AI 中文总结
本文针对垂直参数化多项式方程组，通过关联环面向量丛，给出其达到复非零解最大数量的充要条件，并推导了该最大零点数量的新公式。
AI 中文摘要
我们针对垂直参数化多项式方程组的特化情形，给出其达到复非零解最大数量的充要条件。对每个垂直方程组，我们关联一对对象：一个射影单纯环面簇和一个环面向量丛。该环面向量丛可对环面簇Cox环中的多项式进行齐次化，从而得到一个齐次理想，其在环面上的零点与原方程组的零点一致。我们证明，当且仅当该理想在环面簇的面上无任何解时，孤立解的最大数量得以达到，且该情形对参数的 generic 值成立。此外，我们给出了该 generic 零点数量的新公式，其形式为关联于环面向量丛的多面体族上混合体积的交替和。
英文摘要
We provide necessary and sufficient conditions for the specializations of a vertically parametrized polynomial system to attain the maximal number of complex nonzero solutions. To each vertical system, we attach a pair consisting of a projective simplicial toric variety and a toric vector bundle. The toric vector bundle allows for a homogenization of the polynomials in the Cox ring of the toric variety, providing a homogeneous ideal with the same zeros over the torus as the original system. We show that the maximal number of isolated solutions is attained if and only if this ideal has no solutions in the faces of the toric variety and prove that this happens for generic values of the parameters. In addition, we provide a novel formula for this generic number of zeros as an alternating sum of mixed volumes over a family of polytopes attached to the toric vector bundle.
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