Problems Around Polynomials — Problem 1

Definitions and mathematical statement

Fix an integer $k\geq1$. Consider the constant-coefficient differential equation \[ y^{(k)}+a_1y^{(k-1)}+\cdots+a_ky=0,\qquad a_j\in\mathbb C, \] and its characteristic polynomial $\chi(\lambda)=\lambda^k+a_1\lambda^{k-1}+\cdots+a_k$. Let $\Omega_k$ consist of those equations for which any two distinct characteristic roots have different real parts. Repeated roots are permitted: the restriction concerns distinct values, rather than their multiplicities.

Every solution extends to an entire function and can be written $y(z)=\sum_{\lambda}P_\lambda(z)e^{\lambda z}$, with $\deg P_\lambda$ smaller than the multiplicity of $\lambda$ in $\chi$. A real zero means a real number $x$ at which this complex-valued function vanishes, so that both its real and imaginary parts vanish. Exclude the identically zero solution.

Does there exist a finite integer $B(k)$, depending only on the order $k$, such that every nonzero solution of every equation in $\Omega_k$ has at most $B(k)$ real zeros? One may count these zeros with their analytic multiplicities. Existence of a uniform bound is equivalent for distinct zeros and for multiplicity counting, since a nonzero solution of an order-$k$ equation cannot have a zero of multiplicity $k$. The bound must be independent of all coefficients, imaginary parts of the characteristic roots, separations between their real parts, and initial values. In particular, a bound for each individual equation does not settle the question.

Short English statement

Can the order alone bound the real zeros of every solution when distinct characteristic roots have distinct real parts?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200004 (AMR-021-0004), 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 4 (Problem 1 as printed). Original source indexed by AMR-021-0004. https://arxiv.org/abs/1503.05295.