# Spectroscopy of di-meson bound states in charm and beauty sector

Smruti Patel<EMAIL_ADDRESS>Manan Shah Arpit Parmar P.C.Vinodkumar
p.c<EMAIL_ADDRESS>Deptartment of Physics, Sardar Patel
University,Vallabh Vidyanagar-388120, Gujarat, INDIA

## Introduction

Very recently there exists increasing attention towards the study of four
quark states as di-hadronic molecular states followed by the recent discovery
of $Z_{c}$(3900) state by two seperate experimental groups BES III Ablikim
and BELLE Collabaration Liu . The interpretation of the new state has
triggered a considerable amount of theoretical work, especially due to the
controversies related their internal structure. Moreover, very recently BELLE
Collaboration has made the tantalizing observation of two new charged bottom
resonances, namely Zb(10610) and Zb(10650). Since all the standard bottomonia
are neutrally charged, these two resonances have a flavour only compatible
with $b\bar{b}u\bar{d}$ tetraquarks Adachi ; Bondar .
Motivated by striking observation of tetraquark states, here we wish to
predict the interpretation of these states as di-mesonic molecules composed of
a pair of heavy mesons such as $D\bar{D}$, $D\bar{D}^{*}$, $D^{*}\bar{D}^{*}$,
$D^{+}\bar{D}^{*}$ in the charm sector and $B\bar{B}$, $B^{*}\bar{B}^{*}$,
$B\bar{B}^{*}$ in the bottom sector.

## Phenomenology

Different attempts have been made for the interpretation of exotic hadronic
states of four quark system such as di-mesonic states, hadroquarkonium states
and tetraquark states. Investigations into the existence of multiquark states
have begun in the early days of QCD Jaffe ; Strottman . However, little
success has been achieved in understanding tetraquark states due to the non-
perturbative nature of QCD at the hadronic scale. The hadron molecular
considerations does simplify this difficulty by replacing interquark color
interacion with a residual strong interactions between two color singlet
hadrons. Thus, for the present study of di-mesonic molecules, we employ Woods
Saxon plus coulomb type of potential between two color singlet hadrons of the
form

$V(r)=\frac{V_{0}}{1+\exp^{(\frac{r-R}{a})}}-\frac{B}{r}$ (1)

Table 1: Mass spectra of di-mesonic systems (in MeV). Molecule | $J^{pc}$ | BE | $E_{(j_{1},j_{2};J)}$ | $M_{SA}$ | $M_{J}$ | EXP. | Others
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  3738rai
$D\bar{D}$ | $0^{++}$ | 28.87 | 0.0 | 3758.87 | 3758.87 | - |  3760$\pm 100$Jian
|  |  |  |  |  |  |  $3715^{+24}_{-27}$Hidalgo
|  |  |  |  |  |  | 3878[9]
$D\bar{D}^{*}$ | $1^{++}$ | 28.79 | 0.0 | 3900.8 | 3900.8 |  X(3871.68$\pm 0.17$)Beringer  |  3880$\pm 110$Jian
$D^{+}\bar{D}^{*}$ | $1^{+-}$ | 28.78 | 0.0 | 3904.79 | 3904.79 | $Z_{c}^{+}(3898\pm 5)$ | -
$D^{*}\bar{D}^{*}$ | $2^{++}$ | 28.69 | 0.358 | 4042.69 | 4043.05 | - |  4062[9], $4012^{+4}_{-9}$Hidalgo
| $1^{+-}$ |  | -0.358 |  | 4042.33 | - |  3974rai , $3958^{+24}_{-27}$Hidalgo
| $0^{++}$ |  | -0.72 |  | 4041.97 | - |  3930rai
$B\bar{B}$ | $0^{++}$ | 19.06 | 0.0 | 10577.1 | 10577.1 | - | -
$B\bar{B}^{*}$ | $1^{+-}$ | 18.99 | 0.0 | 10623 | 10623 |  $Z_{b}$(10607$\pm 2.0$)Bondar  | -
$B^{*}\bar{B}^{*}$ | $2^{++}$ | 18.92 | 1.367 | 10668.9 | 10670.28 | - | -
| $1^{+-}$ |  | -1.367 |  | 10667.53 |  $Z_{b}$(10652.2$\pm 1.5$)Bondar  | -
| $0^{++}$ |  | -2.73 |  | 10666.16 | - | -

The potential parameters employed here are as follows: a$=$-0.0387 fm;
$V_{0}$=0.03 GeV; B=0.04; R=0.8875 fm. Binding energy is obtained by
numerically solving Schrödinger equation using mathematica notebook of Range-
Kutta method. The non-relativistic Schrodinger bound-state mass (spin average
mass) of the di-mesonic system is obtained as

$M_{SA}=m{{}_{1}}+m{{}_{2}}+BE$ (2)

We introduce j-j coupling term to obtain the hyperfine splitting of the
different di-meson states. Accordingly, the di-mesonic molecular mass is
obtained as

$M_{J}=M_{SA}+E_{(j_{1},j_{2};J)}$ (3)

Where m1 and m2 are the masses of the constituent mesons, BE represents the
binding energy of the di-mesonic system and $E_{(j_{1},j_{2};J)}$ represents
the spin-dependent term. The hyperfine interaction is computed using the
expression similar to the hyperfine interactions for quarkonia but without
considering color factor and is taken as

$E_{(j_{1},j_{2};J)}=\frac{2<j_{1}.j_{2}>_{J}|R(0)|^{2}}{3m_{1}m_{2}}$ (4)

## Results and conclusion

Table 1 summarizes the binding energies and low lying masses of the di-mesonic
states. The recent experimental exotic states and other theoretical results
are also presented for comparision. In the present work, the mass of
$D\bar{D}^{*}$ is 18 MeV above the experimental value. Overall agreement of
the present results gives us a clue to believe that X(3872) state is nothing
but the loosely bound $D\bar{D}^{*}$ meson molecule and its companion
$D\bar{D}$$(0^{++})$ is predicted to be at 3759 MeV. In the same spirit two
bottomonium-like twin resonances Zb(10610) and Zb(10650) are found to be
$B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$ molecules respectively. And recent
$Z_{c}^{+}$(3900) state is found to be the $D^{+}\bar{D}^{*}$ molecular state.
Other positive parity molecular states ($D^{*}\bar{D}^{*})_{J=0,1,2}$ close to
$\psi$(4040) are predicted around 4042 MeV. Other di-mesonic molecular states
in the charm and beauty sector are also presented in table 1. Many of these
states require further experimental support.

