}···f_{j})_{0}}{(t,f_{1})_{0}} τ + \frac{(f_{1},f_{j+1}···f_{r})_{0}}{(t,f_{1})_{0}} , & h(f_{1}, f_{j+1}; t) ≤ τ < h(f_{1}, f_{j}; t), j < r \\ \frac{(t,f)_{0}}{(t,f_{1})_{0}} τ, & τ < h(f_{1}, f_{r}; t) \end{matrix} .

It is easy to see that the functions q_{i} are piecewise linear, continuous and strictly increasing. Note that q_{i} is determined by the intersection multiplicities of i-th branch f_{i} = 0 with the remaining branches and with the smooth curve t = 0. Let H_{i}(f, t) = { h(f_{i}, f_{j}; t) : j ≠ i } be the set of contact coefficients of the branch f_{i} = 0 with branches f_{j} = 0 (j ≠ i). The main results of this paper is

Theorem 1.3. Let f = f_{1} ⋯ f_{r} be a reduced power series with r > 1 irreducible factors. Then

Q(f, t) = ⋃_{i=1}^{r} q_{i}(Q(f_{i}, t) ∪ H_{i}(f, t)).

We give the proof of (1.3) in Section 4. We put Q_{i}(f, t) = q_{i}(Q(f_{i}, t) ∪ H_{i}(f, t)) and call the elements of Q_{i}(f, t) polar quotients associated with the branch f_{i} = 0.

A polar quotient can be associated with more than one branch. From Theorem 1.3 it follows that the polar quotients of the curve f = 0 with respect to a regular parameter t depend only on the equisingularity type of the curve tf = 0.

If t and f are transverse then the polar quotients of f = 0 do not depend on the regular parameter t and are determined by the equisingularity type of the curve f = 0. Then we write Q_{i}(f) instead of Q_{i}(f, t) and Q(f) instead of Q(f, t).

Using (1.3) we obtain easily the formula for the maximal polar quotient q(f, t) = sup Q(f, t) given in [P]. Obviously q(f, t) = max_{i=1}^{r}{ max Q_{i}(f, t) }, hence it suffices to calculate the maximal polar quotients associated to branches.

Corollary 1.3.

max Q_{i}(f, t) = max\{ q(f_{i}, t), max_{j≠i}\{h(f_{i}, f_{j}; t)\} \} + \frac{1}{(t, f_{i})_{0}} ∑_{j≠i}(f_{i}, f_{j})_{0}

Proof. Let τ_{i} = max(Q(f_{i}, t) ∪ H_{i}(f, t)) = max{ q(f_{i}, t), max_{j≠i}{ h(f_{i}, f_{j}, t) } }. Then J_{i}(τ_{i}) = \{1, …, r\} \ \{i\} and q_{i}(τ_{i}) = τ_{i} + \frac{1}{(t,f_{i})_{0}} ∑_{j≠i}(f_{i}, f_{j})_{0}. We get then max Q_{i}(f, t) = q_{i}(τ_{i}) for q_{i} is an increasing function.

Remark 1.4. For every i ∈ \{1, …, r\} the rational numbers q(f_{i}, t)+\frac{1}{(t,f_{i})_{0}} ∑_{j≠i}(f_{i}, f_{j})_{0} (provided that q(f_{i}, t) ≠ −∞) and max_{j≠i}\{ h(f_{i}, f_{j}; t) \} + \frac{1}{(t,f_{i})_{0}} ∑_{j≠i}(f_{i}, f_{j})_{0} are polar quotients of f = 0.

