Unbounded Gaps in the Tropical Secant Dimension Bound
Manuscript 22 October 2026 · Online 23 October 2026
Abstract
The optimized affine winner-cell bound gives a combinatorial lower bound for the dimension of a secant cone of a monomial image. We show that it need not equal the actual dimension, answering negatively the unrestricted monomial formulation of a question in Draisma's Oberwolfach report. For the six cubic monomials x^3, y^3, z^3, x^2 y, x^2 z, y z^2, the optimized two-cell bound is exactly five, whereas the two-point secant cone is all of C^6. The upper bound follows from a planar separation obstruction; a Jacobian determinant of 216 proves the actual dimension. Disjoint coordinate blocks give, for every r >= 1, a degree-three monomial cone with optimized bound 5r and secant dimension 6r. The affine cones used here are not normal. No counterexample for normal toric embeddings or complete Segre-Veronese systems is claimed. This is an unrefereed preprint; no absolute priority claim is made.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Counterexamples for arbitrary homogeneous monomial maps
- Categories
- math.AG · math.CO
- Manuscript
- 22 October 2026
- Online release
- 23 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Unbounded Gaps in the Tropical Secant Dimension Bound,” EulerSolve Research Papers, OWR-1740-002, 2026. https://doi.org/10.5281/zenodo.23587509.
BibTeX
@misc{ferudun2026owr1740002,
author = {Ferudun, Alper},
title = {Unbounded Gaps in the Tropical Secant Dimension Bound},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-1740-002/},
doi = {10.5281/zenodo.23587509},
note = {OWR-1740-002; unrefereed preprint}
}