## Nihonkai Mathematical Journal

- Nihonkai Math. J.
- Volume 28, Number 1 (2017), 65-68.

### The automorphism theorem and additive group actions on the affine plane

#### Abstract

Due to Rentschler, Miyanishi and Kojima, the invariant ring for a $\mathrm{G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.

#### Note

Partly supported by JSPS KAKENHI Grant Numbers 15K04826, 24340006.

#### Article information

**Source**

Nihonkai Math. J., Volume 28, Number 1 (2017), 65-68.

**Dates**

Received: 14 April 2016

First available in Project Euclid: 7 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.nihmj/1520391684

**Mathematical Reviews number (MathSciNet)**

MR3771369

**Zentralblatt MATH identifier**

06881243

**Subjects**

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)

Secondary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 14R20: Group actions on affine varieties [See also 13A50, 14L30]

**Keywords**

additive group action affine plane polynomial ring polynomial automorphism Jung-van der Kulk theorem Rentschler theorem

#### Citation

Kuroda, Shigeru. The automorphism theorem and additive group actions on the affine plane. Nihonkai Math. J. 28 (2017), no. 1, 65--68. https://projecteuclid.org/euclid.nihmj/1520391684