## Acknowledgments

The work is part of a Major research project No. F. 40-457/2011(SR) funded by
UGC.

## References

  * (1) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 110, 252001 (2013).
  * (2) Z. Q. Liu et al. (Belle Collaboration),Phys. Rev. Lett. 110, 252002 (2013).
  * (3) I. Adachi et al. (Belle Collaboration), arXiv:hep-ex/1105.4583.
  * (4) A. Bondar et al. (Belle Collaboration), Phys. Rev. Lett. 108, 122001 (2012).
  * (5) R L Jaffe, Phys. Rev. D 15, 281 (1977).
  * (6) D Strottman,Phys. Rev. D 20, 748 (1979).
  * (7) J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
  * (8) Jian-rong et al. Phys. Rev. D 80, 056004 (2009).
  * (9) Ajay Kumar rai et al. Nuclear Physics A 782, 406 (2007)
  * (10) C. Hidalgo- Duque et al.Phys. Rev. D 87,076006 (2013)


# Initial Newton polynomial of the discriminant

Beata Gryszka, Janusz Gwoździewicz and Adam Parusiński

###### Abstract

Let $(f,g)\colon(\mathbb{C}^{2},0)\longrightarrow(\mathbb{C}^{2},0)$ be a
holomorphic mapping with an isolated zero. We show that the initial Newton
polynomial of its discriminant is determined, up to rescalling variables, by
the ideals $(f)$ and $(g)$.

00footnotetext:  2020 Mathematics Subject Classification: Primary 32S15;
Secondary 32S45.
Key words and phrases: discriminant curve, Newton diagram, equisingularity

##  1 Introduction

Let $\mathbb{R}_{\geq 0}$ $(\mathbb{Z}_{\geq 0})$ be the set of all non-
negative real (integer) numbers. For a power series
$f=\sum_{(i,j)\in\mathbb{Z}_{\geq
0}^{2}}a_{i,j}x^{i}y^{j}\in\mathbb{C}[[x,y]]$ we define its Newton diagram
$\Delta(f)$ as the convex hull of the union $\bigcup_{a_{i,j}\neq
0}((i,j)+\mathbb{R}_{\geq 0}^{2})$. If $S$ is the union of all compact edges
of $\Delta(f)$, then the polynomial $f|_{S}:=\sum_{(i,j)\in
S}a_{i,j}x^{i}y^{j}$ is called the initial Newton polynomial of $f$. We say
that power series $f_{1},f_{2}\in\mathbb{C}[[x,y]]$ are equal up to rescalling
variables if $f_{2}(x,y)=f_{1}(ax,by)$ for some nonzero constants $a$, $b$.

Let $\phi=(f,g)\colon(\mathbb{C}^{2},0)\rightarrow(\mathbb{C}^{2},0)$ be the
germ of a holomorphic mapping with an isolated zero. To any germ $\xi$ of an
analytic curve in $(\mathbb{C}^{2},0)$ one associates its direct image
$\phi_{*}(\xi)$, see for example [Ca1, Ca2]. The direct image of $\xi$ by
$\phi$ is an analytic curve germ in the target space uniquely determined by
the following two properties:

  * (i)

if $\xi\subset(\mathbb{C}^{2},0)$ is an irreducible curve then $\phi_{*}(\xi)$
is the curve of equation $H^{d}=0$, where $H=0$ is a reduced equation of the
curve $\phi(\xi)$ in the target space and $d$ is the topological degree of the
restriction $\phi|_{\xi}:\xi\to\phi(\xi)$.

  * (ii)

if $h=h_{1}\cdots h_{s}$ is a factorization of a power series $h$ to the
product of irreducible factors in $\mathbb{C}\\{x,y\\}$, then
$\phi_{*}(\\{h=0\\})$ is the curve $H_{1}\cdots H_{s}=0$, where the curves
$H_{i}=0$ are the direct images of the branches $h_{i}=0$ for $i=1,\dots,s$.

The direct image satisfies the projection formula: If $H=0$ is the direct
image of $h=0$ then for any analytic curve $w=0$ in the target space we have
the equality of intersection multiplicities $i_{0}(w\circ\phi,h)=i_{0}(w,H)$.

Let $H=0$ be the direct image of $h=0$. Suppose that the Newton diagram of $H$
has $r$ edges which are not contained in the coordinate axes. Then $H$ can be
written as a product $H_{1}\cdots H_{r}$, where the Newton diagram of each
$H_{i}$ is elementary, i.e. has exactly one edge not contained in the
coordinate axes. A factorization $H=H_{1}\cdots H_{r}$ induces a factorization
$h=h_{1}\cdots h_{r}$ such that $H_{i}=0$ is the direct image of $h_{i}=0$ for
$i=1,\dots,r$.

We will call such a factorization of $h$ the Hironaka factorization, because
it follows from the projection formula applied to the coordinate axes in the
target space that for every ireducible factor $p$ of $h_{i}$ the Hironaka
quotient $i_{0}(g,p)/i_{0}(f,p)$ is the inclination of the edge of
$\Delta(H_{i})$. Moreover the intersection multiplicities $i_{0}(f,h_{i})$,
$i_{0}(g,h_{i})$ determine and are determined by the edges of the Newton
diagram $\Delta(H)$.

In this article we apply this general construction to a more specific
situation. We call ${\rm Jac}(\phi)=\tfrac{\partial f}{\partial
x}\tfrac{\partial g}{\partial y}-\tfrac{\partial f}{\partial y}\tfrac{\partial
g}{\partial x}=0$ the Jacobian curve of $\phi$ and the direct image of the
Jacobian curve is called the discriminant curve. The Newton diagram of the
discriminant, denoted $Q(f,g)$, is called the Jacobian Newton diagram of
$(f,g)$ and it was introduced by Tessier in [Te1, Te2].

