Algebraic conditions for additive functions over the reals and over finite fields

TL;DR

This paper investigates additive functions over real and finite fields constrained by algebraic conditions on affine curves, revealing existence and triviality results depending on the field and curve properties.

Contribution

It characterizes when nonzero additive functions satisfying specific algebraic conditions exist over various fields, including real and finite fields.

Findings

Nonzero additive functions exist over $\,\mathbb{R}$ for the hyperbola $xy=1$.

Such functions exist over fields transcendental over $\,\mathbb{Q}$ or $\,\mathbb{F}_p$.

For large degree curves over finite fields, additive functions must be zero.

Abstract

Let $C$ be an affine plane curve. We consider additive functions $f: K\rightarrow K$ for which $f(x)f(y)=0$, whenever $(x,y)\in C$. We show that if $K=\mathbb{R}$ and $C$ is the hyperbola with defining equation $xy=1$, then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero $f$ exists for a field $K$ if and only if $K$ is transcendental over $\mathbb{Q}$ or over $\mathbb{F}_p$, the finite field with $p$ elements. We also consider the general question when $K$ is a finite field. We show that if the degree of the curve $C$ is large enough compared to the characteristic of $K$, then $f$ must be identically zero.