{"paper":{"title":"Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.AG","authors_text":"Fr\\'ed\\'eric Bihan","submitted_at":"2014-10-29T09:03:10Z","abstract_excerpt":"Given convex polytopes $P_1,...,P_r$ in $R^n$ and finite subsets $W_I$ of the Minkowsky sums $P_I=\\sum_{i \\in I} P_i$, we consider the quantity $N(W)=\\sum_{I \\subset {\\bf [}r {\\bf ]}} {(-1)}^{r-|I|} \\big| W_I \\big|$. We develop a technique that we call irrational mixed decomposition which allows us to estimate $N(W)$ under some assumptions on the family $W=(W_I)$. In particular, we are able to show the nonnegativity of $N(W)$ in some important cases. The quantity $N(W)$ associated with the family defined by $W_I=\\sum_{i \\in I} W_i$ is called discrete mixed volume of $W_1,...,W_r$. We show that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7905","kind":"arxiv","version":2},"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"}