Polynomial maps with constants on split octonion algebras

TL;DR

This paper addresses the surjectivity of polynomial maps with constants on the split octonion algebra $\mathbf{O}(\mathbb{F})$, focusing on maps of the form $A_1X^{k_1}+A_2Y^{k_2}$ with $A_i\in\mathbf{O}(\mathbb{F})$ and $k_i\ge 2$. The authors exploit the action of the exceptional group $G_2(\mathbb{F})$ to reduce the problem to a finite list of orbit representatives for the pair $(A_1,A_2)$ and derive a complete surjectivity criterion. They prove surjectivity whenever at least one coefficient is invertible and classify many non-invertible cases, constructing explicit $X,Y$ in the admissible orbit types and delineating obstructions in the exceptional ones. The work extends Waring-type questions to the octonion setting and provides an orbit-based framework that can be adapted to other exceptional groups and non-associative 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.

Theorems & Definitions (46)

• Theorem A
• Proposition 2.1: Theorem 4.1, LopatinZubkov2025
• Lemma 2.2
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4