Defining normal subgroups of unipotent algebraic groups

Abstract

Let $G$ be a connected unipotent algebraic group defined over the perfect field $k$. We show that polynomial generators ${x_1}, \cdots ,{x_n}$ for the ring $k[G]$ can be chosen so that if $N$ is any connected normal $k$-closed subgroup of $G$, then $I(N)$ can be generated by $\operatorname {co} \dim N$ $p$-polynomials in ${x_1}, \cdots ,{x_n}$ where $p = \operatorname {char} k$. Moreover $k[G/N]$ can also be generated as a polynomial algebra over $k$ by $p$-polynomials.