{
  "abstract": "We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \\sum_{k=1}^{n} c_k \\bigl(P_k(x)\\bigr)^{α_k}, $$ where $c_k, α_k \\in \\mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\\subset \\mathbb{R}$. We improve previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $α_k$. For linear combinations of square roots of positive quadratic polynomials on $\\mathbb{R}$ we prove the linear bound $2n$, answering a question of N.~Alon. A modification of the argument yields a linear bound for a question of A.~Gabrielov, D.~Novikov, and B.~Shapiro related to Maxwell's conjecture. The article describes two independent approaches: an elementary ODE method in the general case, which also gives a polynomial bound for the number of critical points of one dimensional Gaussian mixtures, and a PDE method for the case of positive quadratic polynomials, which connects the problem to the number of nodal domains of solutions to $Δu + λu = 0$ on the punctured hyperbolic plane. As a byproduct of the second approach, we describe a curious relation between axially symmetric harmonic functions on $\\mathbb{R}^3\\setminus\\{(x,0,0)\\}$ and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue $1/4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02871",
  "author_total": 5,
  "authors": [
    {
      "name": "Gal Binyamini"
    },
    {
      "name": "Avner Kiro"
    },
    {
      "name": "Alexander Logunov"
    },
    {
      "name": "Dmitry Novikov"
    },
    {
      "name": "Dmitrii Zakharov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dekr1cw5w3kxa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02871",
  "fulltext_url": null,
  "id": "dekr1cw5w3kxa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02871",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7207878281",
  "pdf_url": "https://arxiv.org/pdf/2609.02871",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dekr1cw5w3kxa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02871",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02871"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Estimating the number of real zeros of linear combinations of radicals of polynomials",
  "title_text": "Estimating the number of real zeros of linear combinations of radicals of polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/dekr1cw5w3kxa",
  "venue": "arXiv",
  "versions": []
}