The images of multilinear non-associative polynomials evaluated on a rock-paper-scissors algebra with unit over an arbitrary field

TL;DR

This paper investigates the images of multilinear non-associative polynomials evaluated on a specific algebra related to rock-paper-scissors, showing they form vector spaces over any field, and explores evaluations on related subalgebras.

Contribution

It proves that the image of any non-associative multilinear polynomial on the rock-paper-scissors algebra with unit is a vector space, extending understanding of polynomial evaluations on such structures.

Findings

Images of polynomials are vector spaces

Results hold over arbitrary fields

Analysis includes subalgebras and evaluation questions

Abstract

Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak M}$ of every non-associative multilinear polynomial over ${\mathbb F}$ is a vector space. The same question we consider for two subalgebras: an algebra of the rock, the paper and the scissors without unit, and an algebra of trace zero elements with zero scalar part. Moreover in this paper we consider the questions of possible evaluations of homogeneous polynomials on these algebras.