{"paper":{"title":"On the Complexity of Computing Gr\\\"obner Bases for Quasi-homogeneous Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"Jean-Charles Faug\\`ere (INRIA Paris-Rocquencourt, LIP6), Mohab Safey El Din (INRIA Paris-Rocquencourt, Thibaut Verron (INRIA Paris-Rocquencourt","submitted_at":"2013-01-23T19:49:36Z","abstract_excerpt":"Let $\\K$ be a field and $(f_1, \\ldots, f_n)\\subset \\K[X_1, \\ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \\ldots, d_n)$ w.r.t a system of weights $(w_{1},\\dots,w_{n})$. Such systems are likely to arise from a lot of applications, including physics or cryptography. We design strategies for computing Gr\\\"obner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi-homogeneous system, the co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.5612","kind":"arxiv","version":3},"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"}