Corollary 1.5. If f = f_{1} ⋯ f_{r} (r > 1) with transverse factors f_{i}, f_{j} for i ≠ j then

Q(f) = ⋃_{i=1}^{r}{ τ_{i} + ord f − ord f_{i} : τ_{i} ∈ Q(f_{i}) } ∪ {ord f}

Proof. Let us calculate Q_{1}(f). It is easy to see that J_{1}(τ) = { j ∈ {2, …, r} : ord f_{1} ≤ τ }. Hence J_{1}(τ) = ∅ if τ < ord f_{1} and J_{1}(τ) = {2, …, r} if τ ≥ ord f_{1}. Obviously H_{1}(f) = {ord f_{1}}. Therefore we get

q_{1}(τ) = \{ \begin{matrix}τ + ord f − ord f_{1}, & τ ≥ ord f_{1} \\ \frac{ord f}{ord f_{1}} τ, & τ < ord f_{1} \end{matrix}

For every τ_{1} ∈ Q(f_{1}) one gets τ_{1} ≥ ord f_{1}. Consequently Q_{1}(f) = { q_{1}(τ_{1}) : τ_{1} ∈ Q(f_{1}) } ∪ q_{1}(ord f_{1}) = { τ_{1} + ord f − ord f_{1} : τ_{1} ∈ Q(f_{1}) } ∪ {ord f} and the corollary follows.

To give another application of the main results we calculate polar quotients of “diagonal curves” introduced by Delgado [D].

Corollary 1.6. Let f = f_{1} ⋯ f_{r} be a reduced power series. Suppose that Γ_{0}(f_{i}) and (t, f_{i})_{0} do not depend on i ∈ {1, …, r}, Γ_{0}(f_{i}) = ⟨b̄_{0}, b̄_{1}, …, b̄_{h}⟩ and (f_{i}, f_{j})_{0} = B_{h−1}b̄_{h} + c with c > 0, for i ≠ j. Then

Q(f, t) = \{ r \frac{B_{k−1}b̄_{k}}{b̄_{0}} : k = 1, …, h \} ∪ \{ r \frac{B_{h−1}b̄_{h} + c}{b̄_{0}}\} .

Proof. By Proposition 1.1 we get Q(f_{i}, t) = { B_{k−1}b̄_{k}/b̄_{0} : k = 1, …, h } on the other hand H_{i}(f, t) = { (B_{h−1}b̄_{h} + c)/b̄_{0} }. It is easy to check that q_{i}(τ) = rτ if τ < (B_{h−1}b̄_{h} + c)/b̄_{0} and q_{i}(τ) = τ + (r − 1)(B_{h−1}b̄_{h} + c)/b̄_{0} if τ ≥ (B_{h−1}b̄_{h} + c)/b̄_{0}. Now we get (1.6) from Theorem 1.3.

2. Zeros of a derivative

Let ℂ{X}^{∗} = ⋃_{n≥1} ℂ{X^{1/n}} be the ring of Puiseux series. If f = 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 the equation f(X, Y) = 0 has in ℂ{X}^{∗} p roots y_{1} = y_{1}(X), …, y_{p} = y_{p}(X) of positive order counted with multiplicities. We put Zer f = ⟨y_{1}, …, y_{p}⟩. Let S ⊂ ℂ{X}^{∗} be a set of Puiseux series. If y_{1}, …, y_{q} ∈ S and y_{q+1}, …, y_{p} ∉ S then we put Zer f ∩ S = ⟨y_{1}, …, y_{q}⟩. Let us recall

The Kuo–Lu lemma. ([KL], lemma 3.3) Suppose that f = f(X, Y) is Y -regular of order p > 1. Let Zer f = ⟨y_{1}, …, y_{p}⟩ and Zer \frac{∂f}{∂Y} = ⟨z_{1}, …, z_{p−1}⟩. Then for every i ∈ { 1, …, p }:

⟨ord(z_{1} −y_{i}), …, ord(z_{p−1} −y_{i})⟩ = ⟨ord(y_{1} −y_{i}), …, { \atop ord(y_{i} − y_{i})}, …, ord(y_{p} −y_{i})⟩.

It is convenient to prove the lemma for formal power series with coefficients in ℂ{X}^{∗}. Replacing f(X, Y) by f(X, Y + y_{i}) we may assume y_{i} = 0. Then the series f(X, Y)/Y and \frac{∂f}{∂Y} have the same Newton diagram (see [Wa], Chapter IV) and the lemma follows from the fact that the Newton diagram determines the order of roots and the number of roots of given order. For more details see [GP1].

