Problems Around Polynomials — Conjecture 7

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.

Fix $d\geq1$ and a sign pattern $\sigma=(\sigma_0,\ldots,\sigma_d)\in\{+,-\}^{d+1}$, ordered from the leading coefficient to the constant coefficient. Let $c(\sigma)$ count sign changes between consecutive entries and $u(\sigma)$ count preservations, so $c(\sigma)+u(\sigma)=d$. A pair $(r_+,r_-)$ of nonnegative integers is admissible if \[r_+\leq c(\sigma),\quad r_+\equiv c(\sigma)\pmod2,\qquad r_-\leq u(\sigma),\quad r_-\equiv u(\sigma)\pmod2.\] A realization is a real polynomial $p(x)=a_0x^d+\cdots+a_d$ with every coefficient nonzero, $\operatorname{sgn}(a_j)=\sigma_j$, and exactly $r_+$ positive and $r_-$ negative roots, counted with multiplicity.

Statement recorded in the supplied source.

For every sign pattern $\sigma$, is each admissible pair $(r_+,r_-)$ with $r_+>0$ and $r_->0$ realizable? Equivalently, can nonrealizability occur only when at least one of the two root counts is zero? [S2]

Short English statement

Are all Descartes-admissible positive and negative root counts attainable whenever both counts are positive?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2200012 (AMR-021-0012). 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 7 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1503.05295.