AI 中文总结
该研究针对Pfaffian方程组非退化解个数的Khovanskii型Bezout定理，给出了避开积分流形一般理论的自包含证明，改进了经典界的依赖条件并得到Pfaffian集连通分支数的改进界。
AI 中文摘要
我们给出了Pfaffian方程组非退化解个数的Khovanskii型Bezout界的直接自包含证明。我们提取了Khovanskii原始论证的核心要素，将其整合为一个避开其专著中发展的积分流形一般理论的证明。我们的表述略微改进了经典结论：该界不再依赖于环境维数，而是取决于Pfaff链中任一函数所依赖的变量的最大个数。作为推论，我们得到了Pfaffian集连通分支数的改进界。
英文摘要
We present a direct and self-contained proof of Khovanskii's Bezout-type bound for the number of nondegenerate solutions of a system of Pfaffian equations. We isolate the ingredients of Khovanskii's original argument and assemble them into a proof that avoids the general theory of integral manifolds developed in his monograph. Our formulation mildly refines the classical statement: rather than depending on the ambient dimension, our bound depends on the maximum number of variables on which any function in the Pfaffian chain depends. As a consequence, we obtain a refined bound on the number of connected components of a Pfaffian set.
Comments16 pages, 2 figures
