FACTORIZATION OF THE POLAR CURVE AND THE NEWTON POLYGON

Andrzej Lenarcik^{∗}, Mateusz Masternak and Arkadiusz Płoski^{∗∗}

IMUJ PREPRINT 2002/15 KRAKÓW

Abstract

Using the Newton polygon we prove a factorization theorem for the local polar curves. Then we give some applications to the polar invariants and pencils of plane curve singularities.

Introduction

Let C{X, Y } be the ring of complex power series in two variables X, Y. We denote by ord f and in f respectively the order and the initial form of a nonzero power series f ∈ C{X, Y }. By definition ord 0 = +∞ and in 0 = 0. Let f be a nonzero power series without constant term. If f = f_{1}^{m_{1}} … f_{r}^{m_{r}} is a decomposition of f into irreducible pairwise different factors f_{i} ∈ C{X, Y } then we put f_{red} = f_{1} … f_{r}. Let t(f) = ord(in f)_{red} . Then t(f) is the number of tangents to the local curve f = 0.

2000 Mathematics Subject Classification: Primary 32S55. Key words and phrases: plane curve singularity, polar invariants, Newton polygon, special value.

∗ Supported in part by the NATO Science Fellowship Program. ∗∗ Supported in part by the KBN grant No 2 P03 A 01522.

Write

in f = (a monomial) ∏_{i=1}^{s}(X − c_{i}Y)^{m_{i}}

with pairwise different c_{i}. We put

d(f) = ∑_{i=1}^{s}(m_{i} − 1)

and call d(f) degeneracy of f. If s = 0 then in f reduces to a monomial and d(f) = 0. Note that d(f) = 0 if and only if all tangents to f = 0 different from the axes are of multiplicity 1.

Assume that f has an isolated singularity at (0, 0) ∈ C^{2} (this is equivalent to the conditions ord f > 1 and f = f_{red} ) and suppose that the line bX−aY = 0 is not tangent to f = 0. The generic polar of f is by definition the series ∂f = a(∂f/∂X) + b(∂f/∂Y).

Let us consider the factorization ∂f = ∏_{i=1}^{u} h_{i} with irreducible h_{i} ∈ C{X, Y } and put (f, h)_{0} = dim_{C}C{X, Y }/(f, h). According to Teissier [Te1] the quotients (f, h_{i})_{0}/ord h_{i} are called polar invariants of the singularity f. The multiplicity m_{q} of the polar quotient q is defined to be m_{q} = ∑_{i∈I_{q}} ord h_{i} where I_{q} = {i : (f, h_{i})_{0}/ord h_{i} = q}.

Teissier’s collection {(q, m_{q})} of polar invariants and their multiplicities is a topological invariant of the singularity (see [Te1] and [Te2]). There are several theorems on the factorization of the polar curve that enable calculation of Teissier’s collection (see [M], [D], [G], [LP]). The aim of this note is to study the factorization of the polar curve ∂f associated with the Newton polygon N_{f}. The main result is a refinement of the factorization theorem given in [LP]. Using our theorem we prove a bound on the number of special values of the pencil (f − tl^{N} : t ∈ C). This bound is analogous to the estimation due to Le Van Thanh and Mutsuo Oka (see [LO], Main result) given in the global affine context.

1 Main result

We use the notation introduced in [LP]. In particular we denote by N_{f} = N (f) the set of all segments of the Newton polygon of the series f = ∑ c_{αβ}X^{α}Y^{β} ∈ C{X, Y }. For every S ∈ N_{f} we denote by |S|_{1} and |S|_{2} the lenghts of the projection of S on the horizontal and vertical axes. We call |S|_{1}/|S|_{2} inclination of the segment S. The power series f is elementary if N_{f} contains only one segment with vertices on the axes. Let ||S|| = min {|S|_{1}, |S|_{2}} and denote a_{S}, b_{S} the distances from S to the axes. Thus the vertices of S are (a_{S}, |S|_{2} +b_{S}) and (|S|_{1} +a_{S}, b_{S}). Let α/α(S)+β/β(S) = 1 be the equation of the line containing S. Clearly α(S), β(S) are rational numbers and α(S)/β(S) = |S|_{1}/|S|_{2}. A segment S ∈ N_{f} is exceptional if 1 = |S|_{1} < |S|_{2} and a_{S} = 0 or 1 = |S|_{2} < |S|_{1} and b_{S} = 0. A segment S ∈ N_{f} (necessarily unique) is principal if |S|_{1} = |S|_{2}. We set N_{f}^{∗} = N_{f} \ {exceptional segments} and N_{f}^{∗∗} = N_{f}^{∗} \ {principal segment}. For every segment S ∈ N_{f}^{∗} we define ε(S) ∈ {−1, 0, 1} by putting ε(S) = −1 is |S|_{1} < |S|_{2} and a_{S} = 0 or |S|_{2} < |S|_{1} and b_{S} = 0. If |S|_{1} = |S|_{2} then ε(S) = 1 − (number of verticies of S lying on the axes). We put ε(S) = 0 for all remaining cases. A segment S ∈ N_{f}^{∗∗} is of the first kind if ε(S) = 0, it is of the second kind if ε(S) = −1.

