Conductors of Abhyankar-Moh semigroups of even degrees

The Abhyankar-Moh theorem for plane valuations at infinity

We introduce the class of plane valuations at infinity and prove an analogue to the Abhyankar-Moh (semigroup) Theorem for it.

One more proof of the Abhyankar-Moh-Suzuki theorem

We extract the Abhyankar-Moh-Suzuki theorem from the Lin-Zaidenberg theorem.

On the Abhyankar-Moh inequality

Abhyankar and Moh in their fundamental paper on the embeddings of the line in the plane proved an important inequality which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. In this note we study the semigroups of integers satisfying the Abhyankar-Moh inequality and give a simple proof of the Abhyankar-Moh embedding theorem.

A Strong Abhyankar-Moh Theorem and Criterion of Embedded Line

The condition of plane polynomial curve to be a line in well-known Abhyankar-Moh Theorem is replaced by weaker ones. A criterion of embedded line is obtained from this strong theorem: Two polynomials can generate the entire polynomial ring iff their derivatives can be generated.

A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki

Using the theory of minimal models of quasi-projective surfaces we give a new proof of the theorem of Lin-Zaidenberg which says that every topologically contractible algebraic curve in the complex affine plane has equation $X^n=Y^m$ in some algebraic coordinates on the plane. This gives also a proof of the theorem of Abhyankar-Moh-Suzuki concerning embeddings of the complex line into the plane. Independently, we show how to deduce the latter theorem from basic properties of $\mathbb{Q}$-acyclic surfaces.

任意超曲面的阿比扬卡尔-莫半群

Abhyankar-Moh Semigroups for arbitrary hypersurfaces

该研究针对任意超曲面奇点构造了与代数闭域相关的半群，扩展了阿比扬卡尔和莫研究的平面曲线值半群，建立其与根选择无关的条件，还指出其他相关扩展是此构造族的特殊情况。