Problems Around Polynomials — Problem 6

Definitions and mathematical statement

Let $n\geq1$ and let $I\subset\mathbb R$ be an open interval. A smooth function $f:I\to\mathbb R$ is polynomial-like of degree $n$ if $f^{(n)}$ never vanishes on $I$. It is real-rooted if it has $n$ distinct zeros in $I$. In that case repeated use of Rolle's theorem shows that $f^{(i)}$ has exactly $n-i$ zeros in $I$, all simple, for $0\leq i<n$. Write these zeros in increasing order as $x^{(i)}_1<\cdots<x^{(i)}_{n-i}$.

Suppose instead that the numbers in such a triangular array are prescribed. The necessary interlacing conditions are \[ x^{(i)}_\ell<x^{(i+1)}_\ell<x^{(i)}_{\ell+1} \quad(0\leq i<n-1,\ 1\leq\ell<n-i). \] They imply all the longer-range inequalities $x^{(i)}_\ell<x^{(j)}_\ell<x^{(i)}_{\ell+j-i}$ whenever the indices are defined. Coincidences between nonadjacent derivative levels are allowed if these strict inequalities permit them.

Characterize precisely the arrays in $I$ satisfying these conditions that arise from some real-rooted polynomial-like function $f$ of degree $n$. Equivalently, determine the additional compatibility conditions among the zero locations of successive derivatives. The realizing function need not be a polynomial, and the numerical locations, rather than only their ordering, are prescribed. Multiplying $f$ by a nonzero constant preserves its array, so one may normalize the sign by requiring $f^{(n)}>0$ throughout $I$.

Short English statement

Which interlacing triangular arrays are exactly the zeros of a smooth function and its successive derivatives?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200016 (AMR-021-0016), 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 16 (Problem 6 as printed). Original source indexed by AMR-021-0016. https://arxiv.org/abs/1503.05295.