Let in(f, S) = ∑_{(α,β)∈S} X^{α}Y^{β}. Clearly X^{a_{S}}Y^{b_{S}} is the monomial of the highest degree dividing in(f, S). Thus we can write in(f, S) = X^{a_{S}}Y^{b_{S}} în(f, S) in C{X, Y }. Note that N (în(f, S)) = {S′} where S′ is the segment with vertices (|S|_{1}, 0) and (0, |S|_{2}). We define the degeneracy d(f, S) of f on S by putting d(f, S) = ord în(f, S) − ord în(f, S)_{red} . Note that d(f, S) = 0 if and only if f is nondegenerate on S that is if the polynomial in(f, S) has no critical points in the set (C \ {0}) × (C \ {0}). If S ∈ N_{f}^{∗∗} is of the second kind then we let v_{S} = X if |S|_{1} < |S|_{2} and v_{S} = Y if |S|_{2} < |S|_{1}. Let S be a segment of a Newton polygon. We call a power series S-elementary if it is elementary and its unique segment is parallel to S. Let us state the main result

Theorem 1.1 Let f = f(X, Y) ∈ C{X, Y } be a power series with an isolated singularity at (0, 0) ∈ C^{2}. Then for every line bX − aY = 0 not tangent to the curve f = 0 there is a factorization of the polar ∂f = a(∂f/∂X) + b(∂f/∂Y):

∂f = A B ∏_{S∈N_{f}^{∗∗}} A_{S}B_{S} in C{X, Y }

such that (i) ord A = t(f) − 1, ord B = d(f). If h is an irreducible factor of A B

then (f, h)_{0}/ord h ≥ ord f with equality if and only if h divides A.

(ii) ord A_{S} = ||S|| + ε(S) − d(f, S), ord B_{S} = d(f, S). If h is an irreducible factor of A_{S}B_{S} then (f, h)_{0}/ord h ≥ max (α(S), β(S)) with equality if and only if h divides A_{S}.

(iii) If ord B_{S} > 0 then B_{S} is S-elementary. If ord A_{S} > 0 and S is of the first kind then A_{S} is S-elementary.

(iv) If ord A_{S} > 0 and S is of the second kind then there is a factorization A_{S} = A′_{S}A′′_{S} such that

· if ord A′_{S} > 0 then A′_{S} is S-elementary.

·· If ord A′′_{S} > 0 then every barier of the Newton diagram of f parallel to a segment of N (A′′_{S}) passes through the vertex of S lying on the axis v_{S} = 0. If |S|_{1} < |S|_{2} (resp. |S|_{2} < |S|_{1}) then for every T ∈ N (A′′_{S}): |T |_{1}/|T |_{2} < |S|_{1}/|S|_{2} (resp. |S|_{1}/|S|_{2} < |T |_{1}/|T |_{2}).

The proof of Theorem 1.1 we give in Section 6 of this note.

Corollary 1.2 Let f = f(X, Y) ∈ C{X, Y } be a power series with an isolated singularity at (0, 0) ∈ C^{2} such that N_{f}^{∗∗} ≠ ∅. Then

(i) Let S ∈ N_{f}^{∗∗} be of the first kind. Then max (α(S), β(S)) is a polar invariant of the curve f = 0. Its multiplicity is at least ||S|| − d(f, S).

(ii) If S ∈ N_{f}^{∗∗} is of the second kind then max (α(S), β(S)) is a polar invariant of f if and only if ord în(f, S)_{red} > 1. Its multiplicity is at least ||S|| − d(f, S) − 1. If ord în(f, S)_{red} = 1 then there is a polar invariant strictly greater than max (α(S), β(S)).

Proof. Fix a segment S ∈ N_{f}^{∗∗}. It is easy to check that ord A_{S} = ||S|| + ε(S) − d(f, S) = ord în(f, S)_{red} + ε(S). If S is of the first kind then ε(S) = 0 and (i) follows. If S is of the second kind then max (α(S), β(S)) is a polar invariant if and only if ord A_{S} = ord în(f, S)_{red} − 1 > 0.

Example 1.3 Let f(X, Y) = (Y − X^{2})^{2} + X^{5}. Then N_{f} = {S} where S is the segment with vertices (0, 2) and (4, 0). Clearly în(f, S)_{red} = Y − X^{2} is of order 1. According to Corollary 1.2 we can only say that the curve f = 0 has a polar invariant greater than max (α(S), β(S)) = max (2, 4) = 4. Taking the new system of coordinates X_{1} = X, Y_{1} = Y −X^{2} we get f_{1}(X_{1}, Y_{1}) = Y_{1}^{2}+X_{1}^{5} and using 1.2 to f_{1} in coordinates (X_{1}, Y_{1}) we get that there is a unique polar invariant equal to 5.

