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

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 $m=3$ and $q$ is even, we confirm the conjecture of Ahmadi and Monsef-Shokri [2] that the polynomial $f(x)=x^3+x^2+1$ is stable over ${\mathbb{F}}_2$. Moreover, when $m=2$ and $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, we improve a lower bound of the number of quadratic stable polynomials by Goméz-Pérez and Nicolás [4].