For every u ∈ ℂ{X}^{∗} and r > 0 we put

S(u, r) = { v ∈ ℂ{X}^{∗} : ord(u − v) = r },

B(u, r) = { v ∈ ℂ{X}^{∗} : ord(u − v) > r },

B̄(u, r) = { v ∈ ℂ{X}^{∗} : ord(u − v) ≥ r }.

According to the given definitions deg(Zer f ∩ S) is equal to the number of roots counted with multiplicities of the equation f(X, Y) = 0 lying in the set S of Puiseux series. If | Zer f| ∩ S = ∅ then we put deg(Zer f ∩ S) = 0.

Lemma 2.1. Suppose that f = f(X, Y) is Y -regular of order p > 1. Then for every y ∈ Zer f and for every r > 0

(i) deg(Zer f ∩ S(y, r)) = deg(Zer \frac{∂f}{∂Y} ∩ S(y, r)), (ii) deg(Zer f ∩ B(y, r)) = deg(Zer \frac{∂f}{∂Y} ∩ B(y, r)) + 1, (iii) deg(Zer f ∩ B̄(y, r)) = deg(Zer \frac{∂f}{∂Y} ∩ B̄(y, r)) + 1.

Proof. The part (i) is a reformulation of the Kuo–Lu lemma, (ii) and (iii) follow from (i).

Proposition 2.2. Under assumptions of Lemma 2.1:

(i) for every z ∈ | Zer \frac{∂f}{∂Y} | there exist y_{1}, y_{2} ∈ Zer f, y_{1} ≠ y_{2} such that ord(z − y_{1}) = ord(y_{1} − y_{2}) ≥ ord(z − y) for all y ∈ | Zer f|,

(ii) for every y_{1}, y_{2} ∈ Zer f, y_{1} ≠ y_{2} there exists z ∈ | Zer \frac{∂f}{∂Y} | such that ord(z−y_{1}) = ord(y_{1} − y_{2}) ≥ ord(z − y) for all y ∈ | Zer f|.

Proof.

(i) Fix z ∈ | Zer \frac{∂f}{∂Y} |. Let y_{1} ∈ | Zer f| be such that ord(z − y_{1}) = max{ ord(z − y) : y ∈ | Zer f| }. By the Kuo–Lu lemma there exists y_{2} ∈ | Zer f| such that ord(z − y_{1}) = ord(z − y_{2}) and we are done.

(ii) Suppose y_{1} ≠ y_{2} are given roots of the equation f(X, Y) = 0. Let B̄ = B̄(y_{1}, ord(y_{1} − y_{2})). Let B_{1}, … B_{s} be a sequence of pairwise disjoint balls such that ⋃_{i=1}^{s} B_{i} = ⋃{ B(y, ord(y_{1} − y_{2})) : y ∈ | Zer f| ∩ B̄ }.

Note that

1) s ≥ 2 for B(y_{1}, ord(y_{1} − y_{2})) ∩ B(y_{2}, ord(y_{1} − y_{2})) = ∅,

2) B_{i} ⊂ B̄ (i = 1, …, s) because the center of B_{i} belongs to B̄ and B_{i}, B̄ have the same radius.

3) ⋃_{i=1}^{s}(B_{i}∩| Zer f|) = B̄∩| Zer f| for B_{i} cover B̄∩| Zer f| and are pairwise disjoint. Using lemma 2.1 we calculate deg((B̄ \ ⋃_{i=1}^{s} B_{i}) ∩ Zer \frac{∂f}{∂Y} ) = deg(B̄ ∩ Zer \frac{∂f}{∂Y} ) −

∑_{i=1}^{s} deg(B_{i} ∩ Zer \frac{∂f}{∂Y} ) = (deg(B̄ ∩ Zer f) − 1) − ∑_{i=1}^{s}(deg(B_{i} ∩ Zer f) − 1) = deg(B̄ ∩ Zer f) − ∑_{i=1}^{s} deg(B_{i} ∩ Zer f) + (s − 1) = s − 1 > 0.

