实域和有限域上可加函数的代数条件
Algebraic conditions for additive functions over the reals and over finite fields
摘要：
设$C$为仿射平面曲线。我们考虑加性函数$f: K\rightarrow K$，其中$f(x)f(y)=0$，每当$(x,y)\in C$。证明了如果$K=\mathbb{R}$和$C$是具有定义方程$xy=1$的双曲线，则存在具有此性质的非零可加函数.此外，我们证明了这样一个非零$f$存在一个域$K$当且仅当$K$是超越的$\mathbb{Q}$或$\mathbb{F}_p$，有限域$p$元素。我们还考虑了$K$是一个有限域时的一般问题。证明了如果曲线$C$的阶数与$K$的特征值相比足够大，则$f$必须恒为零.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
