{"paper":{"title":"Counting Roots of Polynomials Over Prime Power Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.SC"],"primary_cat":"math.NT","authors_text":"Daqing Wan, J. Maurice Rojas, Qi Cheng, Shuhong Gao","submitted_at":"2017-11-03T23:02:22Z","abstract_excerpt":"Suppose $p$ is a prime, $t$ is a positive integer, and $f\\!\\in\\!\\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\\log p)^{O(1)}$. This fixed parameter tractability appears to be new for $t\\!\\geq\\!3$. A consequence for arithmetic geometry is that we can efficiently compute Igusa zeta functions $Z$, for univariate polynomials, assuming the degree of $Z$ is fixed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01355","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"}