The Jacobian curve in the case of $f=0$ smooth and transverse to $g=0$ is
called the generic polar curve of $g$. In this case the Hironaka quotients
$i_{0}(g,h)/i_{0}(f,h)$ where $h$ is an irreducible factor of ${\rm
Jac}(\phi)$, are called the _polar quotients_.

The polar case has been widely studied. The polar quotients are invariants of
singularity as shown in [KL]. Teissier [Te1, Te2] proved that the Jacobian
Newton diagram in polar case is also a singularity invariant. Merle [Me]
described $Q(f,g)$ for $g$ irreducible. Eggers [Eg] gave a description of
$Q(f,g)$ for any $g$.

The case where both curves $f=0$ and $g=0$ are singular is more complicated.
Maugendre [Ma] characterized Hironaka quotients of the Jacobian curve. Michel
[Mi] gave formulas for $i_{0}(f,h_{i})$, $i_{0}(g,h_{i})$ using topological
tools, thus determining the Jacobian Newton diagram $Q(f,g)$. Michel’s result
was reproved in [Gw] where it is shown that $Q(f,g)$ depends only on the
equisingularity type of pairs of curves $f=0$, $g=0$.

Recall that pairs of curves $f=0$, $g=0$ and $\tilde{f}=0$, $\tilde{g}=0$ are
equisingular if there exist factorisations $f=h_{1}\cdots h_{s}$,
$g=h_{s+1}\cdots h_{r}$, $\tilde{f}=\tilde{h}_{1}\cdots\tilde{h}_{\tilde{s}}$,
$\tilde{g}=\tilde{h}_{\tilde{s}+1}\cdots\tilde{h}_{\tilde{r}}$ into the
product of irreducible factors in $\mathbb{C}\\{x,y\\}$ such that

  * •

$s=\tilde{s}$, $r=\tilde{r}$,

  * •

for $i=1,\dots,r$, the semigroups
$\Gamma(h_{i}):=\\{i_{0}(h_{i},w):w\notin(h_{i})\\}$ and
$\Gamma(\tilde{h}_{i}):=\\{i_{0}(\tilde{h}_{i},w):w\notin(\tilde{h}_{i})\\}$
are equal,

  * •

$i_{0}(h_{i},h_{j})=i_{0}(\tilde{h}_{i},\tilde{h}_{j})$ for $1\leq i<j\leq r$.

In this paper we go a step further. Our main result is.

###### Theorem 1.1

Let $f,g,u^{\prime},u^{\prime\prime}\in\mathbb{C}\\{x,y\\}$ be convergent
power series vanishing at zero such that $f$ and $g$ are coprime and let
$\tilde{f}=(1+u^{\prime})f$, $\tilde{g}=(1+u^{\prime\prime})g$. Then the
initial Newton polynomials of discriminants of mappings
$(f,g):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ and
$(\tilde{f},\tilde{g}):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ are equal.

Note that a factorization of the initial Newton polynomial of the discriminant
$D$ of $\phi=(f,g)$ into irreducible factors, gives a factorization
$D=D_{1}\cdots D_{s}$ such that the initial Newton polynomial of each factor
$D_{i}$ is a power of an irreducible polynomial. This induces a factorization
of ${\rm Jac}(\phi)$ that is usually more subtle than the Hironaka
factorization.

The structure of the paper is as follows. In Section 2 we study the initial
weighted form of a nonzero power series. We establish in Lemma 2.1 a relation
between the intersection multiplicities of this power series with certain test
polynomials and the multiplicities of the irreducible factors of its initial
weighted form. In Corollary 3.2, thanks to Casas’ formula, that we recall for
the reader convenience, these multiplicities are expressed in terms of
classical equisingularity invariants (Milnor numbers). This allows us in
Section 4 to show the key lemma, Lemma 4.1. It states that some specific plane
curve singularities, constructed in terms of $f$, $g$, $\tilde{f}$,
$\tilde{g}$, and the test polynomials, are equisingular. The key lemma follows
from a classical Zariski equisingularity criterion of [SEI]. The main result,
Theorem 1.1, is proven Section 4.1. Finally we give several corollaries of the
main result. This includes a characterization of the atypical values of the
pencils $f^{k}=tg^{l}$ and of the asymptotic critical values of the
meromorphic functions $f^{k}/g^{l}$. We also propose an alternative proof of
[GGL, Theorem 6.6] by an argument similar to the proof of the main result.

##  2 Factorization of the initial weighted form

Let $D(u,v)$ be a nonzero complex power series and let $w=(k,l)$ be a weight
vector, where $k$, $l$ are coprime positive integers. Then $D$ can be written
as the sum of quasi-homogeneous polynomials $D=D_{m}+D_{m+1}+{}\cdots$, where
$D_{m}\neq 0$ and $\deg_{w}D_{i}=i$ for $i\geq m$. Write $D_{m}$ as a product

$D_{m}(u,v)=Cu^{\nu_{0}}v^{\nu_{n+1}}\prod_{i=1}^{n}(v^{k}-t_{i}u^{l})^{\nu_{i}},$
(1)

where $t_{i}\neq 0$ and $t_{i}\neq t_{j}$ for $i\neq j$.

The aim of this section is to express the multiplicities $\nu_{i}$ for $1\leq
i\leq n$ by the intersection multiplicities of $D$ with certain polynomials.

###### Lemma 2.1

Let $H_{t}=(v^{k}-tu^{l})^{N}-u^{l(N+1)}$. Then for a sufficiently large
integer $N$ and for every $t\in\mathbb{C}^{*}$ such that $t\neq t_{i}$ for
$1\leq i\leq n$, one has

$\nu_{j}kl=i_{0}(D,H_{t_{j}})-i_{0}(D,H_{t}).$

Proof. For any power series $F(u,v)$ denote by $\mbox{\rm in}_{w}F$ its
weighted initial form with respect to the weight vector $w$. By the quasi-
homogeneous version of Hensel’s lemma (see e.g. [Ar, Lemma A.1]) the power
series $D$ factors to a product $PQ$, where $\mbox{\rm
in}_{w}P=(v^{k}-t_{j}u^{l})^{\nu_{j}}$ and $\mbox{\rm in}_{w}Q$ is not
divisible by $v^{k}-t_{j}u^{l}$.

