{"paper":{"title":"Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.CO"],"primary_cat":"math.LO","authors_text":"Danny Nguyen, John Goodrick, Kevin Woods, Tristram Bogart","submitted_at":"2018-02-03T13:28:13Z","abstract_excerpt":"We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\\ldots,t_k$. A formula in this language defines a parametric set $S_\\mathbf{t} \\subseteq \\mathbb{Z}^{d}$ as $\\mathbf{t}$ varies in $\\mathbb{Z}^k$, and we examine the counting function $|S_\\mathbf{t}|$ as a function of $\\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi-polynomial (there is a period $m$ such that, for sufficiently large $t$, the function is polynomial on each of the residue classes mod $m$). We show that such a nice expression is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00974","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"}