Stable binomials over finite fields

Abstract

In this paper, we study stable binomials over finite fields, i.e., irreducible binomials $x^t-b\in F_q\left[x\right]$ such that all their iterates are also irreducible over $F_q$. We obtain a simple criterion on the stability of binomials based on the forward orbit of 0 under the map $z\mapsto z^t-b$. In particular, our criterion extends the one obtained by Jones and Boston (2011) for the quadratic case. As applications of our main result, we obtain an explicit 1-parameter family of stable quartics over prime fields $F_p$ with $p\equiv 5\left(\mathrm{mod}24\right)$ and also develop an algorithm to test the stability of binomials over finite fields. Finally, building upon a work of Ostafe and Shparlinski (2010), we employ character sums to bound the complexity of such algorithm.