Take any $N>\nu_{j}kl$. In order to compute the intersection multiplicity
$i_{0}(P,H_{t_{j}})$ we will use the classical Zeuten’s Rule
$i_{0}(P,H_{t_{j}})=\sum\mbox{\rm ord}\,P(u,\alpha_{i}(u))$, where the sum
runs over all Newton-Puiseux roots $v=\alpha_{i}(u)$ of $H_{t_{j}}(u,v)=0$.

Solving the equation $(v^{k}-t_{j}u^{l})^{N}-u^{l(N+1)}=0$ with respect to $v$
we get $Nk$ solutions $v=\alpha_{i}(u)$ of the form
$v=\omega\sqrt[k]{t_{j}}u^{l/k}+\mbox{\it higher order terms}$, where
$\omega^{k}=1$ and $\sqrt[k]{t_{j}}$ is a fixed root of a polynomial
$Y^{k}-t_{j}$.

Writing $P(u,v)$ as a power series
$(v^{k}-t_{j}u^{l})^{\nu_{j}}+\sum_{ak+bl>\nu_{j}kl}c_{a,b}u^{a}v^{b}$ and
substituting $v=\alpha_{i}(u)$ we get

$P(u,\alpha_{i}(u))=\epsilon
u^{\nu_{j}l(N+1)/N}+\sum_{ak+bl>\nu_{j}kl}c_{a,b}u^{a}\alpha_{i}(u)^{b},$

where $\epsilon^{N}=1$.

The term $\epsilon u^{\nu_{j}l(N+1)/N}$ on the right-hand side of the above
formula has the smallest order. Indeed, by the inequalities
$ak+bl\geq\nu_{j}kl+1$, $N>\nu_{j}kl$ we get $\mbox{\rm
ord}\,u^{a}\alpha_{i}(u)^{b}=a+b(l/k)=(ak+bl)/k\geq\nu_{j}l(\nu_{j}kl+1)/(\nu_{j}kl)>\nu_{j}l(N+1)/N$.
We have shown that $\mbox{\rm ord}\,P(u,\alpha_{i}(u))=\nu_{j}l(N+1)/N$ for
every Newton-Puiseux root $v=\alpha_{i}(u)$ of $H_{t_{j}}(u,v)=0$. Hence
$i_{0}(P,H_{t_{j}})=(N+1)\nu_{j}kl$.

Computing the intersection multiplicity $i_{0}(Q,H_{t_{j}})$ is simpler. Since
the polynomials $\mbox{\rm in}_{w}Q$ and $\mbox{\rm
in}_{w}H_{t_{j}}=(v^{k}-t_{j}u^{l})^{N}$ are coprime, we have $\mbox{\rm
ord}\,Q(u,\alpha_{i}(u))=\mbox{\rm ord}\,\mbox{\rm
in}_{w}Q(u,\omega\sqrt[k]{t_{j}}u^{l/k})=(1/k)\deg_{w}(\mbox{\rm in}_{w}Q)$
for any Newton-Puiseux root $\alpha_{i}(u)$ of $H_{t_{j}}(u,v)=0$. By Zeuten’s
Rule we get $i_{0}(Q,H_{t_{j}})=N\deg_{w}(\mbox{\rm in}_{w}Q)$.

Analogously we obtain $i_{0}(Q,H_{t})=N\deg_{w}(\mbox{\rm in}_{w}Q)$ and
$i_{0}(P,H_{t})=N\deg_{w}(\mbox{\rm in}_{w}P)=N\nu_{j}kl$.

Finally
$i_{0}(D,H_{t_{j}})-i_{0}(D,H_{t})=(i_{0}(P,H_{t_{j}})+i_{0}(Q,H_{t_{j}}))-(i_{0}(P,H_{t})+i_{0}(Q,H_{t}))=\nu_{j}kl$
which ends the proof.

##  3 Casas’ formula

Consider the following result of Casas-Alvero, which provides a very useful
formula.

######  Theorem 3.1 ([Ca1] Theorem 3.2)

Let $(f,g):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ be the germ of a
holomorphic mapping such that $(f,g)^{-1}(0,0)=\\{(0,0)\\}$. Let $D(u,v)=0$ be
the discriminant of $(f,g)$. Take any curve germ $H(u,v)=0$ and let
$h(x,y)=H(f(x,y),g(x,y))$. Then

$\mu(h)-1=i_{0}(f,g)[\mu(H)-1]+i_{0}(D,H),$

where $\mu(h)$ denotes the Milnor number of the curve $h=0$ at zero.

From the above theorem we obtain a crucial corollary, which is used in the
proof of the main result of this article.

###### Corollary 3.2

Let $(f,g):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ be the germ of a
holomorphic mapping such that $(f,g)^{-1}(0,0)=\\{(0,0)\\}$. Let $D(u,v)=0$ be
the discriminant curve of $(f,g)$ and $h_{t}=(g^{k}-tf^{l})^{N}-f^{l(N+1)}$
for $N>1$. Then, under the notation of (1), for $N\gg 1$ and
$t\in\mathbb{C}^{*}$ different from $t_{1},\dots,t_{n}$, we have
$\nu_{j}kl=\mu(h_{t_{j}})-\mu(h_{t})$.

Proof. It is enough to apply Casas’ formula to $H_{t_{j}}$ and $H_{t}$ as
defined in Lemma 2.1.

##  4 Key lemma and the proof of the main result

Before we present the proof of the main result, we provide the following
lemma. This lemma is interesting not only because of its applications, but
also because of the methods, which are used in its proof. At the end of this
paper we show another application of these methods, presenting a new proof of
some known fact.

######  Lemma 4.1 (Key Lemma)

Let $f,g,u^{\prime},u^{\prime\prime}\in\mathbb{C}\\{x,y\\}$ be convergent
power series vanishing at zero such that $f$ and $g$ are coprime and let
$\tilde{f}=(1+u^{\prime})f$, $\tilde{g}=(1+u^{\prime\prime})g$. Then for
sufficiently large integer $N$ the curves $(g-f)^{N}-f^{N+1}=0$ and
$(\tilde{g}-\tilde{f})^{N}-\tilde{f}^{N+1}=0$ are equisingular.

