Research Problems in Function Theory — Problem 2.72

Definitions and mathematical statement

For a homogeneous linear equation of order $n$, a fundamental system is a collection of $n$ linearly independent solutions, equivalently with nonzero Wronskian. Finitely many nonreal zeros means that the zeros in $\mathbb C\setminus\mathbb R$ form a finite set; there is no finiteness requirement on real zeros.

Source target (S. Hellerstein and J. Rossi). Replace finitely many zeros by finitely many nonreal zeros in the reported Frank characterization of polynomial-coefficient differential equations having a fundamental system of solutions with that property. Does the same equivalence with reduction to constant coefficients continue to hold? The source says the unresolved range may be taken as $n\geq3$, citing Hellerstein--Rossi and Gundersen for the lower-order background.

Two formula defects prevent a unique precise equivalence. The extracted operator is \[ L_n(w)=w^{(n)}+a_{n-1}(z)w^{(n-1)}+\cdots+a_0(z), \] with polynomial $a_j$, but its final term lacks a factor $w$. A standard homogeneous linear version would instead have $\sum_{j=0}^{n-1}a_j(z)w^{(j)}$, where $w^{(0)}=w$. This is an explicitly identified possible repair, not the supplied equation as written. The extracted transformation is \[ w(z)=\exp\big(q(z)u(z)\big), \] where $q$ is a polynomial. That is different from the usual linear gauge substitution $w=e^{q}u$ and does not by itself specify a linear conjugation of differential operators.

Once the intended operator and substitution are recovered, the constant-coefficient target would need to be specified as an equation such as $u^{(n)}+\sum_{j=0}^{n-1}c_ju^{(j)}=0$ with constants $c_j$. One must also fix whether the finite-nonreal-zero condition concerns the particular displayed fundamental system or existence of some such system. No repaired theorem or equivalence is asserted here.

Short English statement

Does Frank's finite-zero reduction theorem extend to solutions with only finitely many nonreal zeros? The extracted equation and transformation are malformed, and the fundamental-system quantifier also needs clarification.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 2.72, attributed to S. Hellerstein and J. Rossi, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The source label 2.72 is retained as the local equation/problem reference.
• Frank, reported finite-zero characterization; Hellerstein and Rossi, and Gundersen, lower-order background. The extracted citation entries give no titles, years, or theorem locations for these results.
• Local registry data/unsolvedmath/problems.json, permanent ID $2302072$, source code catalogue entry, Research Problems in Function Theory --- Problem 2.72. The record labels this open and OPEN-TRIAGE, reporting no recent resolution located, with NEEDS_REVIEW source-status and rights qualifications.