Rational maps $H$ for which $K(tH)$ has transcendence degree 2 over $K$

TL;DR

This paper classifies rational maps with transcendence degree at most 2 over a field, extends classical theorems to arbitrary fields, and explores properties of certain subalgebras of rational functions.

Contribution

It generalizes a theorem of Gordan and N"other to arbitrary fields and classifies rational maps with specific transcendence degree and Jacobian conditions.

Findings

Classified rational maps with transcendence degree ≤ 2 over any field.

Extended Gordan-N"other theorem beyond characteristic zero.

Derived results on subalgebras of rational functions with transcendence degree 1.

Abstract

We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another indeterminate. Furthermore, we classify all such maps for which additionally $JH \cdot H = {\rm tr} JH \cdot H$ (where $JH$ is the Jacobian matrix of $H$), i.e. $$ \sum_{i=1}^n H_i \frac{\partial}{\partial x_i} H_k = \sum_{i=1}^n H_k \frac{\partial}{\partial x_i} H_i $$ for all $k \le n$. This generalizes a theorem of Paul Gordan and Max N\"other, in which both sides and the characteristic of $K$ are assumed to be zero. Besides this, we use some of our tools to obtain several results about $K$-subalgebras $R$ of $K(x)$ for which ${\rm trdeg}_K L = 1$, where $L$ is the fraction field of $R$. We start with some observations about to what extent, L\"uroth's theorem can be generalized.