Example 1.4 Let f(X, Y) = Y^{11} + XY^{8} − 2X^{2}Y^{6} + X^{3}Y^{4} − 2X^{4}Y^{3} + X^{5}Y^{2} − 2X^{7}Y + X^{9}. Then N_{f} = {E, S, U, T } where |E|_{1}/|E|_{2} < |S|_{1}/|S|_{2} < |U|_{1}/|U|_{2} < |T |_{1}/|T |_{2}. Here E is exceptional, U is principal and N_{f}^{∗∗} = {S, T } where S is of the first kind (ε(S) = 0) and T is of the second kind (ε(T) = −1). According to Theorem 1.1 there is a factorization ∂f = ABA_{S}B_{S}A_{T} B_{T} in C{X, Y } where ord A = t(f) − 1 = 2, ord B = d(f) = 1, ord A_{S} = ||S|| − d(f, S) = 1, ord B_{S} = d(f, S) = 1, ord A_{T} = ||T || − 1 − d(f, T) = 0, ord B_{T} = d(f, T) = 1. We may assume that A_{T} = 1 in C{X, Y } for A_{T} is a unit. The polar ∂f = 0 consists of the curve A = 0 of order 2 transverse to the curve f = 0 and of four nonsingular branches A_{S} = 0, B = 0, B_{S} = 0 and B_{T} = 0. The polar invariants are ord f = 7 (of multiplicity 2), (f, A_{S})_{0} = max (α(S), β(S)) = 10 and the numbers (f, B)_{0} > 7, (f, B_{S})_{0} > 10 and (f, B_{T}) > max (α(T), β(T)) = 9. The theorem does not give information if the invariants (f, B)_{0}, (f, B_{S})_{0} and (f, B_{T})_{0} are equal or not.

2 Contact exponent and minimal polar invariant

Let f = f_{1} … f_{r} be an isolated singularity with branches f_{i} = 0 and l = 0 be a smooth curve (that is l is a series of order 1). Then we consider the contact exponent of l = 0 with f = 0

δ(f, l) = \begin{matrix}r \\ min \\ i = 1 \end{matrix} \{\frac{(f_{i}, l)_{0}}{ord f_{i}} \}

and the contact exponent of f = 0:

δ(f) = max {δ(f, l) : l = 0 runs over the set of nonsingular curves different from the branches f_{i} = 0}

(see [BK] for Hironaka’s theory of maximal contact).

Clearly δ(f) ≥ 1 and δ(f) = 1 if and only if t(f) > 1.

Theorem 2.1 Let f = f(X, Y) ∈ C{X, Y } be a power series with an isolated singularity at (0, 0) ∈ C^{2}. Then

(i) if t(f) > 1 then the minimal polar invariant of f = 0 is equal to ord f and its multiplicity is t(f) − 1.

(ii) Suppose that t(f) = 1 and δ(f, Y) = δ(f). Let F be the first segment of the Newton polygon N_{f}. Then the minimal polar invariant of f = 0 is equal to α(F) and its multiplicity is ||F || − ε(F) − d(F, f).

(iii) The minimal polar invariant of the singularity f = 0 is equal to (ord f)δ(f).

Proof. Part (i) of the theorem follows from Theorem 1.1 (i). To check (ii) observe that from the assumptions it follows that the axis X = 0 is transverse to the curve f = 0. The Newton diagram of f has the vertex (0, ord f) and lies strictly above the line α + β = ord f. Hence all segments of N_{f} have the inclination strictly greater than 1. In particular |F |_{1} > |F |_{2}. Recall that |F |_{1}/|F |_{2} = δ(f, Y) = δ(f) and consider two cases.

Case 1. The power series f is not elementary. Then the segment F is of the first kind and α(F) is a polar invariant of f = 0. Using Theorem 1.1 we check that all polar invariants of f different from α(F) are strictly greater than α(F). Thus α(F) is the minimal polar invariant of f and its multiplicity equals ||F || + ε(F) − d(f, F).

Case 2. The power series f is elementary. Then F is the unique segment of N_{f}. Using the criterion of maximal contact (see [BK], Lemma 5, p. 649) we get that in(f, F) is not of the form (bY − aX^{k})^{m}, ab ≠ 0. Therefore by our main result α(F) = max (α(F), β(F)) is the minimal polar invariant of f and its multiplicity is ||F || + ε(F) − d(f, F).

To check (iii) we note that α(F)/ord f = α(F)/β(F) = |F |_{1}/|F |_{2} = δ(f) and use (ii).

Example 2.2 Suppose f is an irreducible power series with characteristic β_{0}, β_{1}, …, β_{g} (see for example [M]). If the axis Y = 0 has the maximal contact with f = 0 then N_{f} = {F } where F is the segment with vertices (0, β_{0}) and (β_{1}, 0). Let e_{1} = GCD(β_{0}, β_{1}). Then in(f, S) = (bX^{β_{1}/e_{1}} − cY^{β_{0}/e_{1}})^{e_{1}} with bc ≠ 0 and an easy calculation shows that the minimal polar invariant equals max (β_{0}, β_{1}) = β_{1} and is of multiplicity β_{1}/e_{1} − 1 (here ε(F) = −1). Thus we have got the first of Merle’s formulas [M].

The reasoning like that in the proof of Theorem 2.1 shows

Theorem 2.3 Suppose that f = 0 has exactly one polar invariant. If δ(f, Y) = δ(f) and (f, X)_{0} = ord f then N_{f} = {F } and f is nondegenerate on F. The segment F has vertices (0, n) and (m, 0) or (0, n) and (m, 1) with m ≥ n.

Corollary 2.4 ([E], p. 16) If f = 0 has exactly one polar invariant then f = 0 is topologically equivalent to a plane curve singularity of type Y^{n} − X^{m} = 0 or of the type Y^{n} − Y X^{m} = 0.

