Pith Number

pith:77ZXZ2UV

pith:2026:77ZXZ2UVAL7JYTWSTLMEVO43XA

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Euclidean distance degree defect of singular projective varieties

Botong Wang, Jose Israel Rodriguez, Lauren\c{t}iu G. Maxim

A constructible enhancement and topological formula compute the ED degree defect for arbitrary singular projective varieties.

arxiv:2605.13726 v1 · 2026-05-13 · math.AG · math.AT

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting.

C2weakest assumption

The topological formula derived for the smooth case continues to hold after the introduction of singularities, with the constructible enhancement correctly capturing the additional contributions from singular strata.

C3one line summary

A topological formula computes the Euclidean distance degree defect for arbitrary singular complex projective varieties, extending the smooth case.

References

22 extracted · 22 resolved · 0 Pith anchors

[1] P. Aluffi and C. Harris , The E uclidean distance degree of smooth complex projective varieties , Algebra Number Theory 12 (2018), no. 8, 2005--2032. https://doi.org/10.2140/ant.2018.12.2005 DOI 2018 · doi:10.2140/ant.2018.12.2005

[2] D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler , Numerically solving polynomial systems with B ertini , Software, Environments, and Tools, 25. Society for Industrial and Applied Mathe 2013 · doi:10.1137/1.9781611972702.ch1

[3] J.-P. Brasselet, D. T. L\^ e , and J. Seade , Euler obstruction and indices of vector fields , Topology 39 (2000), no. 6, 1193--1208. https://doi.org/10.1016/S0040-9383(99)00009-9 DOI 2000 · doi:10.1016/s0040-9383(99)00009-9

[4] J.-P. Brasselet, D. Massey, A. J. Parameswaran, and J. Seade , Euler obstruction and defects of functions on singular varieties , J. London Math. Soc. (2) 70 (2004), no. 1, 59--76. https://doi.org/10. 2004 · doi:10.1112/s0024610704005447

[5] P. Breiding, F. Sottile, and J. Woodcock , Euclidean distance degree and mixed volume , Found. Comput. Math. 22 (2022), no. 6, 1743--1765. https://doi.org/10.1007/s10208-021-09534-8 DOI 2022 · doi:10.1007/s10208-021-09534-8

Receipt and verification

First computed	2026-05-18T02:44:16.605438Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

fff37cea9502fe9c4ed29ad84abb9bb838bd84a79073364aecc9637511566476

Aliases

arxiv: 2605.13726 · arxiv_version: 2605.13726v1 · doi: 10.48550/arxiv.2605.13726 · pith_short_12: 77ZXZ2UVAL7J · pith_short_16: 77ZXZ2UVAL7JYTWS · pith_short_8: 77ZXZ2UV

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/77ZXZ2UVAL7JYTWSTLMEVO43XA \
  | 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: fff37cea9502fe9c4ed29ad84abb9bb838bd84a79073364aecc9637511566476

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "13fd7db06ccfb7d9d184fe924e75e2afe31050fc3aa3d7dab9f8a6ab18fb1d35",
    "cross_cats_sorted": [
      "math.AT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-13T16:07:00Z",
    "title_canon_sha256": "f174e9c69b816785d33d7973c87926006c5e527b928a5b18320c40460b8a87f4"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13726",
    "kind": "arxiv",
    "version": 1
  }
}