Problems Around Polynomials — Problem 7

Definitions and mathematical statement

Fix $n\geq1$ and consider real polynomials $P$ of degree $n$ having $n$ distinct real zeros. Each derivative $P^{(j)}$, for $0\leq j<n$, then has exactly $n-j$ simple real zeros. To form an unambiguous word, restrict to polynomials for which zeros belonging to different derivative orders do not coincide.

Arrange all $\binom{n+1}{2}$ zeros on the real line, and replace a zero of $P^{(j)}$ by the symbol $j$. The resulting word has length $n(n+1)/2$, with exactly $n-j$ occurrences of $j$. Rolle's theorem requires the subword using just $j,j+1$ to alternate, starting and ending with $j$. Equivalently, between successive occurrences of $j$ there is exactly one occurrence of $j+1$.

Which words satisfying these necessary conditions occur for a polynomial $P$? Give a characterization valid for arbitrary $n$, with a polynomial realization of each admitted word and an obstruction for every excluded word. Only the total order of the zeros is prescribed, not their spacings; thus this is different from prescribing numerical locations for every derivative zero. Polynomials differing by a positive affine change of the variable or a nonzero scalar have the same word. Reflection of the variable reverses the word and preserves realizability. Degenerate configurations with coincident zeros from different derivative orders are boundary cases, rather than words in this strict-order formulation.

Short English statement

Characterize the possible orderings of all zeros of a real-rooted polynomial and its derivatives.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200017 (AMR-021-0017), 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 17 (Problem 7 as printed). Original source indexed by AMR-021-0017. https://arxiv.org/abs/1503.05295.