Characterization of 2-Pisot Elements in the Field of Laurent Series over a Finite Field

Let $\mathbb{F}_q$ be a finite field, and let $\mathbb{F}_q[X]$ be the ring of polynomials with coefficients in $\mathbb{F}_q$. A 2-Pisot element is a pair of algebraic integers of formal Laurent series over $\mathbb{F}_q[X]$ with absolute value strictly greater than $1$ and such that all remaining conjugates have an absolute value strictly smaller than $1$. Our paper is devoted to characterize 2-Pisot elements in the case $q\neq2^r$.