Estimating the number of real zeros of linear combinations of radicals of polynomials
Abstract
We obtain upper bounds for the number of real zeros of functions of the form f(x) = ∑k=1n ck (Pk(x))αk, where ck, αk ∈ ℝ and each Pk is a real polynomial of degree at most d that is non-negative on an interval I⊂ 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 αk. For linear combinations of square roots of positive quadratic polynomials on 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 ℝ3∖{(x,0,0)} and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue 1/4.
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BibTeX
@misc{binyamini2026estimating,
title = {{Estimating the number of real zeros of linear combinations of radicals of polynomials}},
author = {Gal Binyamini and Avner Kiro and Alexander Logunov and Dmitry Novikov and Dmitrii Zakharov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02871},
eprint = {2609.02871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02871},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Estimating the number of real zeros of linear combinations of radicals of polynomials
AU - Gal Binyamini
AU - Avner Kiro
AU - Alexander Logunov
AU - Dmitry Novikov
AU - Dmitrii Zakharov
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02871
UR - https://arxiv.org/abs/2609.02871
AB - We obtain upper bounds for the number of real zeros of functions of the form f(x) = ∑_k=1^n c_k (P_k(x))^α_k, where c_k, α_k ∈ ℝ and each P_k is a real polynomial of degree at most d that is non-negative on an interval I⊂ 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 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 ℝ^3∖{(x,0,0)} and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue 1/4.
ER -