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PreprintarXiv licencearXiv

A note on the expectation of zeros of random harmonic polynomials: The Kac model

  • Dawei Lu
  • Yuchen Wang
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Abstract

Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials Hn,m(z)=pn(z)+qm(z)̅ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is O(n) when °p=α°q, where 0≤α<1. This result extends the previous estimates when m is a fixed constant or m=n to more general case.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026note,
  title = {{A note on the expectation of zeros of random harmonic polynomials: The Kac model}},
  author = {Dawei Lu and Yuchen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12741},
  eprint = {2609.12741},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12741},
}
RIS
TY  - UNPB
TI  - A note on the expectation of zeros of random harmonic polynomials: The Kac model
AU  - Dawei Lu
AU  - Yuchen Wang
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12741
UR  - https://arxiv.org/abs/2609.12741
AB  - Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials H_n,m(z)=p_n(z)+q_m(z)̅ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is O(n) when °p=α°q, where 0≤α<1. This result extends the previous estimates when m is a fixed constant or m=n to more general case.
ER  -