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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.7905","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-10-29T09:03:10Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"5b1c6f9972941afb31aa0a4fa750947ab047b5571beba26db7c6e456247c9c8d","abstract_canon_sha256":"edfd10421ca5c219de04b46431d5bc6dd6a78cd2402670f96aa2ed284bcd526d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:51.300047Z","signature_b64":"6NTts6pQUD1QUD+C+mdu3r1Ct9yKXZ1ajtWxU2HISBRIg/UjmwGb9omprmN38ueJ6s+ysKFHxzi8Z3FnZVglDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67bc64476499edc4210dc0432a370fd96bbdbc103ec23da42e098d2be739bf1e","last_reissued_at":"2026-05-18T02:37:51.299536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:51.299536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $W_1,...,W_r$. 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