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Arkadiusz Ploski – One of the best experts on this subject based on the ideXlab platform.

a bound for the milnor number of plane curve singularities
Open Mathematics, 2014CoAuthors: Arkadiusz PloskiAbstract:Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

a bound for the milnor number of plane curve singularities
arXiv: Algebraic Geometry, 2013CoAuthors: Arkadiusz PloskiAbstract:Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d1)^2\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which $\mu_0(f)=(d1)^2\left[\frac{d}{2}\right]$.
Y Polatoglu – One of the best experts on this subject based on the ideXlab platform.

bounded log harmonic functions with positive real part
Journal of Mathematical Analysis and Applications, 2013CoAuthors: Y PolatogluAbstract:Abstract Let H ( D ) be the linear space of all analytic functions defined on the open unit disc D = { z   z  1 } , and let B be the set of all functions w ( z ) ∈ H ( D ) such that  w ( z )  1 for all z ∈ D . A logharmonic mapping is a solution of the nonlinear elliptic partial differential equation f z ¯ ¯ = w ( z ) ( f ¯ / f ) f z , where w ( z ) is the second dilatation of f and w ( z ) ∈ B . In the present paper we investigate the set of all logharmonic mappings R defined on the unit disc D which are of the form R = H ( z ) G ( z ) ¯ , where H ( z ) and G ( z ) are in H ( D ) , H ( 0 ) = G ( 0 ) = 1 , and R e ( R ) > 0 for all z ∈ D . The class of such functions is denoted by P L H .
Arnaud Chéritat – One of the best experts on this subject based on the ideXlab platform.

Upper bound for the size of quadratic Siegel disks
Inventiones mathematicae, 2004CoAuthors: Xavier Buff, Arnaud ChéritatAbstract:If α is an irrational number, we let { p _ n / q _ n }_ n ≥0, be the approximants given by its continued fraction expansion. The Bruno series B (α) is defined as $$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial P _α: z ↦ e ^2 i πα z + z ^2 has an indifferent fixed point at the origin. If P _α is linearizable, we let r (α) be the conformal radius of the Siegel disk and we set r (α)=0 otherwise. Yoccoz proved that if B (α)=∞, then r (α)=0 and P _α is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all irrational number α with B (α)

On the Size of Quadratic Siegel Disks: Part I
arXiv: Dynamical Systems, 2003CoAuthors: Xavier Buff, Arnaud ChéritatAbstract:If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2i\pi \a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all irrational number $\a$ with $B(\a)