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Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face $F \\subsetneq P$ intersects at least $\\dim(F) + 1$ of the polytopes $D_i$. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems.\n  Motivated by relaxing the bound $\\dim(F) + 1$ to $\\dim(F)$, we prove a combinatorial formula for the "},"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":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.13410","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T12:05:38Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"0635a6fead9326aa63378a198c37843be4fbe549f122c477ce506ab03ec36770","abstract_canon_sha256":"8de9022ad4a7a3b6e2543d68cb1aa3f0c965c63578da324664acd3bcfc6308f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:47.459180Z","signature_b64":"2F/8mN0YhzSbfo1DfSoK5rDpFPL1gADD3PfNYr6sPaXnQL7L2qRPrqMfb+1abJlEGvKcOzW1kqwxzhohRYMBBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5abf1079db234130e6f0a9f2251ee05ca192455436c088c7c3e63a5a74af2d32","last_reissued_at":"2026-05-18T02:44:47.458626Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:47.458626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semi-interlaced polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A combinatorial formula is proven for the mixed volume of semi-interlaced polytopes, including those arising in algebraic degree computations via Kouchnirenko-Bernshtein theory.","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Fedor Selyanin","submitted_at":"2026-05-13T12:05:38Z","abstract_excerpt":"The Minkowski mixed volume of $n$ subpolytopes $D_1, \\dots, D_n$ of a polytope $P \\subset {\\mathbb R}^n$ clearly does not exceed the normalized volume $n! \\text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face $F \\subsetneq P$ intersects at least $\\dim(F) + 1$ of the polytopes $D_i$. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems.\n  Motivated by relaxing the bound $\\dim(F) + 1$ to $\\dim(F)$, we prove a combinatorial formula for the "},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"We prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. 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