Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory

平面分支的Zariski模问题的构造性解

Constructive solution of Zariski's Moduli Problem for Plane Branches

该研究以算法方式计算不可约全纯平面曲线芽的Kähler微分集合，讨论其值半模的多种生成概念并给出对应基的几何性质与算法构造，得到平面分支Zariski模问题的显式构造性解。

On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approach

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).

Constructive solutions to Pólya-Schur problems

We present constructive solutions to the following Pólya-Schur problems concerning linear operators on the space of univariate polynomials: Given subsets $Ω_1$ and $Ω_2$ of the complex plane, determine operators that map all polynomials having no zeros in $Ω_1$ to polynomials having no zeros in $Ω_2$, or to the zero polynomial. We describe an explicit class consisting of rank 1 operators and product-composition operators that solve the stated problems for arbitrary $Ω_1$ and $Ω_2$; and this class is shown to comprise all solutions when $Ω_1$ is bounded and $Ω_2$ has non-empty interior. The lat…

The Geometric Dynamical Northcott Property For Regular Polynomial Automorphisms of the Affine Plane

We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a si…