Root Vector Majorization for Doubly Stochastic Stable Polynomials

Definitions and mathematical statement

For a positive integer $d$ and a vector $x\in{\mathbb{R}}^d$, let $x^{\downarrow}$ denote its coordinates arranged in nonincreasing order. For $x,y\in{\mathbb{R}}^d$, write $x\prec y$, and say that $x$ is majorized by $y$, if \[ \sum_{i=1}^{k}x_i^{\downarrow}\leq\sum_{i=1}^{k}y_i^{\downarrow}\quad(1\leq k<d), \qquad \sum_{i=1}^{d}x_i=\sum_{i=1}^{d}y_i. \] The supplied record refers to a doubly stochastic stable-polynomial setting with a class $D_{m,d}$, an object $P_\infty$, a root vector $\lambda(P)=(\lambda_1(P),\ldots,\lambda_d(P))$, a symmetric subclass denoted by $D_{m,d}\cap\operatorname{Sym}_m$, and an operator $T$. It does not supply their definitions. In particular, it does not identify the polynomial whose zeros form $\lambda(P)$, establish that these zeros are real, or specify the admissible values of $m,d$.

Subject to recovering those definitions and ensuring that all displayed root vectors belong to ${\mathbb{R}}^d$, the supplied conjecture asks whether, for every admissible choice of $m,d$, \[ \lambda(P)\prec\lambda(P_\infty) \qquad\text{for every }P\in D_{m,d}. \] It further asks whether, for every $P\in D_{m,d}\cap\operatorname{Sym}_m$ and every pair of admissible parameters $r,s$ with $0\leq r\leq s$, \[ \lambda\left(\Psi_r(P)\right) \prec \lambda\left(\Psi_s(P)\right), \qquad \Psi_t=\exp\left(t(I-T)\right). \] Here $I$ denotes the identity on the intended, unspecified operator space. The exponential and its sign are retained exactly as supplied. The record does not specify the parameter domain, the meaning or domain of $T$, the definition of the exponential, or conditions ensuring that $\lambda(\Psi_t(P))$ is defined and real. Thus these formulas preserve the intended comparisons but do not constitute a self-contained conjecture until the missing setting is recovered.

Short English statement

The proposed comparisons say that the root vector of every polynomial in the intended doubly stochastic stable-polynomial class is majorized by the reference root vector, and that the root vectors increase in majorization along the stated evolution within the symmetric subclass. Essential definitions are absent from the supplied record, so this is a conditional, statement-only transcription requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30003443$, title Root Vector Majorization for Doubly Stochastic Stable Polynomials, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year 2017 and no named proposer.
• Supplied report citation: Real Algebraic Geometry With a View Toward Moment Problems and Optimization (2018), Oberwolfach Reports, pp. 771--862, DOI 10.4171/owr/2017/14. Supplied source URL: https://doi.org/10.4171/owr/2017/14. The report text and the preceding notation referenced by the statement were not supplied.
• Supplied literature reference labeled "Branden, stable-polynomial conjectures": https://doi.org/10.4171/owr/2017/14. This reference label does not establish a named proposer for this particular conjecture.
• Supplied literature reference labeled "Slofstra, related Kadison--Singer background": https://arxiv.org/abs/1606.03140. Its supplied qualification is related background; no direct support for the precise conjecture is established by the record.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes majorization of the full zero vector as conjectural, and reports only a largest-root inequality in the symmetric subclass. It calls the unresolved claim the strong Van der Waerden conjecture. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry supplies the editorial label corrected_verified, explained as preserving mathematical content while repairing notation, typography, and sentence structure. This supplied label does not resolve the missing definitions or establish mathematical correctness.