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

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.