Proof. By Zariski [SEI] two plane curve singularities (at the origin) are
equivalent if there is a bijection between their branches that makes both
their proper and total transforms by the blowing-up of the origin equivalent.
In order to apply this criterion, we construct a common resolution of
$h:=(g-f)^{N}-f^{N+1}=0$ and
$\tilde{h}:=(\tilde{g}-\tilde{f})^{N}-\tilde{f}^{N+1}=0$ with locally
equivalent singularities (for both the proper and the total transforms). Then
it follows that $h$ and $\tilde{h}$ are equisingular.

Let $R:M\to(\mathbb{C}^{2},0)$ be a resolution of singularities of the curve
$fg(g-f)(\tilde{g}-\tilde{f})=0$. The total transform
$R^{-1}(\\{fg(g-f)(\tilde{g}-\tilde{f})=0\\})$ can be written as the union of
irreducible components $E_{1}\cup\dots\cup E_{n}\cup E_{n+1}\cup\dots\cup
E_{m}$, where $E=E_{1}\cup\dots\cup E_{n}$ is the exceptional divisor
$R^{-1}(0)$ and $E_{n+1}$, …, $E_{m}$ are the components of the proper
transform of the curve $fg(g-f)(\tilde{g}-\tilde{f})=0$. By abuse of notation
we will use the same symbols for germs of functions on $(\mathbb{C}^{2},0)$
and for their pull-backs to $M$.

For $i=1,\dots,m$ we denote the orders of
$f,g,g-f,\tilde{f},\tilde{g},\tilde{g}-\tilde{f}$ along $E_{i}$ by
$a_{i},b_{i},c_{i},\tilde{a}_{i},\tilde{b}_{i},\tilde{c}_{i}$, respectively.

Take any point $P$ of $E_{i}$, where $i\in\\{1,\dots,m\\}$. If $P$ is a smooth
point of the total divisor, and $E_{i}$ in a neighborhood of $P$ has a local
equation $x=0$, then $f=Ax^{a_{i}}$ and
$\tilde{f}=(1+u^{\prime})f=\tilde{A}x^{\tilde{a}_{i}}$, where $A$ and
$\tilde{A}$ do not vanish at $P$. Hence $\tilde{a}_{i}=a_{i}$. Moreover, if
$E_{i}$ is a component of the exceptional divisor, then
$\tilde{A}|_{E_{i}}=A|_{E_{i}}$, since $u^{\prime}|_{E_{i}}=0.$ Similarly we
obtain $\tilde{b}_{i}=b_{i}$.

Choose $N$ big enough so that for all $i\in\\{1,\dots,n\\}$
$Nc_{i}>(N+1)a_{i}$ if $c_{i}>a_{i}$ and $N\tilde{c}_{i}>(N+1)a_{i}$ if
$\tilde{c}_{i}>a_{i}$. This is for instance the case if $N>\max\\{a_{i}:1\leq
i\leq n\\}$.

Then the orders of meromorphic functions $F=(g-f)^{N}/f^{N+1}$ and
$\tilde{F}=(\tilde{g}-\tilde{f})^{N}/\tilde{f}^{N+1}$ along the components of
the exceptional divisor are different from zero. Hence the proper preimage of
the curve $h=0$ (resp. $\tilde{h}=0$), which, in the complement of the total
transform, is given by $F=1$ (resp. $\tilde{F}=1$) does not intersect the
exceptional divisor at the smooth points of the total transform.

Take the intersection point $P$ of a component $E_{i}$ of the exceptional
divisor with $E_{j}$, where $1\leq j\leq m$, $j\neq i$. Choose a local
analytic coordinate system $(x,y)$ centered at $P$ such that $E_{i}$ has the
equation $x=0$ and $E_{j}$ has the equation $y=0$. In these coordinates
$f=Ax^{a_{i}}y^{a_{j}}$, $g=Bx^{b_{i}}y^{b_{j}}$, $g-f=Cx^{c_{i}}y^{c_{j}}$,
$\tilde{f}=\tilde{A}x^{a_{i}}y^{a_{j}}$,
$\tilde{g}=\tilde{B}x^{b_{i}}y^{b_{j}}$, and
$\tilde{g}-\tilde{f}=\tilde{C}x^{\tilde{c}_{i}}y^{\tilde{c}_{j}}$, where
$A,B,C,\tilde{A},\tilde{B},\tilde{C}$ are germs of holomorphic functions that
do not vanish at zero. It follows from [Za], the first paragraph of proof of
Proposition 2.1, see also [BM, Lemma 4.7], that the set of pairs
$\\{(a_{i},a_{j}),(b_{i},b_{j}),(c_{i},c_{j})\\}$ is totally ordered, with
respect to the partial order $(a,a^{\prime})\leq(b,b^{\prime})$ if $a\leq b$
and $a\leq b^{\prime}$, and two of these pairs are equal and are less than or
equal to the third one.

In the sequel we denote by $\,o\\!\\!\\!\\!$— the germ of any holomorphic
function that does not vanish at the origin.

Let us write the equation of

$h=(g-f)^{N}-f^{N+1}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Nc_{i}}y^{Nc_{j}}-\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{(N+1)a_{i}}y^{(N+1)a_{j}}$

in a neighborhood of $P$.

We have the following possibilities:

(I) $(b_{i},b_{j})>(a_{i},a_{j})$

Then $(c_{i},c_{j})=(\tilde{c}_{i},\tilde{c}_{j})=(a_{i},a_{j})$ and we get

$h=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}\tilde{h}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Na_{i}}y^{Na_{j}}.$

(II) $(b_{i},b_{j})<(a_{i},a_{j})$

Then $(c_{i},c_{j})=(\tilde{c}_{i},\tilde{c}_{j})=(b_{i},b_{j})$ and we get

$h=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}\tilde{h}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Nb_{i}}y^{Nb_{j}}.$

(III) $(b_{i},b_{j})=(a_{i},a_{j})$

Then $(c_{i},c_{j})\geq(a_{i},a_{j})$ and
$(\tilde{c}_{i},\tilde{c}_{j})\geq(a_{i},a_{j})$ and we get

$h=x^{Na_{i}}y^{Na_{j}}(\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{N(c_{i}-a_{i})}y^{N(c_{j}-a_{j})}-\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{a_{i}}y^{a_{j}})$

and similarly for $\tilde{h}$.

