{"paper":{"title":"Footprint and minimum distance functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.AG","math.CO","math.IT"],"primary_cat":"math.AC","authors_text":"Luis N\\'u\\~nez-Betancourt, Rafael H. Villarreal, Yuriko Pitones","submitted_at":"2017-12-01T16:09:28Z","abstract_excerpt":"Let $S$ be a polynomial ring over a field $K$, with a monomial order $\\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $\\delta_I$ and the footprint function ${\\rm fp}_I$. It is shown that $\\delta_I$ is positive and that ${\\rm fp}_I$ is positive if the initial ideal of $I$ is unmixed. Then we show that if $I$ is radical and its associated primes are generated by linear forms, then $\\delta_I$ is strictly decreasing until it reaches the asymptotic value $1$. If $I$ is the edge ideal of a Cohen--Macaulay bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00387","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"}