3 Special values of plane curve pencils

When studying the singularities at infinity of a polynomial in two complex variables of degree N > 0 one considers the pencils of plane curves of the form f_{t} = f − tl^{N}, t ∈ C where f and l = bX − aY are coprime (such pencils are called in [C] Iomdin Lê deformations). Let μ_{0}(f) = (∂f/∂X, ∂f/∂Y)_{0} be the Milnor number of the local curve f = 0. Recall that μ_{0}(f) = +∞ if and only if f has a multiple factor. The number t_{0} ∈ C is a special value of the pencil (f_{t}, t ∈ C) if μ_{0}(f_{t_{0}}) > inf{μ_{0}(f_{t}) : t ∈ C}. The set of special values is finite. Using our main result we will give a bound on the number of special values in terms of the Newton diagram of the series. Let r(f, S) be the number of irreducible factors of în(f, S) and put r(S) = GCD(|S|_{1}, |S|_{2}).

Lemma 3.1 One has r(S) − r(f, S) = (r(S)/||S||)d(f, S). In particular r(f, S) ≤ r(S) with equality if and only if f is nondegenerate on S.

Proof. Write

în(f, S) = ∏_{i=1}^{r}(b_{i}X^{|S|_{1}/r(S)} − a_{i}Y^{|S|_{2}/r(S)})^{m_{i}}

with pairwise linearly independent (a_{i}, b_{i}) ∈ C. Then r(f, S) = r and r(S) = ∑_{i=1}^{r} m_{i}. Now

d(f, S)=ord în(f, S) − ord în(f, S)_{red}
=∑_{i=1}^{r} m_{i} \frac{||S||}{r(S)} − ∑_{i=1}^{r} \frac{||S||}{r(S)} = \frac{||S||}{r(S)}(r(S) − r(f, S))

and the lemma follows.

The following result is a local counterpart of the Le Van Thanh and Oka theorem giving an estimation for the number of critical values at infinity (see [LO], Main Theorem).

Let q(S) = max (α(S), β(S)) for any S ∈ N_{f}^{∗}. We put l = bX − aY and suppose that the line l = 0 is not tangent to f = 0.

Theorem 3.2 Let N ≠ ord f be a strictly positive integer. The number of the nonzero special values of the pencil (f − tl^{N} : t ∈ C) is less than or equal to

∑_{S: q(S)<N}(r(S) − r(f, S)) + ∑_{S: q(S)=N} r(f, S) .

We use the convention that a sum over the empty set equals zero. If f is a nondegenerate power series then the sum above reduces to

∑_{S: q(S)=N} r(S) .

Note also the bound for all series with the given Newton polygon.

Corollary 3.3 The number of the nonzero special values of (f −tl^{N} : t ∈ C) (N ≠ ord f) is less than or equal to

∑_{S: q(S)≤N} r(S) .

In connection with the above corollary recall the following well-known fact: the number of branches of the curve f = 0 different from the axes is less or equal to ∑_{S} r(S) (with equality for nondegenerate curves).

To get Theorem 3.2 from the main result we need a few lemmas. The lemma below is a local version of the description of critical values at infinity given in [LO] (pp. 410–411). Let f/h be a meromorphic fraction with coprime f, h ∈ C{X, Y } and let p = p(X, Y) ∈ C{X, Y } be irreducible power series such that p does not divide h. Let (x(u), y(u)) ∈ C{u}^{2}, (x(0), y(0)) = (0, 0) be a parametrization of the branch p = 0. Then we put

(\frac{f}{h} ) (p) = \frac{f(x(u), y(u))}{l(x(u), y(u))}|_{u=0} ∈ C ∪ \{∞\}.

Lemma 3.4 The set of nonzero special values of the pencil (f − tl^{N} : t ∈ C) is equal to the set

{(f/l^{N})(p) : p is irreducible factor of j(f, l) such that (f, p)_{0}/(l, p)_{0} = N} .

Proof. See [MM] Théorème 1 or [GB-P] Proposition 2.2.

Let r_{0}(ϕ) be the number of irreducible factors of the series ϕ.

Lemma 3.5 Suppose that ϕ is S-elementary. Then

r_{0}(ϕ) ≤ (ord ϕ)\frac{r(S)}{||S||} .

Proof. Let r = r_{0}(ϕ). Then ϕ = ∏_{i=1}^{r} ϕ_{i} with irreducible ϕ_{i}. The power series ϕ_{i} are S-elementary. Therefore the unique segment of N (ϕ_{i}) joins the points (k_{i}|S|_{1}/r(S), 0) and (0, k_{i}|S|_{2}/r(S)) for an integer k_{i} ≥ 1. Consequently

ord ϕ_{i} = min (\frac{|S|_{1}}{r(S)}k_{i}, \frac{|S|_{2}}{r(S)}k_{i}) ≥ \frac{||S||}{r(S)}

for all i = 1, …, r. We get

ord ϕ = ∑_{i=1}^{r} ord ϕ_{i} ≥ \frac{||S||}{r(S)}r_{0}(ϕ)

and the lemma follows.

Lemma 3.6 Let us keep the notation from Theorem 1.1. Then

(i) If ord B > 0 then N_{f} has the principial segment U and ord B = r(U)− r(f, U).

(ii) If ord B_{S} > 0 then r_{0}(B_{S}) ≤ r(S) − r(f, S).

