Pith Number

pith:ILOYKWEN

pith:2026:ILOYKWENC7G4CJPEKCB7SUGLIW

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Zero asymptotics for successive derivatives of hyperexponential functions with finite essential singularities

Boris Shapiro, Christian H\"agg

After removing forced singular factors from the numerator, the zeros of high-order derivatives of hyperexponential functions converge in the plane to the Voronoi edges of all finite singular points plus weighted atoms at the essential ones.

arxiv:2604.09333 v2 · 2026-04-10 · math.CV · math.CA

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Claims

C1strongest claim

After the forced singular factors are removed from the numerator of f^{(n)}, the normalized zero-counting measures converge in the original z-plane to the classical Voronoi edge measure generated by all finite singular sites, augmented by explicitly weighted atoms at the finite essential singularities.

C2weakest assumption

The function belongs to the hyperexponential class f=(P/Q)exp(S/T) whose finite essential singularities are of arbitrary but finite order and whose exponent S/T satisfies the stated polynomial-growth condition; the proof relies on this form to separate forced singular factors and to apply potential-theoretic limits without additional growth restrictions.

C3one line summary

The authors extend Pólya's shire theorem to hyperexponential functions f=(P/Q)exp(S/T), showing that normalized zero-counting measures of derivatives converge to a Voronoi edge measure augmented by weighted atoms at essential singularities, with explicit microscopic cluster laws.

References

28 extracted · 28 resolved · 0 Pith anchors

[1] R. Bøgvad and C. Hägg,A refinement for rational functions of Pólya’s method to construct Voronoi diagrams, J. Math. Anal. Appl.452(2017), no. 1, 312–334. 1, 2 2017

[2] R. Bøgvad, B. Shapiro, G. Tahar, and S. Warakkagun,The translation geometry of Pólya’s shires, arXiv:2503.07895 (2025), to appear inDuke Mathematical Journal. 1, 2 2025

[3] S. Chen, R. Feng, and M. F. Singer,Parallel Telescoping and Parameterized Picard–Vessiot Theory, Proceedings of ISSAC 2014, ACM, New York, 2014, pp. 137–144. 2 2014

[4] J. G. Clunie and A. Edrei,Zeros of successive derivatives of analytic functions having a single essential singularity II, J. Anal. Math.56(1991), 141–185. 1, 2 1991

[5] Deift,Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert Approach, Courant Lecture Notes in Mathematics3, Amer 1999

Receipt and verification

First computed	2026-05-20T00:01:41.219383Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

42dd85588d17cdc125e45083f950cb458acdc168a373dc53daa4bf80b95e60c1

Aliases

arxiv: 2604.09333 · arxiv_version: 2604.09333v2 · doi: 10.48550/arxiv.2604.09333 · pith_short_12: ILOYKWENC7G4 · pith_short_16: ILOYKWENC7G4CJPE · pith_short_8: ILOYKWEN

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ILOYKWENC7G4CJPEKCB7SUGLIW \
  | 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: 42dd85588d17cdc125e45083f950cb458acdc168a373dc53daa4bf80b95e60c1

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b4acae87c90691fcbb138b23fc34f16ac7bb127e40bf97f35cfdb72aeb1e8e8e",
    "cross_cats_sorted": [
      "math.CA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CV",
    "submitted_at": "2026-04-10T14:01:04Z",
    "title_canon_sha256": "24a2af0a163a96d37783dc74e2440cc946d90b9af5304715c01e51bd3c919713"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.09333",
    "kind": "arxiv",
    "version": 2
  }
}