Problems Around Polynomials — Problem 5

Definitions and mathematical statement

Fix a degree $d\geq1$ and a sign pattern $\sigma=(\sigma_0,\ldots,\sigma_d)\in\{+,-\}^{d+1}$, read from the leading coefficient down to the constant term. Let $c(\sigma)$ be the number of changes between consecutive signs and let $s(\sigma)=d-c(\sigma)$ be the number of preservations. An ordered pair $(u,v)$ of nonnegative integers is admissible when \[ u\leq c(\sigma),\quad u\equiv c(\sigma)\pmod2,\qquad v\leq s(\sigma),\quad v\equiv s(\sigma)\pmod2. \]

Call this pair realizable if there is a real polynomial $P(x)=a_0x^d+a_1x^{d-1}+\cdots+a_d$, with every coefficient nonzero and $\operatorname{sign}(a_j)=\sigma_j$, having exactly $u$ positive and $v$ negative real roots, counted with algebraic multiplicities. Its remaining $d-u-v$ roots must be nonreal and occur in conjugate pairs; zero itself cannot be a root because $a_d\ne0$.

Determine, for every sign pattern in every degree, precisely which admissible pairs are realizable. The displayed conditions are the necessary restrictions coming from applying Descartes' rule to $P(x)$ and $P(-x)$; the problem is to characterize any additional obstructions. A solution should supply realizations for the allowed pairs as well as proofs of impossibility for excluded pairs. The classification includes pairs with either coordinate zero and pairs with both coordinates positive. No claim that either of these subclasses is automatically realizable is built into the question.

Short English statement

Which positive and negative root counts can occur for a specified pattern of coefficient signs?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200011 (AMR-021-0011), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015), source-order item 11 (Problem 5 as printed). Original source indexed by AMR-021-0011. https://arxiv.org/abs/1503.05295.
• S3 V. P. Kostov, Three realization problems about univariate polynomials, 2026. https://arxiv.org/abs/2601.10529.