{"paper":{"title":"On two problems of Carlitz and their generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ioulia N. Baoulina","submitted_at":"2016-09-15T19:59:24Z","abstract_excerpt":"Let $N_q$ be the number of solutions to the equation $$ (a_1^{}x_1^{m_1}+\\dots+a_n^{}x_n^{m_n})^k=bx_1^{k_1}\\cdots x_n^{k_n} $$ over the finite field $\\mathbb F_q=\\mathbb F_{p^s}$. Carlitz found formulas for~$N_q$ when $k_1=\\dots=k_n=m_1=\\dots=m_n=1$, $k=2$, $n=3$ or $4$, $p>2$; and when ${m_1=\\dots=m_n=2}$, $k=k_1=\\dots=k_n=1$, $n=3$ or $4$, $p>2$. In earlier papers, we studied the above equation with $k_1=\\dots=k_n=1$ and obtained some generalizations of Carlitz's results. Recently, Pan, Zhao and Cao considered the case of arbitrary positive integers $k_1,\\dots,k_n$ and proved the formula $N"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04807","kind":"arxiv","version":1},"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"}