arXiv · 2311.05408
We prove the statement in the title for $n\geq 24$.
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J. Jelisiejew, M. Kool, R. F. Schmiermann. 2025-12-09. Behrend's function is not constant on $\mathrm{Hilb}^n(\mathbb{A}^3)$. https://doi.org/10.2140/gt.2025.29.4469
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Let $Ω\subset\mathbb R^n$ be a compact convex domain. An $Ω$-tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $Ω$ that vanishes on $\partialΩ$. For a finite set $P\subsetΩ^\circ$, we study the least such function above prescribed initial data whose corner locus contains $P$. It is obtained by repeatedly applying one-point shrinking operators $G_p$. We prove that every fair order of these operators stabilizes after finitely many nontrivial steps. We also describe an event-driven implementation that records the lowest monomials at each point and updates only affected watcher lists. Finally, we show that, on every compact subset of $Ω^\circ$, the resulting dynamics can be approximated by a finite path whose intermediate tropical hypersurfaces have only mild singularities on that compact set; equivalently, the corresponding local cells of the dual regular subdivision contain no lattice points other than their vertices.
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