{"paper":{"title":"Sparse Univariate Polynomials with Many Roots Over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC"],"primary_cat":"math.NT","authors_text":"Daqing Wan, J. Maurice Rojas, Qi Cheng, Shuhong Gao","submitted_at":"2014-11-24T04:20:25Z","abstract_excerpt":"Suppose $q$ is a prime power and $f\\in\\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2(q-1)^{\\frac{t-2}{t-1}}$ on the number of cosets in $\\mathbb{F}^*_q$ needed to cover the roots of $f$ in $\\mathbb{F}^*_q$. Here, we give explicit $f$ with root structure approaching this bound: For $q$ a $(t-1)$-st power of a prime we give an explicit $t$-nomial vanishing on $q^{\\frac{t-2}{t-1}}$ distinct cosets of $\\mathbb{F}^*_q$. Over prime fields $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6346","kind":"arxiv","version":3},"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"}