Research Problems in Function Theory — Problem 4.6

Definitions and mathematical statement

For each integer $\nu\geq0$, use the Hermite polynomial normalization \[ H_\nu(z)=(-1)^\nu e^{z^2} \frac{d^\nu}{dz^\nu}e^{-z^2}; \qquad H_0=1,\quad H_1=2z. \] These are real polynomials of degree $\nu$, regarded as functions of the complex variable $z$. For arbitrary integers $2\leq n<m$ and arbitrary complex numbers $a,b$, set $F_{n,m,a,b}(z)=1+H_1(z)+aH_n(z)+bH_m(z)$. The choices $a=0$ or $b=0$ are permitted. The fixed linear term ensures that this polynomial is never constant.

Is there a finite constant $C\geq0$ such that, for every choice of $n,m,a,b$, some zero $\zeta$ of $F_{n,m,a,b}$ satisfies $|\operatorname{Im}\zeta|\leq C$? Equivalently, is \[ \sup_{\substack{2\leq n<m\\a,b\in\mathbb C}} \min_{F_{n,m,a,b}(\zeta)=0}|\operatorname{Im}\zeta|<\infty? \] The strip is centered on the real axis, and its width must be independent of both polynomial degrees and both coefficients. Neither a bound on the real part of the selected zero nor a positive proportion of zeros in the strip is requested. The normalization of the Hermite family is part of the problem: rescaling the variable separately at each degree would change the uniform-strip assertion. A negative answer requires parameters for which the distance of every zero from the real axis tends to infinity.

Short English statement

Must a combination of the constant and linear Hermite polynomials with two higher Hermite polynomials always have a zero in one fixed horizontal strip?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2304006 (AMR-022-4006), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 4.6. Original source indexed by AMR-022-4006. https://arxiv.org/abs/1809.07200.