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 $$f(x)=\sum _{k=1}^n{c}_k(P_k(x){)}^{{\alpha }_k},$$ where $c_k,{\alpha }_k\in \mathbb{R}$ 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 ${\alpha }_k$. 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 $\Delta u+\lambda u=0$ 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 $1/4$.