Vector invariants for two-dimensional orthogonal groups over finite fields

Let ${\mathbb{F}}_q$ be a finite field of characteristic $2$ and $O_2^{+}({\mathbb{F}}_q)$ be the $2$-dimensional orthogonal group of plus type over ${\mathbb{F}}_q$. Consider the standard representation $V$ of $O_2^{+}({\mathbb{F}}_q)$ and the ring of vector invariants ${\mathbb{F}}_q[mV{]}^{O_2^{+}({\mathbb{F}}_q)}$ for any $m\in {\mathbb{N}}^{+}$. We prove a first main theorem for $(O_2^{+}({\mathbb{F}}_q),V)$, i.e., we find a minimal generating set for ${\mathbb{F}}_q[mV{]}^{O_2^{+}({\mathbb{F}}_q)}$. As a consequence, we derive the Noether number ${\beta }_{mV}(O_2^{+}({\mathbb{F}}_q))=\max \{q-1,m\}$. We construct a free basis for ${\mathbb{F}}_q[2V{]}^{O_2^{+}({\mathbb{F}}_q)}$ over a suitably chosen homogeneous system of parameters. We also obtain a generating set of the Hilbert ideal for ${\mathbb{F}}_q[mV{]}^{O_2^{+}({\mathbb{F}}_q)}$ which shows that the Hilbert ideal can be generated by invariants of degree $⩽q-1=\frac{\mathrm{\mid }{O}_2^{+}({\mathbb{F}}_q)\mathrm{\mid }}{2}$, positively confirming a conjecure of Derksen and Kemper for this particular case.