A two-sided quaternion polynomial is a finite sum of words 
with quaternion
coefficients 
on both sides of the variable 
. Such expressions define polynomial maps from 
to 
. Degree refers to the total degree of this evaluated real
coordinate map, after cancellations. Two-sided polynomials need not have the root
behavior of polynomials whose coefficients all occur on the same side (Janovská
and Opfer 2010).
Laniewski (2026) constructed a two-sided quartic with 132 simple roots and proved that this root count persists under sufficiently small perturbations within the quartic coefficient space. A root is simple when the real derivative at that root is invertible. Laniewski (2026) reports AI assistance with the mathematical construction and verification.