(iii) If ord A_{S} > 0 and S is of the first kind then r_{0}(A_{S}) ≤ r(f, S).

Proof.

(i) It is easy to see that if d(f) > 0 then N_{f} has the principal segment U and d(f) = r(U) − r(f, U). Use Theorem 1.1 (i).

(ii) Suppose that ord B_{S} > 0. Then by Theorem 1.1 (ii) we get ord B_{S} = d(f, S). Now Lemmas 3.5 and 3.1 give

r_{0}(B_{S}) ≤ (ord B_{S})\frac{r(S)}{||S||} = d(f, S)\frac{r(S)}{||S||} = r(S) − r(f, S) .

(iii) Suppose that ord A_{S} > 0 and S is of the first kind. Then ordA_{S} = ||S|| − d(f, S) by Theorem 1.1 (ii) and using Lemmas 3.5 and 3.1 we get

r_{0}(A_{S}) ≤ (ord A_{S})\frac{r(S)}{||S||} = (||S|| − d(f, S))\frac{r(S)}{||S||} = r(f, S) .

Lemma 3.7 Suppose that ord A_{S} > 0 for a segment S ∈ N_{f}^{∗∗} of the second kind. Let N = max (α(S), β(S)). Let A_{S} = A′_{S}A′′_{S} be the factorization of A_{S} such that in Theorem 1.1 (iv). Then

(i) r_{0}(A′_{S}) ≤ r(f, S) − 1,

(ii) for every prime factor p of A′′_{S}: (f/l^{N})(p) = (f/l^{N})(v_{S}).

Proof. By Theorem 1.1 (ii) we get ord A_{S} = ||S||−1−d(f, S) (ε(S) = −1 for the segments of second kind) and consequently, like in the proof of Lemma 3.7 we obtain

r_{0}(A′_{S}) ≤ (ord A′_{S})\frac{r(S)}{||S||} ≤ (ord A_{S})\frac{r(S)}{||S||} = r(f, S) − \frac{r(S)}{||S||} < r(f, S) .

Since r_{0}(A′_{S}) and r(f, S) are integers we get r_{0}(A′_{S}) ≤ r(f, S) − 1. To prove the second part of Lemma 3.7 assume that S = F is the first segment of N_{f} (if S = L is the last segment the proof is similar). Then v_{S} = v_{F} = X. Let p be a prime factor of A′′_{F}. We may assume that the branch p = 0 is different from the axis X = 0. Note that \frac{|F_{1}|}{|F_{2}|} < 1 and N = β(F). Let (x(u), y(u)) be the injective parametrization of the branch p = 0. Put m = ord x(u) and n = ord y(u). The series p is elementary, the unique segment of N_{p} joins the points (n, 0) and (0, m) and is of inclination n/m ≤ |F |_{1}/|F |_{2} < 1 by Theorem 1.1 (iv). The line supporting the Newton diagram of f of slope −m/n passes through the point (0, β(F)) = (0, N) and, consequently, has the equation mα + nβ = nN. It intersects the Newton diagram of f exactly at point (0, N). Therefore

f(X, Y) = c_{0N}Y^{N} + ∑_{mα+nβ>nN} c_{αβ}X^{α}Y^{β} with c_{0N} ≠ 0 .

The line l(X, Y) = bX − aY is not tangent to f = 0. Then a ≠ 0 and

f(x(u), y(u))=c_{0N}y(u)^{N} + terms of order > nN
l(x(u), y(u))=(−a)^{N}y(u)^{N} + terms of order > nN

Consequently

(\frac{f}{l^{N}} ) (p) = \frac{c_{0N}}{(−a)^{N}} = (\frac{f}{l^{N}} ) (X) .

Now we give the proof of Theorem 3.2.

Let

∂f = AB ∏_{S∈N_{f}^{∗∗}} A_{S}B_{S}

be a factorization of ∂f such that in Theorem 1.1.

According to Lemmas 3.4 and 3.7 the number of nonzero special values of (f − tl^{N} : t ∈ C) is equal to

#{(f/l^{N})(p) : p is a prime factor of ∂f and (f, p)_{0}/ord p = N}

≤ ord B + ∑_{S: q(S)<N} r_{0}(B_{S}) + ∑_{S: q(S)=N}^{I} r_{0}(A_{S}) + ∑_{S: q(S)=N}^{II} (r_{0}(A′_{S}) + 1)

where the symbols ∑^{I} resp (∑^{II}) mean that the summation is carried over the segments of the first kind (of the second kind). The theorem follows from Lemmas 3.6 and 3.7.

Remark 3.8 An obvious modification of the above proof shows that the pencil (f − tl^{ordf} : t ∈ C) has at most t(f) − 1 nonzero special values.

4 Preliminary lemmas

Let φ = φ(X, Y) ∈ C{X, Y } be a nonzero power series without constant term and λ = bX − aY be a linear form. Put ∂φ = a(∂φ/∂X) + b(∂φ/∂Y) (we do not assume that inφ(a, b) ≠ 0!) and note that ord ∂φ = ord φ − 1 if and only if in φ ≠ const.λ^{ord φ}.

Moreover ∂φ = 0 in C{X, Y } if and only if φ = φ^{0}(λ) where φ^{0} is a power series of one variable. In particular if ∂φ = 0 then λ divides φ.