Therefore the set (B̄ \ ⋃_{i=1}^{s} B_{i}) ∩ | Zer \frac{∂f}{∂Y} | is nonempty. Let z ∈ (B̄ \ ⋃_{i=1}^{s} B_{i}) ∩ | Zer \frac{∂f}{∂Y} |. Then ord(z − y_{1}) ≥ ord(y_{1} − y_{2}) ≥ ord(z − y) for all y ∈ | Zer f| ∩ B̄. In particular ord(z − y_{1}) = ord(y_{1} − y_{2}). If y ∈ | Zer f| \ B̄ then ord(y − y_{1}) < ord(y_{1} − y_{2}) = ord(z − y_{1}). Hence we get ord(z − y_{1}) ≥ ord(z − y) for y ∈ | Zer f|.

For given y_{1}, y_{2} ∈ | Zer f|, y_{1} ≠ y_{2} we put

L_{y_{1},y_{2}} = | Zer f| ∩ B̄(y_{1}, ord(y_{1} − y_{2})),
L_{y_{1},y_{2}}^{c} = | Zer f| \ L_{y_{1},y_{2}},

l_{y_{1},y_{2}} = ∑_{y∈L_{y_{1},y_{2}}^{c}} ord(y_{1} − y) + (#L_{y_{1},y_{2}}) ord(y_{1} − y_{2}).

Theorem 2.3. Let f = f(X, Y) be reduced Y -regular with ord f(0, Y) > 1. Then \{ l_{y_{1},y_{2}} : y_{1}, y_{2} ∈ | Zer f|, y_{1} ≠ y_{2} \} = \{ ord f(X, z(X)) : z ∈ | Zer \frac{∂f}{∂y} | \}.

Proof. For every z = z(X) ∈ ℂ{X}^{∗} without constant term one has

(1) ord f(X, z(X)) = ∑_{y∈| Zer f|} ord(z−y) = ∑_{y∈Ly_{1},y_{2}^{c}} ord(z−y)+ ∑_{y∈Ly_{1},y_{2}} ord(z−y).

Let y_{1}, y_{2} ∈ Zer f and z ∈ |Zer \frac{∂f}{∂Y} | be such that ord(z − y_{1}) = ord(z − y_{2}) ≥ ord(z − y) for all y ∈ | Zer f|. We will check that

(2)  ord(z − y) = \{ \begin{matrix}ord(y − y_{1}), & y ∈ L_{y_{1},y_{2}}^{c} \\ ord(y_{2} − y_{1}), & y ∈ L_{y_{1},y_{2}} \end{matrix}

Indeed, if y ∈ L_{y_{1},y_{2}}^{c} then ord(z − y) = ord(y − y_{1}) for ord(y − y_{1}) < ord(y_{2} − y_{1}) = ord(z − y_{1}).

If y ∈ L_{y_{1},y_{2}} then ord(z − y) ≥ min{ ord(z − y_{1}), ord(y_{1} − y) } = min{ ord(y_{2} − y_{1}), ord(y_{1} − y) } = ord(y_{2} − y_{1}). On the other hand ord(z − y) ≤ ord(z − y_{1}) for all y ∈ | Zer f| and ord(z − y) ≤ ord(y_{2} − y_{1}). Hence ord(z − y) = ord(y_{2} − y_{1}) for y ∈ L_{y_{1},y_{2}}. Now, Theorem 2.3 follows from (1), (2) and Proposition 2.2.

3. Puiseux’ date

We use the notations and definitions from [P], Section 3. Let f = f(X, Y) ∈ ℂ{X, Y } be an irreducible power series Y -regular of order p > 0. Let (b_{0}, …, b_{h}) be the characteristic of f. We put B_{k} = GCD(b̄_{0}, …, b̄_{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 = 0, 1, …, h. Then the sequence b̄_{0} = p, b̄_{1}, …, b̄_{h} is the p-minimal system of generators of the semigroup Γ_{0}(f) (see [P], Proposition 3.2). We put P (f) = ⟨\frac{b_{1}}{b_{0}} : B_{0} − B_{1}, …, \frac{b_{h}}{b_{0}} : B_{h−1} − B_{h}⟩ and call P (f) Puiseux’ date of the branch f = 0. Clearly |P (f)| = \{ \frac{b_{1}}{b_{0}} , …, \frac{b_{h}}{b_{0}} \}.

