{"paper":{"title":"On coefficients of powers of polynomials and their compositions over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.NT","authors_text":"Amela Muratovi\\'c-Ribi\\'c, Gary L. 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 $k \\geq 0$. As a consequence, we prove that the rank of $A(f)$ gives th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07487","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}