Write $g-f=(B-A)x^{a_{i}}y^{a_{j}}$ and
$\tilde{g}-\tilde{f}=(\tilde{B}-\tilde{A})x^{a_{i}}y^{a_{j}}$.

Let us consider two subcases of (III).

First, assume that $a_{j}=b_{j}>0$. Then $E_{j}$ is a component of the
exceptional divisor, since otherwise one of $a_{j}$, $b_{j}$ would vanish.
Since $A$ and $\tilde{A}$ are equal on the exceptional divisor, we have
$A(0,y)=\tilde{A}(0,y)$ and $A(x,0)=\tilde{A}(x,0)$. The same equations hold
for $B$ and $\tilde{B}$. It follows that the Newton diagrams of $B-A$ and
$\tilde{B}-\tilde{A}$ have the same intersection points with the coordinate
axes. Moreover, $B-A$ and $\tilde{B}-\tilde{A}$ are factors of $g-f$ or
$\tilde{g}-\tilde{f}$ and hence their Newton diagrams have only one vertex.

If $(0,0)$ is the vertex of the Newton diagram $\Delta$ of
$\tilde{B}-\tilde{A}$, then $c_{i}=\tilde{c}_{i}=a_{i}$ and
$c_{j}=\tilde{c}_{j}=a_{j}$, what implies that
$h=\hbox{$\,o\\!\\!\\!\\!$\lower 0.25832pt\hbox{---}\,}x^{Na_{i}}y^{Na_{j}}$.
If $(a,0)$ is the vertex of $\Delta$ for some $a>0$, then
$c_{i}=\tilde{c}_{i}>a_{i}$ and $c_{j}=\tilde{c}_{j}=a_{j}$, what means that
$h=x^{(N+1)a_{i}}y^{a_{j}}(\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{N(c_{i}-a_{i})-a_{i}}-\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}y^{a_{j}}).$ Similarly we have in the case, when $(0,b)$
is the vertex of $\Delta$ for some $b>0$. The last possibility is, when the
vertex of $\Delta$ is of the form $(a,b)$ for some $a,b>0$. In this case we
have that $c_{i}>a_{i}$, $\tilde{c}_{i}>a_{i}$, $c_{j}>a_{j}$ and
$\tilde{c}_{j}>a_{j}$. Thus $h=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{(N+1)a_{i}}y^{(N+1)a_{j}}.$

Next, assume that $a_{j}=b_{j}=0$. Then $E_{j}$ is a component of the proper
transform of $(g-f)(\tilde{g}-\tilde{f})=0$. Without loss of generality we may
assume that $E_{j}\subset\\{g-f=0\\}$. Then $c_{j}>0$ and $\tilde{c}_{j}\geq
0$. Write $B-A=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{a}y^{c_{j}}$, where $a=c_{i}-a_{i}$, and
$\tilde{B}-\tilde{A}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{\tilde{a}}y^{\tilde{c}_{j}}$, where
$\tilde{a}=\tilde{c}_{i}-\tilde{a}_{i}$. Since
$A|_{E_{i}}=\tilde{A}|_{E_{i}}$, we have $A(0,y)=\tilde{A}(0,y)$ and similarly
$B(0,y)=\tilde{B}(0,y)$. Therefore $a=0$ if and only if $\tilde{a}=0$ and if
this is the case $c_{j}=\tilde{c}_{j}$. Then
$h=x^{Na_{i}}(\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}y^{Nc_{j}}-\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{a_{i}})$ and a similar formula holds for $\tilde{h}$.
If $a$ and $\tilde{a}$ are both strictly positive then, by the assumption on
$N$, $h=x^{(N+1)a_{i}}(\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{N(c_{i}-a_{i})-a_{i}}y^{Nc_{j}}-\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,})=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{(N+1)a_{i}}$.

###### Corollary 4.2

Let $f,g,u^{\prime},u^{\prime\prime}\in\mathbb{C}\\{x,y\\}$ be convergent
power series vanishing at zero such that $f$ and $g$ are coprime and let
$\tilde{f}=(1+u^{\prime})f$, $\tilde{g}=(1+u^{\prime\prime})g$. Then for
positive integers $k$, $l$, $t\neq 0$ and sufficiently large integer $N$ the
curves $h_{t}=(g^{k}-tf^{l})^{N}-f^{l(N+1)}=0$ and
$\tilde{h}_{t}=(\tilde{g}^{k}-t\tilde{f}^{l})^{N}-\tilde{f}^{l(N+1)}=0$ are
equisingular.

Proof. It is enough to apply Lemma 4.1 to $f_{1}=tf^{l}$, $g_{1}=g^{k}$ and
$\tilde{f}_{1}=t\tilde{f}^{l}$, $\tilde{g}_{1}=\tilde{g}^{k}$.

Now, we are ready to prove the main result.

###  4.1 Proof of Theorem 1.1

Let $D(u,v)=0$ be the discriminant of
$(f,g):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ and let $\tilde{D}(u,v)=0$ be
the discriminant of
$(\tilde{f},\tilde{g}):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$. Let $w=(k,l)$
be an arbitrary weight vector, where $k$, $l$ are coprime positive integers.
Write $\mbox{\rm
in}_{w}D=Cu^{\nu_{0}}v^{\nu_{n+1}}\prod_{i=1}^{n}(v^{k}-t_{i}u^{l})^{\nu_{i}}$
and $\mbox{\rm
in}_{w}\tilde{D}=\tilde{C}u^{\eta_{0}}v^{\eta_{n+1}}\prod_{i=1}^{n}(v^{k}-t_{i}u^{l})^{\eta_{i}}$.
By [Gw, Theorem 2.1] the Newton diagrams of $D$ and $\tilde{D}$ coincide.
Hence $\nu_{0}=\eta_{0}$, $\nu_{n+1}=\eta_{n+1}$ and it is enough to prove
that $\nu_{i}=\eta_{i}$ for $1\leq i\leq n$. This follows from Corollary 3.2
since by Corollary 4.2 for $t\neq 0$ one has $\mu(h_{t})=\mu(\tilde{h}_{t})$.
This ends the proof.