Proposition 3.1. For every i ∈ { 1, …, p }:

⟨ord(y_{1} − y_{i}), …, { \atop ord(y_{i} − y_{i})}, …, ord(y_{p} − y_{i})⟩ = P (f).

Proof. It is a reformulation of Proposition 3.1 from [P].

The following notation will be useful. Let α ∈ ℝ_{sym}^{p} and let K > 0 be a real number. Write α = ⟨a_{1}, …, a_{q}, a_{q+1}, …, a_{p}⟩ where a_{1}, …, a_{q} < K and K ≤ a_{q+1}, …, a_{p}. Then we put α_{K} = ⟨a_{1}, …, a_{q}, K, …, K⟩ ∈ ℝ_{sym}^{p+1} with K repeated p − q + 1 times. We let α_{(+∞)} = α.

Let g = g(X, Y) ∈ ℂ{X, Y } be an irreducible Y -regular power series. Let Zer g = ⟨z_{1}(X), …, z_{q}(X)⟩. Then the order of contact of f and g is defined to be

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

Note that for every K > 0

P (f)_{K} = ⟨\frac{b_{1}}{b_{0}} : B_{0} − B_{1}, …, \frac{b_{k−1}}{b_{0}} : B_{k−2} − B_{k−1}, K : B_{k−1}⟩

where k > 0 is the smallest integer such that K ≤ \frac{b_{k}}{b_{0}} .

Proposition 3.2. Let z = z(X) ∈ | Zer g|. Then

⟨ord(z − y_{1}), …, ord(z − y_{p})⟩ = P (f)_{cont(f,g)}.

Proof. It is easy to check that max{ ord(z − y_{1}), …, ord(z − y_{p}) } = cont(f, g). We may assume that cont(f, g) = ord(z − y_{p}). For every i ∈ { 1, …, p } ord(z − y_{i}) < ord(z − y_{p}) and then ord(z − y_{i}) = ord(y_{i} − y_{p}) or ord(z − y_{i}) = ord(z − y_{p}). Therefore (3.2) follows from (3.1).

For every α = ⟨a_{1}, …, a_{p}⟩ ∈ ℝ_{sym}^{p} we put ∑ α = ∑_{i=1}^{p} a_{i}. Clearly ∑(α ⊕ β) = ∑ α + ∑ β.

Proposition 3.3. Let f = f(X, Y) ∈ ℂ{X, Y } and g = g(X, Y) ∈ ℂ{X, Y } be irreducible Y -regular power series. Suppose f is of characteristic (b_{0}, …, b_{h}). Then (i) ∑ P (f)_{b_{k}/b_{0}} = \frac{B_{k−1}b̄_{k}}{b̄_{0}} ,

(ii) ∑ P (f)_{cont(f,g)} = \frac{(f,g)_{0}}{(X,g)_{0}} , (iii) if K ≤ cont(f, g) then ∑ P (g)_{K} = \frac{(X,g)_{0}}{(X,f)_{0}} ∑ P (f)_{K}.

Proof. Property (i) follows from the definitions. To check (ii) recall that \frac{(f,g)_{0}}{(X,g)_{0}} = ord f(X, z(X)) where z = z(X) ∈ | Zer g|.

