光滑锥对分的极大元

对于第一象限中的光滑锥在对分下的极大元的个数问题给出了极大元的判别方法,证明了该问题对应于实代数簇在被超平面界定的区域内的整点等价类个数.在二维的情形下,证明了极大元唯一;在三维和更高维数的情形下证明了极大元个数无穷多,并分别给出了具体例子.

Extremal element of smooth cone under bisection

We study the number of extremal elements of smooth cones in the first orthant under bisection. We give a criterion for a smooth cone to be an extremal element under bisection, and prove that the number is equal to the number of equivalent classes of integer points in the region of a real algebraic variety cut by hyperplanes. In 2 dimensional case, we prove the uniqueness of extremal element. In 3 dimensional and higher cases, we can only prove that there are infinite extremal elements.