发表机构
Baruch College & The Graduate Center, City University of New York（纽约市立大学巴鲁克学院及研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了R⁴中无法被四个仿射超平面等分为16份的光滑正密度，否定了Grünbaum 1960年猜想的最后案例及Ramos的超平面等分一般性猜想，通过计算机辅助方法完成证明。
AI 中文摘要
我们在R⁴中构造了一个光滑严格正密度，它无法被四个仿射超平面分成16个等大的部分。这解决了Grünbaum在1960年提出猜想的最后一个未解决案例，并否定了Ramos关于超平面等分的一般性猜想。该证明将问题简化为：证明六个变量中的五个显式多项式在[-1,1]⁶中没有公共零点，验证依赖于结合凸性和Bernstein系数的计算机辅助细分论证。
英文摘要
We construct a smooth strictly positive density in $\mathbb{R}^4$ that cannot be divided into $16$ parts of the same size by four affine hyperplanes. This settles the last open case of Grünbaum's 1960 conjecture and disproves Ramos' general conjecture on hyperplane equipartitions. We reduce the construction to finding two homogeneous polynomials in four variables, of degrees three and four, whose multilinear coefficients cannot vanish simultaneously after any orthogonal change of coordinates. We give two proofs of this nonvanishing result. The first uses a local perturbation argument. The second reduces it to the absence of a common zero for five explicit polynomials on $[-1,1]^6$, verified by a computer-assisted Bernstein subdivision argument.
Comments18 pages, 1 figure, ancillary files verify computer-assisted component. v2: Added a non-computational proof of the main tensor nonvanishing theorem and reorganized the manuscript to present this proof first. The original computer-assisted proof and verification files are retained as an independent proof
