{"paper":{"title":"An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.CO","authors_text":"John Chiarelli, Michael Saks, Pooya Hatami","submitted_at":"2018-01-25T19:14:02Z","abstract_excerpt":"We prove that there is a constant $C\\leq 6.614$ such that every Boolean function of degree at most $d$ (as a polynomial over $\\mathbb{R}$) is a $C\\cdot 2^d$-junta, i.e. it depends on at most $C\\cdot 2^d$ variables. This improves the $d\\cdot 2^{d-1}$ upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)]. Our proof uses a new weighting scheme where we assign weights to variables based on the highest degree monomial they appear on.\n  The bound of $C\\cdot 2^d$ is tight up to the constant $C$ as a lower bound of $2^d-1$ is achieved by a read-once decision tree of depth $d$. We slight"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08564","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"}