Problems Around Polynomials — Conjecture 5

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a nonzero real-rooted polynomial $p$ of degree at least two, its mesh is the minimum distance between consecutive real roots, listing roots with multiplicity. Thus a repeated root gives mesh zero. Nonzero constants and linear polynomials have mesh $+\infty$. As usual for linear real-rootedness preservers, the zero polynomial is an allowed zero output. Write $\mathcal H_m$ for zero together with real-rooted polynomials of degree at most $m$ and mesh at least one.

For $m\geq1$, put $(x)_m=\prod_{j=0}^{m-1}(x-j)$. Let $T$ be any constant-coefficient finite difference operator \[(Tp)(x)=\sum_{j=0}^s b_jp(x-j),\qquad b_j\in\mathbb R,\quad s\geq0.\]

Statement recorded in the supplied source.

Is $T(\mathcal H_m)\subseteq\mathcal H_m$ if and only if $T((x)_m)\in\mathcal H_m$, for every $m\geq1$ and every such $T$? [S2] [S3]

Short English statement

Can preservation of real roots separated by at least one, for all polynomials of bounded degree, be tested on one falling-factorial polynomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2200009 (AMR-021-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boris Shapiro, Problems Around Polynomials -- The Good, The Bad and The Ugly, Arnold Mathematical Journal 1 (2015), pp.91--104, Conjecture 5 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1503.05295.
• S3 Petter Brändén, Ilia Krasikov and Boris Shapiro, Elements of Pólya--Schur theory in finite difference setting (2013), Section 1 mesh definition and Section 4, Conjectures 2 and 3. https://arxiv.org/pdf/1204.2963.