Some Results on Linearized Trinomials that Split Completely

Linearized polynomials over finite fields have been much studied over the last several decades. Recently there has been a renewed interest in linearized polynomials because of new connections to coding theory and finite geometry. We consider the problem of calculating the rank or nullity of a linearized polynomial $L(x)={\sum }_{i=0}^d{a}_i{x}^{q^i}$ (where $a_i\in {\mathbb{F}}_{q^n}$) from the coefficients $a_i$. The rank and nullity of $L(x)$ are the rank and nullity of the associated ${\mathbb{F}}_q$-linear map $$\mathbb{F}${q^n} \longrightarrow $\mathbb{F}${q^n}$\mathrm{.}McGuireandSheekeydefineda$d\times d$matrix$A_L$withthepropertythat$$\mbox{nullity} (L)=\mbox{nullity} (A_L -I).$$ We present some consequences of this result for some trinomials that split completely, i.e., trinomials $L(x)=x^{q^d}-b{x}^q-ax$ that have nullity $d$. We give a full characterization of these trinomials for $n\le d^2-d+1$.