{"paper":{"title":"Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.CO","math.IT"],"primary_cat":"math.NT","authors_text":"Daniel J. Katz, Philippe Langevin","submitted_at":"2014-09-08T18:41:30Z","abstract_excerpt":"We consider Weil sums of binomials of the form $W_{F,d}(a)=\\sum_{x \\in F} \\psi(x^d-a x)$, where $F$ is a finite field, $\\psi\\colon F\\to {\\mathbb C}$ is the canonical additive character, $\\gcd(d,|F^\\times|)=1$, and $a \\in F^\\times$. If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.2459","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"}