Hence \frac{(f,g)_{0}}{(X,g)_{0}} = ∑_{i=1}^{p} ord(z(X)−y_{i}(X)) = ∑ P (f)_{cont(f,g)} by Proposition 3.2. Now fix K ≤ cont(f, g). Let (b′_{0}, …, b′_{h′}) be the characteristic of g. Let B′_{k} = GCD(b′_{0}, …, b′_{k}) and let l > 0 be the smallest integer such that K ≤ b′_{l}/b′_{0}. Then

∑ P (g)_{K} = ∑_{i=1}^{l−1} \frac{b′_{i}}{b′_{0}} (B′_{i−1} − B′_{i}) + B′_{l−1}K

Let k > 0 be the smallest integer such that cont(f, g) ≤ b_{k}/b_{0}. Then k ≤ h′, cont(f, g) ≤ b′_{k}/b′_{0} and b_{i}/b_{0} = b′_{i}/b′_{0}, B_{i}/b_{0} = B′_{i}/b′_{0} for i < k and i < l for l ≤ k (recall that K ≤ cont(f, g)).

Hence we get

∑ P (g)_{K} = (∑_{i=1}^{l−1} \frac{b_{i}}{b_{0}} (B_{i−1} − B_{i}) + B_{l−1}K)\frac{b′_{0}}{b_{0}} = \frac{(X, g)_{0}}{(X, f)_{0}} ∑ P (f)_{K}.

Consider a reduced power series f = f_{1} ⋯ f_{r} with r > 1 factors. For every i ∈ { 1, …, r } we put K_{ij} = cont(f_{i}, f_{j}) if i ≠ j and K_{ii} = +∞. We define Puiseux’ date P_{i}(f) of f with respect to the irreducible factor f_{i} by putting

P_{i}(f) = P (f_{1})_{K_{i1}} ⊕ ⋯ ⊕ P (f_{r})_{K_{ir}}.

Proposition 3.4.

|P_{i}(f)| = |P (f_{i})| ∪ { K_{i1}, …, K_{ii}, …, K_{ir} }

Proof. We have |P_{i}(f)| = ⋃_{j=1}^{r} |P (f_{j})_{K_{ij}}|. Let b_{0}, …, b_{h} be the characteristic of the branch f_{i}. Then |P (f_{i})| = { b_{1}/b_{0}, …, b_{h}/b_{0} } and |P (f_{j})_{K_{ij}}| = { b_{1}/b_{0}, …, b_{k(j)−1}/b_{0}, cont(f_{i}, f_{j}) } where k(j) is the smallest integer k > 0 such that cont(f_{i}, f_{j}) ≤ b_{k}/b_{0}.

Proposition 3.5. Let y_{i} ∈ | Zer f_{j}| for some j ∈ { 1, …, r }. Then

⟨ord(y_{i} − y_{1}), …, { \atop ord(y_{i} − y_{i})}, …, ord(y_{i} − y_{p})⟩ = P_{j}(f)

Proof. Proposition 3.5 follows from (3.1), (3.2) and form the definition of P_{j}(f).

4. Proof

A local isomorphism Φ is a pair of power series without constant term such that Jac Φ(0, 0) ≠ 0. It is easy to check the following

Lemma 4.1. Let Φ be a local isomorphism. A rational number q is a polar quotient of f with respect to a regular parameter t if and only if q is a polar quotient of f ∘Φ with respect to t ∘ Φ.

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

Proposition 4.2. Let f_{1} ⋯ f_{r} be a reduced power series with irreducible factors f_{i}. Suppose that f is Y -regular of order p > 0 and let p_{i} : ℝ_{+} → ℝ be given by the formula

p_{i}(K) = ∑ P_{i}(f)_{K} for K > 0.

Then Q(f, X) = ⋃_{i=1}^{r} p_{i}(|P_{i}(f)|).

Proof. Let Zer f = ⟨y_{1}, …, y_{p}⟩ and Zer \frac{∂f}{∂Y} = ⟨z_{1}, …, z_{p−1}⟩. It is easy to see that Q(f, X) = { ord f(X, z_{1}(X)), …, ord f(X, z_{p−1}(X)) }. Indeed, if ϕ is an irreducible factor of \frac{∂f}{∂Y} and z_{i} = z_{i}(X) ∈ Zer ϕ then \frac{(f,ϕ)_{0}}{(X,ϕ)_{0}} = ord f(X, z_{i}(X)).

