{"paper":{"title":"Prescribed Initial Behavior of $\\mu(I^k)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Reza Abdolmaleki, Shinya Kumashiro","submitted_at":"2026-06-22T07:10:04Z","abstract_excerpt":"It is well known that for every graded ideal $I$, the numbers of minimal generators $\\mu(I^k)$ of its powers of $I$ are eventually increasing. However, its initial behavior can be surprisingly flexible. We prove that any prescribed finite pattern of increases, decreases, and equalities can occur among the first differences $\\mu(I^{k+1})-\\mu(I^k)$ of a suitable monomial ideal $I$ in $K[x,y]$. This provides a broad positive answer to a previously posed sign-realization problem and unifies several known constructions exhibiting unusual behavior of $\\mu(I^k)$. Moreover, as a consequence, analogous"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22933","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.22933/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}