Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields

TL;DR

This paper introduces a new class of hyperfields, explores their properties, and addresses questions about solutions to homogeneous linear equations over these hyperfields, including counterexamples to previous claims.

Contribution

It defines a new class of hyperfields, provides partial answers to a question about solutions to linear equations, and refutes a prior claim about non-quotient hyperfields.

Findings

Existence of nonzero solutions in certain hyperfields with more unknowns than equations

Counterexamples to previous claims about non-quotient hyperfields

New class of hyperfields including several known constructions

Abstract

We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false.