##  5 Corollaries

As a direct consequence we obtain the following result.

###### Corollary 5.1

Under assumptions of Theorem 1.1 the discriminants of $(f,g)$ and
$(\tilde{f},\tilde{g})$ have the same tangents.

The next corollary concerns the atypical values of the pencils
$g^{k}-tf^{l}=0$ and the asymptotic critical values of the meromorphic
functions $g^{k}/f^{l}$.

###### Corollary 5.2

Under the assumptions of Theorem 1.1, for every pair of coprime positive
integers $k,l$ we have

  * (i)

the pencils $g^{k}-tf^{l}=0$ and $\tilde{g}^{k}-t\tilde{f}^{l}=0$, where
$t\in\mathbb{C}$ is a parameter, have the same sets of atypical values.

  * (ii)

the meromorphic functions $g^{k}/f^{l}$ and $\tilde{g}^{k}/\tilde{f}^{l}$ have
the same asymptotic critical values.

  * (iii)

the generic fibers of $g^{k}/f^{l}$ and $\tilde{g}^{k}/\tilde{f}^{l}$ are
equisingular.

Proof. To verify (i) we apply Casas’ formula to $H_{t}=v^{k}-tu^{l}$ to show
that these families are $\mu$-constant for $t\neq t_{i}$, $i=1,\ldots,n$.

To show (ii) let us recall after [KMP, Section 5], that
$t_{0}\in\mathbb{C}\cup\infty$ is _an assymptotic critical value_ of $F=f/g$
at the origin if there is a sequence of points
$(x_{k},y_{k})\xrightarrow{{k\to\infty}}(0,0)$, $(x_{k},y_{k})\neq(0,0)$, such
that $\|(x_{k},y_{k})\|\|\nabla F\|\to 0$ and $F(x_{k},y_{k})\to t_{0}$.
Therefore if $\tilde{F}=uF$, with $u(0)=1$, then an elementary computation
shows that the asymptotic critical values of $\tilde{F}$ and $F$ coincide.

Now we show the equivalence of (i) and (ii) thus providing for both of them
alternative proofs. That is we show that the set of asymptotic critical values
of $F=f/g$ coincide with the atypical values $t$ of the pencil $f-tg$. Indeed,
an elementary computation, see [KMP, Proposition 5.1], shows that $t_{0}$ is
the asymptotic critical value if and only if the Kuo-Verdier condition (w)
fails at $(0,0,t_{0})$ for the strata $(RegX,T)$, where $RegX$ is the regular
part of $X=V(f-tg)$ and $T$ is the $t$-axis. On the other hand it is well
known that for families of isolated plane curve singularities that their
equisingularity is equivalent to Whitney equisingularity, see [SEII, Theorem
8.1], and in the complex domain the Kuo-Verdier condition is equivalent to
Whitney’s conditions, see [Te3]. For a direct elementary proof that
$\mu$-constant condition is equivalent to Verdier condition for the families
of curve singularities $f-tg$, see [KP, Theorem 4.1].

Finally, to show (iii) we may again apply Casas’ formula to the deformations
$s(g^{k}-tf^{l})+(1-s)(\tilde{g}^{k}-t\tilde{f}^{l})$ with the parameter
$s\in\mathbb{C}$.

###### Corollary 5.3

Let $(f,g)\colon(\mathbb{C}^{2},0)\longrightarrow(\mathbb{C}^{2},0)$ be a
holomorphic mapping with an isolated zero. Then the initial Newton polynomial
of its discriminant is determined, up to rescalling variables, by the ideals
$(f)$ and $(g)$ in $\mathbb{C}\\{x,y\\}$.

Proof. Let $u_{1},u_{2}\in\mathbb{C}\\{x,y\\}$ be power series with nonzero
constant terms. Let $a=u_{1}(0,0)$ and $b=u_{2}(0,0)$. If $D$ (resp. $D_{1}$)
is the discriminant of
$(f,g)\colon(\mathbb{C}^{2},0)\longrightarrow(\mathbb{C}^{2},0)$ (resp.
$(af,bg)\colon(\mathbb{C}^{2},0)\longrightarrow(\mathbb{C}^{2},0)$) then
$D(au,bv)=D_{1}(u,v)$. Hence by Theorem 1.1 applied to $(af,bg)$ and
$(u_{1}f,u_{2}g)$, the initial Newton polynomials of the discriminants of
$(f,g)$ and $(u_{1}f,u_{2}g)$ are equal up to rescalling variables.

As an application of the methods used in this paper we present a new proof of
[GGL, Theorem 6.6].

######  Theorem 5.4 ([GGL, Theorem 6.6])

Let $h=0$ be a unitangent singular curve and let $\ell_{1}=0$, $\ell_{2}=0$ be
smooth curves transverse to $h=0$. Then there exists a nonzero constant
$d\in\mathbb{C}$ such that the initial Newton polynomials of discriminants of
mappings $(d\ell_{1},h):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ and
$(\ell_{2},h):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ are equal.

Keep the assumptions of Theorem 5.4 and let $d$ be the limit at the origin of
the meromorphic function $\ell_{2}/\ell_{1}$ restricted to the tangent to the
curve $h=0$. Fix positive integers $k$, $l$ and a nonzero complex constant
$t$. Let $f=th^{l}$, $g=(d\ell_{1})^{k}$, and $\tilde{g}=\ell_{2}^{k}$. In
order to prove Theorem 5.4 it is enough to have the following counterpart of
Lemma 4.1.

###### Lemma 5.5

For sufficiently large integer $N$ the curves $(g-f)^{N}-f^{N+1}=0$ and
$(\tilde{g}-f)^{N}-f^{N+1}=0$ are equisingular.

