Eremenko's disconjugacy conjecture for Wronskians

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stated as proved in arXiv version 2, while the earlier conjectural formulation and the dimension-at-most-two result are superseded.