Lemma 4.1 Suppose that ord ∂φ = ord φ − 1 and let k ≥ 0 be the greatest integer such that λ^{k} divides φ. Let φ = λ^{k}φ_{1}^{m_{1}} … φ_{s}^{m_{s}} with irreducible and pairwise coprime φ_{i} ∈ C{X, Y }. Then ∂φ = λ^{k}φ_{0}φ_{1}^{m_{1}−1} … φ_{s}^{m_{s}−1} in C{X, Y } and φ, φ_{0} are coprime.

Proof. It is easy to check that λ^{k}, φ_{1}^{m_{1}−1}, …, φ_{s}^{m_{s}−1} divide ∂φ and are pairwise coprime. Thus we can write ∂φ = λ^{k}φ_{0}φ_{1}^{m_{1}−1} … φ_{s}^{m_{s}−1} in C{X, Y }. If φ_{i} were a factor of φ_{0} then we would get that φ_{i} didides ∂φ_{i}. This implies ∂φ_{i} = 0. Impossible because λ does not divide φ_{i}. Similarly we check that λ does not divide φ_{0}.

Remark It is easy to check that ord φ_{0} = ∑_{i=1}^{s} ord φ_{i} − 1 .

The following is well-known

Lemma 4.2 If in f = φ_{1}^{m_{1}} … φ_{t}^{m_{t}} with φ_{i} linear pairwise linearly independent then f = f_{1} … f_{t} in C{X, Y } and in f_{i} = φ_{i}^{m_{i}} for i = 1, …, t.

Lemma 4.3 Let ∂f = a(∂f/∂X) + b(∂f/∂Y) with (a, b) ∈ C^{2} such that in f(a, b) ≠ 0. Then ∂f = A Ã in C{X, Y } where ord A = t(f) − 1, ord Ã = ord f − t(f) and for every irreducible factor h of ∂f: (f, h)_{0}/ord h = ord f if and only if h divides A.

Proof. Let in f = φ_{1}^{m_{1}} … φ_{t}^{m_{t}}, φ_{i} linear and t = t(f). Then in(∂f) = ∂(in f) = φ_{0}φ_{1}^{m_{1}−1} … φ_{t}^{m_{t}−1} in C{X, Y } with coprime φ_{0}, in f. By Lemma 4.2 we get the factorization ∂f = g_{0}g_{1} … g_{t} where in g_{0} = φ_{0} and in g_{i} = φ_{i}^{m_{i}−1} for i = 1, …, t. By Remark to Lemma 4.1 we get

ord g_{0} = ∑_{i=1}^{t} ord φ_{i} − 1 = t − 1 = t(f) − 1 .

Put A = g_{0}, Ã = g_{1} … g_{t} .

5 Newton polygon and factorization of polar series

Let us keep the notation introduced above. The following two lemmas are well-known.

Lemma 5.1 Let f = f(X, Y) be a nonzero power series without constant term. Then there is a factorization

f = uX^{d_{1}}Y^{d_{2}} ∏_{S∈N_{f}} f_{S} in C{X, Y }

where u is a unit, such that

(i) N (f_{S}) = {S′} where S′ is the segment with vertices (|S|_{1}, 0) and (0, |S|_{2}),

(ii) in(f_{S}, S′) = const.în(f, S)

Lemma 5.2 Suppose that N (f) = {S} where S is a segment with vertices on the axes. Suppose that in(f, S) = ψ_{1} … ψ_{m} with coprime ψ_{i}. Then there is a factorization

f = f_{1} … f_{m}

such that

(i) N (f_{i}) = {S^{(i)}} where S^{(i)} is a segment parallel to S,

(ii) in(f_{i}, S^{(i)}) = ψ_{i} for i = 1, …, m .

6 Proof of the main result

We will prove our theorem for polars ∂f = a(∂f/∂X) + b(∂f/∂Y) such that ab in f(a, b) ≠ 0. If a = 0 or b = 0 but in f(a, b) ≠ 0 then the proof needs some modifications (see [LP],p. 318). By Lemma 5.1 we may write

(1) ∂f = uX^{δ_{1}}Y^{δ_{2}} ∏_{T ∈N (∂f)}(∂f)_{T} in C{X, Y } where u is a unit and

(2) (∂f)_{T} is an elementary power series; N ((∂f)_{T}) = {T′} where T′ is the segment with vertices (|T |_{1}, 0) and (0, |T |_{2}),

(3) in((∂f)_{T} , T′) = const. în(∂f, T).

The proposition below is already proved in [LP] but it is not stated there explicitly.

Proposition 6.1 Suppose that f ∈ C{X, Y } has an isolated singularity at (0, 0) ∈ C^{2}. Then there is a factorization

∂f = g ∏_{S∈N_{f}^{∗∗}} g_{S} in C{X, Y }

such that

(i) ord g = d(f) + t(f) − 1, ord g_{S} = ||S|| + ε(S) for S ∈ N_{f}^{∗∗},

(ii) If S ∈ N_{f}^{∗∗} is a segment of the first kind then there is a segment T ∈ N_{∂f} (necessarily unique) parallel to S. We have g_{S} = (∂f)_{T}.