Let l_{y_{i},y_{j}} (i ≠ j) be the quantities introduced in Section 3 of this paper. It is easy to see that

l_{y_{i},y_{j}} = ∑⟨ord(y_{i} − y_{1}), …, { \atop ord(y_{i} − y_{i})}, …, ord(y_{i} − y_{p})⟩_{ord(y_{i}−y_{j})}

for i ≠ j.

Now suppose that y_{i} ∈ | Zer f_{1}| and let K_{1} = ord(y_{i} − y_{j}) for a j ≠ i. By Proposition 3.5 we get

l_{y_{i},y_{j}} = ∑ P_{1}(f)_{K_{1}} = p_{1}(K_{1})

and consequently { l_{y_{i},y_{j}} : j ≠ i } = { p_{1}(K_{1}) : K_{1} ∈ |P_{1}(f)| } = p_{1}(|P_{1}(f)|).

Similarly { l_{y_{i},y_{j}} : j ≠ i } = p_{2}(|P_{2}(f)|) if y_{i} ∈ | Zer f_{2}|, …, { l_{y_{i},y_{j}} : j ≠ i } = p_{r}(|P_{r}(f)|) if y_{i} ∈ | Zer f_{r}|. Therefore by Theorem 2.3 we get Q(f, X) = { l_{y_{i},y_{j}} : j ≠ i } = p_{1}(|P_{1}(f)|) ∪ ⋯ ∪ p_{r}(|P_{r}(f)|).

Now we can give

Proof of Proposition 1.1. We assume t = X. Let f be an irreducible power series Y -regular of order p > 1. Let (b_{0}, b_{1}, …, b_{h}) be the characteristic of f. According to Proposition 4.2 we have Q(f, X) = p(|P (f)|) where p : ℝ_{+} → ℝ is given by the formula p(K) = ∑ P (f)_{K} for K > 0. Recall that |P (f)| = { b_{1}/b_{0}, …, b_{h}/b_{0} } and p(b_{k}/b_{0}) = ∑ P (f)_{b_{k}/b_{0}} = \frac{B_{k−1}b̄_{k}}{b̄_{0}} by Proposition 3.3(i). Therefore

Q(f, X) = \{ \frac{B_{k−1}b̄_{k}}{b̄_{0}} : k = 1, …, h \}.

Proof of Theorem 1.3. We assume t = X. We need to calculate p_{i}(|P_{i}(f)|) for i = 1, …, r. Without restriction of generality we assume i = 1. Let K > 0. By definition of Puiseux’ date we get

P_{1}(f)_{K} = P (f_{1})_{min(K,K_{11})} ⊕ ⋯ ⊕ P (f_{r})_{min(K,K_{1r})}

Therefore (1)

p_{1}(K) = ∑ P_{1}(f)_{K} = ∑_{j=1}^{r}(∑ P (f_{j})_{min(K,K_{1j})}) = ∑_{j=1}^{r} \frac{(X, f_{j})_{0}}{(X, f_{1})_{0}} (∑ P (f_{1})_{min(K,K_{1j})})

by Proposition 3.3(iii).

It is easy to see that K_{1j} ≤ K if and only if ∑ P (f_{1})_{K_{1j}} ≤ ∑ P (f_{1})_{K}. On the other hand by Proposition 3.3(ii) we get

∑ P (f_{1})_{K_{1j}} = \frac{(f_{1}, f_{j})_{0}}{(X, f_{j})_{0}} .

