Problems Around Polynomials — Conjecture 3

Definitions and mathematical statement

Let $N\geq1$, let $(x_i,y_i)\in{\mathbb{R}}^2$, let $\xi_i\in{\mathbb{R}}$ for $1\leq i\leq N$, and let $\alpha\geq\tfrac12$. No distinctness of the points or sign restriction on the charges is stated. Define \[ q_i(x)=(x-x_i)^2+y_i^2,\qquad D=\{x\in{\mathbb{R}}:q_i(x)>0\text{ for all }i\},\qquad \Psi(x)=\sum_{i=1}^{N}\frac{\xi_i}{q_i(x)^\alpha}\quad(x\in D). \] For $x\in D$ the real powers are well defined. If all $y_i\ne0$, then $D={\mathbb{R}}$; otherwise the source's formula has possible singularities on the real line and no value there is supplied. For arbitrary real $\alpha$ this function is generally not rational, despite that word in the source.

Recorded conjecture (A. Gabrielov, D. Novikov, B. Sh.). The function $\Psi$ has at most $N$ local maxima on the whole real line. More explicitly, once the intended domain and notion of local maximum are fixed, the proposed bound is \[ \#\{x:x\text{ is an intended local maximum of }\Psi\}\leq N. \] An ordinary non-strict local maximum at $x\in D$ means $\Psi(x)\geq\Psi(z)$ for every $z\in D$ sufficiently near $x$; a strict local maximum requires $>$ for $z\ne x$. The source does not select between these notions or restrict to isolated maxima. Since arbitrary real charges include the all-zero choice, the non-strict point-count interpretation makes every point of $D$ a maximum in that permitted degenerate case. Thus a nondegeneracy or counting convention is essential; none is silently added.

The source appends the appraisal "seems good, but no progress." The supplied research summary reports the conjecture still open and even the case $N=3$, $\alpha=1$, $\xi_1=\xi_2=\xi_3=1$ unresolved, and relates it to Maxwell's conjecture and Gabrielov--Novikov--Shapiro critical-point bounds. These are supplied claims requiring verification, not a resolution of the domain and degeneracy issues above.

Short English statement

For arbitrary real planar charges and $\alpha\geq1/2$, bound the number of local maxima of the sum of inverse powers of squared distances along the real axis by the number of charges. The source leaves real-axis poles and degenerate or non-strict maxima unresolved, and calls the expression rational although general real exponents need not be rational.

Sources

• Local UnsolvedMath registry, permanent ID 2200003, source code catalogue entry, "Problems Around Polynomials --- Conjecture 3"; source attribution A. Gabrielov, D. Novikov, B. Sh., with "seems good, but no progress." Supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Shapiro, Problems Around Polynomials: The Good, The Bad and The Ugly (2015), source-order item 3, Conjecture 3 as printed, https://arxiv.org/abs/1503.05295. The record reports source-TeX extraction and access on 2026-07-29. It says the source presents an open item and current status requires release review. Rights: NEEDS_REVIEW; public arXiv source TeX inspected according to the record.
• The supplied research summary mentions Maxwell's conjecture and Gabrielov--Novikov--Shapiro critical-point bounds but supplies no specific publications or theorem references for its status assessment.