(iii) Suppose that S ∈ N_{f}^{∗∗} is a segment of the second kind. Then

(α) for every T ∈ N (g_{S}) the power series (∂f)_{T} divides g_{S}.

(β) If |S|_{1} < |S|_{2} (resp. |S|_{2} < |S|_{1}) then for every T ∈ N (g_{S}) :

|T |_{1}/|T |_{2} ≤ |S|_{1}/|S|_{2} (resp. |S|_{1}/|S|_{2} ≤ |T |_{1}/|T |_{2}).

(γ) Every barier of the Newton diagram of f parallel to a segment of N (g_{S}) passes through the vertex of S lying on the axis v_{S} = 0. (δ) If there is no segment of N (g_{S}) parallel to a segment of N_{f}^{∗∗} then

d(f, S) = 0.

Proof. We may assume that N_{f}^{∗∗} ≠ ∅. According to [LP], Theorem 1.1 p. 310 there is a factorization

(4) ∂f = v ∏_{S∈N_{f}^{∗∗}} g_{S} in C{X, Y }

such that

(5) ord g_{S} = ||S|| + ε(S) for every S ∈ N_{f}^{∗},

(6) if ord v > 0 then ord v = 1 and (f, v)_{0} = ord f.

Moreover, by the definition of v given in [LP], p. 317 we have

(7) ordv > 0 if and only if inf = const. X^{α_{0}}Y^{β_{0}} for some α_{0}, β_{0} > 0.

To define the power series g we consider two cases.

Case 1. The initial form inf is not a monomial. Then there exists the principal segment U ∈ N_{f}^{∗} and inf = in(f, U). It is easy to see that ord în(f, U)_{red} = t(f) − 1 − ε(U). Consequently d(f) = d(f, U) = ||U|| − ord în(f, U)_{red} = ||U|| + ε(U) − (t(f) − 1) and ord g_{U} = ||U|| + ε(U) = d(f) + t(f) − 1 by (5). Put g = vg_{U}. Note that v is a unit by (7) hence ord g = ord g_{U} = d(f) + t(f) − 1.

Case 2. The initial form inf is a monomial. If in f = const.X^{ordf} or in f = const. Y^{ord f} then d(f) + t(f) − 1 = 0. We put g = v. By (7) v is a unit and consequently ord g = 0. If in f = const. X^{α_{0}}Y^{β_{0}} with α_{0} > 0 and β_{0} > 0 then d(f) + t(f) − 1 = 1. We put g = v. By (6) and (7) we get ord g = 1.

By the definition of the series g we can rewrite (4) in the form

(8) ∂f = g ∏_{S∈N_{f}^{∗∗}} g_{S}, ordg = d(f) + t(f) − 1

The conditions (ii) and (iii) (α), (β), (γ) follow immediately from the definition of g_{S} given in [LP] p. 317. To check (iii) (δ) we observe that by [LP], Lemma 2.3 the initial form in(f, S) is the sum of two monomials. Thus d(f, S) = 0.

Proposition 6.2 Let S ∈ N_{f}^{∗∗} and T ∈ N_{∂f} be parallel. Then there is a factorization

(∂f)_{T} = A_{S}B_{S} in C{X, Y }

such that

(i) If ord A_{S} > 0 then A_{S} is S-elementary. Suppose ord A_{S} > 0 and let N (A_{S}) = {S̃}. Then the power series in(A_{S}, S̃) and in(f, S) are coprime.

(ii) We have ord B_{S} = d(f, S). If ord B_{S} > 0 then B_{S} is S-elementary. Suppose ord B_{S} > 0 and let N (B_{S}) = { ˜S̃}. Then the power series in(B_{S}, ˜ S̃) divides in(f, S).

Proof. Let ∂ = a(∂/∂X) + b(∂/∂X) with ab inf(a, b) ≠ 0. We may assume that |S|_{1} < |S|_{2}. By [LP], Theorem 2.1 (5), p. 313 we get

(9) in(∂f, T) = a\frac{∂}{∂X} in(f, S) .

Let us consider the factorization

(10) in(f, S) = X^{a_{S}}Y^{b_{S}}ψ_{1}^{k_{1}} … ψ_{m}^{k_{m}} with irreducible, pairwise coprime ψ_{i} ∈ C{X, Y } and let λ = Y.

We have assumed that |S|_{1} < |S|_{2} thus the lower vertex of S does not lie on the vertical axis and we may apply Lemma 3.1 to in(f, S) and λ:

(11) \frac{∂}{∂X} in(f, S) = Y^{b_{S}}ψ_{0}X^{max(a_{S}−1,0)}ψ_{1}^{k_{1}−1} … ψ_{m}^{k_{m}−1}

where ψ_{0} and in(f, S) are coprime. By (2), (3) and (11) we get

(12) in((∂f)_{T} , T′) = ψ_{0}ψ_{1}^{k_{1}−1} … ψ_{m}^{k_{m}−1} . Clearly the power series ψ_{1}, …, ψ_{m} and ψ_{0} (if ordψ_{0} > 0) are S −elementary. According to Lemma 4.2 we get the factorization

(13) (∂f)_{T} = g_{0}g_{1} … g_{m} in C{X, Y } such that · ordg_{0} > 0 if and only if ord ψ_{0} > 0. If ord g_{0} > 0 then g_{0} is

S − elementary, N (g_{0}) = {S^{(0)}} and in(g_{0}, S^{(0)}) = ψ_{0},

·· ord g_{j} > 0 if and only if k_{j} > 0 for (j = 1, …, m). If ord g_{j} > 0 then g_{j} is S-elementary, N (g_{j}) = {S^{(j)}} and N (g_{j}, S^{(j)}) = ψ_{j}^{k_{j}−1}.