Let p_{11}(K) = ∑ P (f_{1})_{K}. Thus K_{1j} ≤ K if and only if \frac{(f_{1},f_{j})_{0}}{(X,f_{j})_{0}} ≤ p_{11}(K). By definition of the set J_{1}(τ) we see that K_{1j} ≤ K if and only if j ∈ J_{1}(p_{11}(K)). Then, we can rewrite (1) as follows

p_{1}(K) =

∑_{j∈J_{1}^{c}(p_{11}(K))} \frac{(X, f_{j})_{0}}{(X, f_{1})_{0}} (∑ P (f_{1})_{min(K,K_{1j})}) + ∑_{j∈J_{1}(p_{11}(K))}(∑ P (f_{1})_{min(K,K_{1j})}) =

∑_{j∈J_{1}^{c}(p_{11}(K))} \frac{(X, f_{j})_{0}}{(X, f_{1})_{0}} p_{11}(K) + ∑_{j∈J_{1}(p_{11}(K))} \frac{(X, f_{j})_{0}}{(X, f_{1})_{0}} \frac{(f_{1}, f_{j})_{0}}{(X, f_{j})_{0}} =

( ∑_{j∈J_{1}^{c}(p_{11}(K))} \frac{(X, f_{j})_{0}}{(X, f_{1})_{0}} )p_{11}(K) + ∑_{j∈J_{1}(p_{11}(K))} \frac{(f_{1}, f_{j})_{0}}{(X, f_{1})_{0}} = q_{1}(p_{11}(K)).

Recall that |P_{1}(f)| = |P (f_{1})| ∪ { K_{12}, …, K_{1r} }. Thus p_{11}(|P_{1}(f)|) = Q(f_{1}, X) ∪ H_{1}(f, X). Therefore p_{1}(|P_{1}(f)|) = q_{1}(p_{11}(|P_{1}(f)|)) = q_{1}(Q(f_{1}, X) ∪ H_{1}(f, X)) and the theorem follows from Proposition 4.2.

References

[CA] E. Casas–Alvero, Singularities of plane curves, Cambridge Univ. Press (to appear).

[D] F. Delgado de la Mata, A factorization theorem for the polar of a curve with two branches, Compositio Math. 92 (1994), 327–375.

[E] F. H. Eggers, Polarinvarianten un die Topologie von Kurvensingularitaen, Bonner Mathematische Schriften 147 (1983).

[Eph] R. Ephraim, Special polars and curves with one place at infinity, Proc. of Symp. in Pure Math. 40 Part 1 (1983), 353–359.

[Ga] E. Garcia–Barroso, Sur les courbes polaires d’une courbe plane réduite, Proc. of the London Math. Soc. 3(81) (2000), 1–28.

[GP1] J. Gwoździewicz, A. Płoski, On the Merle formula for polar invariants, Bull. Soc. Sci. Letters Łódź 41(7) (1991), 61–67.

[GP2] J. Gwoździewicz, A. Płoski, On the approximate roots of polynomials, Ann. Pol. Math. LX.3 (1995), 199–210.

[KL] T. C. Kuo, Y. C. Lu, On analytic function germs of two complex variables, Topology 16 (1987), 299–310.

[LMW1] Lê Dung Trang, F. Michel, C. Weber, Sur le comportement des polaires associées aux germes des courbes planes, Compositio Math. 72 (1989), 88–113.

[LMW2] Lê Dung Trang, F. Michel, C. Weber, Courbes polaires et topologie des courbes planes, Ann. scient Éc. Norm. Sup. 4^{e} série, t.24 (1991), 141–169.

[LP] A. Lenarcik, A. Płoski, Polar invariants of plane curves and the Newton polygon, Kodai. Math. J. 23 (2000), 309–319.

[M] M. Merle, Invariants polaires des courbes planes, Invent. Math. 41 (1977), 103–111.

[P] A. Płoski, On the maximal polar quotient of an analytic plane curve, Kodai. Math. J. (to appear).

[T] B. Teissier, Variétés polaires, Invent. Math 40 (1977), 267–292.

[Wa] R. Walker, Algebraic curves, Princeton, 1953.

[Wh] H. Whitney, Complex analytic varieties, Addison-Wesley, 1972.

A.P. Technical University, Al. 1000 LPP 7, 25–314 Kielce, Poland E-mail address: matap@tu.kielce.pl

J.G. Technical University, Al. 1000 LPP 7, 25–314 Kielce, Poland E-mail address: matjg@tu.kielce.pl