Polynomial group laws. II

Abstract:

Let V be a Zariski-open (i.e., cofinite) subset of an infinite field K. Call a map $m:V \times V \to V$ separately polynomial if for each $x \in V$ the two partial maps $y \to m(x,y),y \to m(y,x)$ are polynomial. If $m:V \times V \to V$ is a separately polynomial group law, then either $V = K$ and $m(x,y) = x + y + k$ for some $k \in K$ or $V = K - \{ k\}$ and $m(x,y) = b(x - k)(y - k) + k$ for some $k \in K$ and $b \in {K^\ast } = K - \{ 0\}$.