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 $k_1,k_2\ge 2$, we study surjectivity of the map $A_1(x^{k_1})+A_2(y^{k_2})\in \mathbf{O}(\mathbb{F})⟨x,y⟩$ 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.