{"paper":{"title":"Density function for the second coefficient of the Hilbert-Kunz function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Mandira Mondal, Vijaylaxmi Trivedi","submitted_at":"2018-01-22T07:07:05Z","abstract_excerpt":"We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $\\beta$-density function $g_{R, {\\bf m}}:[0,\\infty)\\longrightarrow {\\mathbb R}$, where $(R, {\\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\\int_0^{\\infty}g_{R, {\\bf m}}(x)dx = \\beta(R, {\\bf m}),$$ where $\\beta(R, {\\bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {\\bf m})$, as constructed by Huneke-McDermott-Monsky.\n  Moreover we prove, (1) the function $g_{R, {\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.06977","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"}