Proof. Let $\sigma:M\to(\mathbb{C}^{2},0)$ be the blowing-up of
$\mathbb{C}^{2}$ at the origin. Then the proper transforms of the curves
$f=0$, $g=0$, $\tilde{g}=0$ intersect the exceptional divisor $E$ at points
$P$, $Q$, $\tilde{Q}$ respectively (if $g=0$ and $\tilde{g}=0$ have the same
tangent, then $Q=\tilde{Q}$).

Let $R:M_{1}\to M$ be a resolution of singularities of the curve
$fg(g-f)(\tilde{g}-f)=0$ at $P$. By Zariski equisingularity criterion it is
enough to prove that the total and the proper transforms of the curves
$(g-f)^{N}-f^{N+1}=0$ and $(\tilde{g}-f)^{N}-f^{N+1}=0$ by $R\circ\sigma$ are
equisingular. To prove the equisigularity on $R^{-1}(P)\setminus E$ it is
enough to use the arguments from the proof of Lemma 4.1 since the meromorphic
function $\tilde{g}/g$ is constant and equal to 1 on the set $R^{-1}(P)$. Thus
it remains to show that the equisingularity classes of these curves are the
same on the component $E$ of the exceptional divisor.

The point $P$ is the intersection of $E$ and another component $E^{\prime}$ of
the exceptional divisor of $R\circ\sigma$. Choose a local analytic coordinate
system $(x,y)$ centered at $P$ such that $E$ has the equation $x=0$ and
$E^{\prime}$ has the equation $y=0$. In these coordinates
$f=\hbox{$\,o\\!\\!\\!\\!$\lower 0.25832pt\hbox{---}\,}x^{a}y^{a^{\prime}}$,
$g=\hbox{$\,o\\!\\!\\!\\!$\lower 0.25832pt\hbox{---}\,}x^{b}y^{b^{\prime}}$,
$\tilde{g}=\tilde{\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}}x^{b}y^{b^{\prime}}$, and
$g-f=\hbox{$\,o\\!\\!\\!\\!$\lower 0.25832pt\hbox{---}\,}x^{c}y^{c^{\prime}}$.
The set $\\{(a,a^{\prime}),(b,b^{\prime}),(c,c^{\prime})\\}$ is again totally
ordered as in the proof of Key Lemma, and two of these pairs are equal and are
less than or equal to the third one.

First, consider the case $a=b$. Then the meromorpic function $g/f$ restricted
to $E$ is nonzero on $E\setminus\\{P,Q\\}$, has order $b^{\prime}-a^{\prime}$
at $P$ and has order $b$ at $Q$. Since the sum of orders of a meromorphic
function defined on a Riemann manifold at zeroes and poles must be zero, we
get $b+(b^{\prime}-a^{\prime})=0$ which implies that $b^{\prime}<a^{\prime}$.
It follows that $(a,a^{\prime})\neq(b,b^{\prime})$.

If $(a,a^{\prime})<(b,b^{\prime})$ then $(c,c^{\prime})=(a,a^{\prime})$ and we
get

$(g-f)^{N}-f^{N+1}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Na}y^{Na^{\prime}}.$

If $(b,b^{\prime})<(a,a^{\prime})$ then $(c,c^{\prime})=(b,b^{\prime})$ and we
get

$(g-f)^{N}-f^{N+1}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Nb}y^{Nb^{\prime}}.$

Now, we will determine the class of equisingularity of $(g-f)^{N}-f^{N+1}=0$
at points of $E$ different from $P$ and $Q$. Choose a local analytic
coordinate system $(x,y)$ centered at one of such a point such that $E$ has
the equation $x=0$. Then $f=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{a}$ and $g=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{b}$.

If $a<b$ then

$(g-f)^{N}-f^{N+1}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Na}.$

If $b<a$ then

$(g-f)^{N}-f^{N+1}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Nb}.$

If $a=b$ then the meromorphic function $F=g/f$ restricted to $E$ has exactly
one zero of order $b$ at $Q$ and exactly one pole at $P$. Hence every nonzero
complex number, in particular 1, is a regular value of $F|_{E}$. As a
consequence the set $(F|_{E})^{-1}(1)$ consists of $b$ points $P_{i}$, $1\leq
i\leq b$ and at each of these points the curve $F=1$ intersects $E$
transversally. Thus, for every $i\in\\{1,\dots b\\}$ we may find a local
analytic coordinate system $(x,y)$ centered at $P_{i}$ such that $g-f=x^{a}y$.
We get in the neighborhood of $P_{i}$ a local equation

$(g-f)^{N}-f^{N+1}=x^{Na}(y^{N}-\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{a}).$

Finally, we will determine the equisingularity class of $(g-f)^{N}-f^{N+1}=0$
at $Q$. Choose a local analytic coordinate system $(x,y)$ centered at $Q$ such
that $f=x^{a}$ and $g=x^{b}y^{b}$ in the neighborhood of $Q$.

If $a\leq b$ then

$(g-f)^{N}-f^{N+1}=\hbox{$\,o\\!\\!\\!\\!$\lower
0.25832pt\hbox{---}\,}x^{Na}.$

If $a>b$ then

$(g-f)^{N}-f^{N+1}=x^{Nb}[(y^{b}-x^{a-b})^{N}-x^{(a-b)N+a}]$

and its equisingularity type is uniquely determined by $a,b,N$.

###### Corollary 5.6

Let $f=0$ be a unitangent singular curve and let $\ell=0$ be a smooth curve
transverse to $f=0$. Then the initial Newton polynomial of the discriminant of
the mapping $(\ell,f):(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ is determined,
up to rescalling variables by the ideal $(f)\subset\mathbb{C}\\{x,y\\}$.

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Beata Gryszka
Institute of Mathematics
Pedagogical University of Cracow
Podchora̧żych 2
PL-30-084 Cracow, Poland
e-mail<EMAIL_ADDRESS>
Janusz Gwoździewicz
Institute of Mathematics
Pedagogical University of Cracow
Podchora̧żych 2
PL-30-084 Cracow, Poland
e-mail<EMAIL_ADDRESS>
Adam Parusiński
Université Côte d’Azur
CNRS, LJAD, UMR 7351
06108 Nice, France
e-mail<EMAIL_ADDRESS>