{"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. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees -- such as Maximum Likelihood, Euclidean Distance, and Polar degrees -- via the Kouchnirenko--Bernshtein theory.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The subpolytopes satisfy the semi-interlaced intersection condition that each proper face F intersects at least dim(F) of the polytopes D_i (rather than the stricter dim(F)+1 required for full interlacing).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A combinatorial formula is proven for the mixed volume of semi-interlaced polytopes, including those arising in algebraic degree computations via Kouchnirenko-Bernshtein theory.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"bc0724c3ec9fd77970dc89cdc4c7b1c42058a20f6b54141bbed7f19c617fd1f2"},"source":{"id":"2605.13410","kind":"arxiv","version":1},"verdict":{"id":"b8364e31-ba15-4b4e-b478-a0e8929ae9cf","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:13:29.770104Z","strongest_claim":"We prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees -- such as Maximum Likelihood, Euclidean Distance, and Polar degrees -- via the Kouchnirenko--Bernshtein theory.","one_line_summary":"A combinatorial formula is proven for the mixed volume of semi-interlaced polytopes, including those arising in algebraic degree computations via Kouchnirenko-Bernshtein theory.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The subpolytopes satisfy the semi-interlaced intersection condition that each proper face F intersects at least dim(F) of the polytopes D_i (rather than the stricter dim(F)+1 required for full interlacing).","pith_extraction_headline":""},"references":{"count":47,"sample":[{"doi":"","year":2000,"title":"B. Beler, A. Enge, K. Fukuda, Exact Volume Computation for Polytopes: A Practical Study. In: G. Kalai, G.M. Ziegler (eds.) Polytopes Combinatorics and Computation, no. 29 in DMV Seminar, pp. 131–154. ","work_id":"5853d1bf-abcc-4ebd-82fd-798ac251e50a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"Arnold, Arnold’s problems, Springer-Verlag, Berlin; PHASIS, Moscow (2004)","work_id":"d0c8a148-fe02-4457-ab6f-2cd01b0d9608","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1975,"title":"Bernshtein, The number of roots of a system of equations, Functional Analysis and Its Applications, 9:3 (1975), 183–185","work_id":"04056d0f-ff33-4393-a493-00f9bc42f288","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/978-3-031-51462-3","year":2024,"title":"P. Breiding, K. Kohn, B. Sturmfels, Metric Algebraic Geometry, Oberwolfach Seminars (OWS, volume 53), 2024, link.springer.com/book/10.1007/978-3-031-51462-3 https://link.springer.com/book/10.1007/978-","work_id":"fdee40f0-3517-4497-b043-0d905598e4e5","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"C. Borger, A. Kretschmer, B. Nill, Thin polytopes: Lattice polytopes with vanishing local h^* -polynomial, Int. Math. Res. Not. IMRN (2023). arXiv:2207.09323 https://arxiv.org/abs/2207.09323","work_id":"160ae66f-0af5-465f-b50f-d0fd14288514","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":47,"snapshot_sha256":"27d99f6feba8b443ca949130e79ed0732ea28dc0f6bce436a97d30600153aef5","internal_anchors":9},"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"}