Eremenko–Gabrielov secant conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $m,p\geq2$ and $d=m+p-1$. Define $F:\mathbb R\to\mathbb R^{m+p}$ by $F(x)=(1,x,\ldots,x^d)$. For $j=1,\ldots,mp$, choose pairwise disjoint real intervals $I_j$ and distinct parameters $x_{j,1},\ldots,x_{j,p}\in I_j$, and put \[X_j=\operatorname{span}_{\mathbb C}\{F(x_{j,1}),\ldots,F(x_{j,p})\}\subset\mathbb C^{m+p}. \] An $m$-plane $Y\subset\mathbb C^{m+p}$ meets $X_j$ when $Y\cap X_j\neq\{0\}$, and is real when it has a basis in $\mathbb R^{m+p}$ (equivalently $\overline Y=Y$).

Statement recorded in the supplied source.

Must every complex $m$-plane $Y$ meeting each $X_j$ be real? These complementary-plane incidence conditions are the divisor form of the secant conjecture, proved in the cited Theorem 1.8; no incidence conditions of higher Schubert codimension are imposed here. [S2]

Short English statement

Are all subspaces satisfying the stated disjoint-interval secant incidences real? The cited theorem proves this particular form.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1800002 (AMR-017-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 S. N. Karp and K. Purbhoo, Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus, arXiv:2309.04645v2 (2026), §1.3.2, Theorem 1.8; compare §5.7.2. https://arxiv.org/abs/2309.04645v2.