Pith Number

pith:LK7RA6O3

pith:2026:LK7RA6O3ENATBZXQVHZCKHXALS

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Semi-interlaced polytopes

Fedor Selyanin

A combinatorial formula is proven for the mixed volume of semi-interlaced polytopes, including those arising in algebraic degree computations via Kouchnirenko-Bernshtein theory.

arxiv:2605.13410 v1 · 2026-05-13 · math.CO · math.AG

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Claims

C1strongest 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.

C2weakest 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).

C3one 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.

References

47 extracted · 47 resolved · 9 Pith anchors

[1] 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. 2000

[2] Arnold, Arnold’s problems, Springer-Verlag, Berlin; PHASIS, Moscow (2004) 2004

[3] Bernshtein, The number of roots of a system of equations, Functional Analysis and Its Applications, 9:3 (1975), 183–185 1975

[4] 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- 2024 · doi:10.1007/978-3-031-51462-3

[5] 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 2023

Receipt and verification

First computed	2026-05-18T02:44:47.458626Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5abf1079db234130e6f0a9f2251ee05ca192455436c088c7c3e63a5a74af2d32

Aliases

arxiv: 2605.13410 · arxiv_version: 2605.13410v1 · doi: 10.48550/arxiv.2605.13410 · pith_short_12: LK7RA6O3ENAT · pith_short_16: LK7RA6O3ENATBZXQ · pith_short_8: LK7RA6O3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LK7RA6O3ENATBZXQVHZCKHXALS \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 5abf1079db234130e6f0a9f2251ee05ca192455436c088c7c3e63a5a74af2d32

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8de9022ad4a7a3b6e2543d68cb1aa3f0c965c63578da324664acd3bcfc6308f1",
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      "math.AG"
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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T12:05:38Z",
    "title_canon_sha256": "0635a6fead9326aa63378a198c37843be4fbe549f122c477ce506ab03ec36770"
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  "source": {
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