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Mullen, Qiang Wang","submitted_at":"2015-03-25T18:46:32Z","abstract_excerpt":"For any given polynomial $f$ over the finite field $\\mathbb{F}_q$ with degree at most $q-1$, we associate it with a $q\\times q$ matrix $A(f)=(a_{ik})$ consisting of coefficients of its powers $(f(x))^k=\\sum_{i=0}^{q-1}a_{ik} x^i$ modulo $x^q -x$ for $k=0,1,\\ldots,q-1$. This matrix has some interesting properties such as $A(g\\circ f)=A(f)A(g)$ where $(g\\circ f)(x) = g(f(x))$ is the composition of the polynomial $g$ with the polynomial $f$. In particular, $A(f^{(k)})=(A(f))^k$ for any $k$-th composition $f^{(k)}$ of $f$ with $k \\geq 0$. 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