Eremenko's disconjugacy conjecture for Wronskians

From papers

Let $V\in\operatorname{Gr}_{k,n}(\mathbb{R})$ . Suppose that all complex zeros of the Wronskian $\operatorname{Wr}(V)$ are real, and let $I\subseteq\mathbb{R}$ be any interval on which $\operatorname{Wr}(V)$ is nonzero. Eremenko's disconjugacy conjecture. Every nonzero polynomial $f\in V$ has at most $k-1$ zeros in $I$ . This conjecture, together with the reality theorem, implies the secant conjecture. Its case $k=2$ is known, but the general case remains open.

Progress summary

Solved

A recent arXiv paper states and proves the conjecture in full, so the formerly open problem now appears settled.

The conjecture asserts that if a polynomial subspace has a Wronskian whose complex zeros are all real, then every nonzero element has at most one fewer zeros than the subspace dimension on any interval avoiding those zeros.

Known results

The case of dimension at most two was known from work of Eremenko, Gabrielov, Shapiro, and Vainshtein; the conjecture also yields the divisor case of the secant conjecture together with the Shapiro–Shapiro reality theorem.

General proof in arXiv version 2

Theorem 1.9 of the newer paper states the full disconjugacy conjecture: every nonzero $f\in V$ has at most $\dim V-1$ zeros, counted with multiplicity, on intervals avoiding the real zeros of $\operatorname{Wr}(V)$ . The source presents this as proved and derives the divisor form of the secant conjecture.

Current status (as of August 2026): The general conjecture is stated as proved in arXiv version 2, while the earlier conjectural formulation and the dimension-at-most-two result are superseded.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Steven N. Karp, “Wronskians, total positivity, and real Schubert calculus”, arXiv:2110.02301 (2023).

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