On stable polynomials of degrees $2,3,4$

Tong Lin, Qiang Wang

Let $q$ be a prime power. We construct stable polynomials of the form $b^{m-1}(x+a{)}^m+c(x+a)+d$ over a finite field ${\mathbb{F}}_q$ for $m=2,3,4$ by Capelli's lemma. When $m=3$ an…