Let A_{S} = g_{0} and B_{S} = g_{1} … g_{m}. By (13) we get (∂f)_{T} = A_{S}B_{S}. Suppose that ordA_{S} > 0. Then N (A_{S}) = {S̃} where S̃ = S^{(0)} and in(A_{S}, S̃) = in(g_{0}, S^{(0)}) = ψ_{0}, consequently in(f, S) are coprime. On the other hand

ord B_{S} = ∑_{j=1}^{m} ord g_{j} = ∑_{j=1}^{m}(k_{j} − 1)ord ψ_{j}
= ∑_{j=1}^{m} k_{j}ord ψ_{j} − ∑_{j=1}^{m} ψ_{j}
=ord în(f, S) − în(f, S)_{red} = d(f, S) .

Suppose that ord B_{S} > 0. The power series B_{S} is S-elementary as a product of S-elementary power series. If N (B_{S}) = { ˜S̃} then in(B_{S}, ˜ S̃) = ∏_{j=1}^{m} ψ_{j}^{k_{j}−1} divides in(f, S).

Proposition 6.3 Let S ∈ N_{f}^{∗∗} and let h be an irreducible factor of g_{S}. Then (f, h)_{0}/ord h ≥ max (α(S), β(S)). The inequality (f, h)_{0}/ord h > max (α(S), β(S)) holds if and only if the following two conditions are fulfiled

(a) the power series h is S-elementary,

(b) if S̃ is the unique segment of N_{h} then the system of equations in(h, S̃) = 0, in(f, S) = 0 has a solution in (C \ {0}) × (C \ {0}).

Proof. Proposition 6.3 follows from [LP], Lemma 3.2, p. 316 which remains true when we replace the sentence “if the pair f, h is nondegenerate” by “if and only if the pair f, h is nondegenerate”.

Now we can give the proof of Theorem 1.1. Let us consider the factorization

∂f = g ∏_{S∈N_{f}^{∗∗}} g_{S} in C{X, Y }

such that in Proposition 6.1.

Let S ∈ N_{f}^{∗∗} be a segment of the first kind. Then by Proposition 6.1 (ii) we have g_{S} = (∂f)_{T} where T ∈ N_{∂f} is a segment parallel to S. Let g_{S} = (∂f)_{T} = A_{S}B_{S} be the factorization of (∂f)_{T} such that in Proposition 6.2. Then ord B_{S} = d(f, S) by Proposition 6.2 (ii) and consequently ord A_{S} = ord g − ord B_{S} = ||S|| + ε(S) − d(f, S) by Proposition 6.1 (i). Moreover if ord A_{S} > 0 (resp. ord B_{S} > 0) then A_{S} (resp. B_{S}) is S-elementary. Let h be an irreducible factor of A_{S}B_{S} = g_{S}. Then (f, h)_{0}/ord h ≥ max (α(S), β(S)) by Proposition 6.3. Using Proposition 6.2 we check that h divides B_{S} if and only if h fulfils the conditions (a) and (b) from Proposition 6.3. Thus (f, h)_{0}/ord h > max (α(S), β(S)) if and only if h is a divisor of B_{S}. Summing up, we have checked that the factorization g_{S} = A_{S}B_{S} where S ∈ N_{f}^{∗} is of the first kind, satisfies all conditions stated in Theorem 1.1. Now suppose that S ∈ N_{f}^{∗∗} is of the second kind. We consider two cases.

Case 1. There is no segment T ∈ N_{∂f} parallel to S. Then we put A_{S} = A′_{S} = g_{S}, A′′_{S} = 1 and B_{S} = 1.

Case 2. There is a segment T ∈ N_{∂f} parallel to S. Then by Proposition 6.2 we get

(∂f)_{T} = A′_{S}B_{S} in C{X, Y } .

On the other hand (∂f)_{T} divides g_{S} by Proposition 6.1 (iii) and we can write

g_{S} = A′′_{S}(∂f)_{T} in C{X, Y } .

Thus we get g_{S} = A_{S}B_{S} with A_{S} = A′_{S}A′′_{S}. As in the case of the segment of the first kind we check that the factorizations g_{S} = A_{S}B_{S}, A_{S} = A′_{S}A′′_{S} have all properties stated in Theorem 1.1. To finish the proof it suffices to check that there is a factorization g = AB in C{X, Y } with ord A = t(f) − 1 such that h is an irreducible factor of g with property (f, h)_{0}/ord h = ord f if and only if h divides A. To this purpose we apply Lemma 4.3 to the series ∂f and observe that if h is an irreducible factor of ∏_{S∈N_{f}^{∗∗}} g_{S} then (f, h)_{0}/ord h ≥ max (α(S), β(S)) for a segment S ∈ N_{f}^{∗∗} and max (α(S), β(S)) > ord f.

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

e-mail: {ztpal,matmm,matap}@tu.kielce.pl