Polynomial maps with constants on split octonion algebras

Abstract

Let $\mathbf{O}(\mathbb{F})$ be the split octonion algebra over an algebraically closed field $\mathbb{F}$. For positive integers $k_1, k_2\geq 2$, we study surjectivity of the map $A_1(x^{k_1}) + A_2(y^{k_2}) \in \mathbf{O}(\mathbb{F})\langle x, y\rangle$ on $\mathbf{O}(\mathbb{F})$. For this, we use the orbit representatives of the ${G}_2(\mathbb{F})$-action on $\mathbf{O}(\mathbb{F}) \times \mathbf{O}(\mathbb{F}) $ for the tuple $(A_1, A_2)$, and characterize the ones which give a surjective map.