For every n≥ 1, real coefficients c_1,…,c_n, and polynomials P_1,…,P_n∈R[x] satisfying deg P_k≤ 2 and P_k(x)>0 for all x∈R, if f(x)=∑_k=1^nc_k√P_k(x) is not identically zero, must f have at most 2n distinct real zeros?

N. Alon’s question on zeros of sums of square roots; Maxwell’s conjecture-related zero bound

For every $n\ge 1$ , real coefficients $c_1,\ldots,c_n$ , and polynomials $P_1,\ldots,P_n\in\mathbb{R}[x]$ satisfying $\deg P_k\le 2$ and $P_k(x)>0$ for all $x\in\mathbb{R}$ , if $f(x)=\sum_{k=1}^{n}c_k\sqrt{P_k(x)}$ is not identically zero, must $f$ have at most $2n$ distinct real zeros?

References

Primary source

arXiv

Additional references

Estimating the number of real zeros of linear combinations of radicals of polynomials — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A September 2026 preprint claims to settle the question, but its proof has not been independently verified.

N. Alon asked whether a nonzero sum of $n$ square roots of positive quadratic polynomials can have more than $2n$ distinct real zeros. The question is equivalent, after normalization, to the canonical form used in the recent preprint.

Known results

Kayal and Saha, 2010: a related power-series problem has the bound $\operatorname{ord}(S)\le dn^2+n$ , but this does not imply the stated real-zero bound.

September 2026 preprint

Estimating the number of real zeros of linear combinations of radicals of polynomials claims that sums of square roots of positive quadratics either vanish identically or have at most $2n$ distinct real zeros. Its argument reduces the claim to a $2(n-1)$ bound for combinations of $((x-a_k)^2+b_k^2)^{-3/2}$ , and also states a related linear bound in the Maxwell-conjecture setting. The preprint is unrefereed, so these remain unverified claims.

Community submission (unverified)

Posted September 3, 2026, a submitted proof argues for the same reduction through normalization, two differentiations, and Rolle’s theorem, invoking the claimed sharp negative-power estimate. Its correctness is unverified.

Current status (as of September 2026): The $2n$ bound is claimed in a new preprint, but the question remains mathematically unsettled pending verification.

Sources

ProofA 2n Bound for Real Zeros of Linear Combinations of Square Roots of Positive Quadratics A publication-ready mathematical manuscript based on the working document and the September 2026 preprint arXiv:2609.02871. Status note. The central negative-power zero estimate is treated here as the deep external input supplied by Binyamini–Kiro–Logunov–Novikov–Zakharov (2026). The elementary reduction and the final Rolle argument are given explicitly. This distinction is essential because the cited work isSee full solutionHide full solution

We study real zeros of linear combinations of square roots of positive quadratic polynomials. The formulation of Alon’s question concerns functions of the form f(x)=Σ c_k√P_k(x), where P_k∈R[x], deg P_k≤2, and P_k(x)>0 for every real x. The main bound is that every nonzero such function has at most 2n distinct real zeros. The proof separates into an elementary normalization-and-differentiation argument and a deep zero estimate for linear combinations of negative powers of positive quadratic forms. After reducing each nonconstant positive quadratic to the canonical form (x−a)^2+b^2, two differentiations produce an exponent −3/2. A sharp bound of 2(n−1) for the resulting negative-power family, together with two applications of Rolle’s theorem, yields the 2n estimate. We also record the relationship with Coulomb-type restrictions and the Gabrielov–Novikov–Shapiro problem related to Maxwell’s conjecture.

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