真实和复杂世界中的Rolle模型
Rolle models in the real and complex world
摘要：
许多问题的分析（真正的和复杂的）和几何（分析，代数，丢番图等）可以简化为计算方程组的“解的数目”，方程组由代数等式和具有代数右端的微分方程定义（普通和Pfiran）。在纯代数的情况下，范例是由Bézout定理：孤立的解决方案系统的多项式方程的次数$\le d$在$n$维空间不超过$d^n$，约束多项式在$d$和指数在$n$。这个界限是最优的：如果我们正确地计算解决方案（即，具有多重性，包括复数解和无限超平面上的解），则等式成立。这个范例可以被推广到如上所述的先验情况。事实证明，一些计数问题承认类似的界限只取决于度和维度，而其他问题只能局部处理，即，在某些有限大小的域中，允许解的个数有界。只有一类计数问题是有界的，但除了度和维数外，其答案还依赖于相应的Pfidian系统的高度。这些结果的统一特征是其证明的核心事实。这一事实可以被视为各种遥远的推广罗尔定理已知从本科微积分，声称任何两个根之间的一个单变量可微函数的一段必须谎言的根源，其衍生物。我们讨论了向量值和复杂的解析函数（没有一个是直接的）和芽的全纯映射的罗尔定理的推广。
英文摘要：
Numerous problems of analysis (real and complex) and geometry (analytic, algebraic, Diophantine e.a.) can be reduced to calculation of the ``number of solutions'' of systems of equations, defined by algebraic equalities and differential equations with algebraic right hand sides (both ordinary and Pfaffian). In the purely algebraic context the paradigm is given by the Bézout theorem: the number of isolated solutions of a system of polynomial equations of degree $\le d$ in the $n$-dimensional space does not exceed $d^n$, the bound polynomial in $d$ and exponential in $n$. This bound is optimal: if we count solutions properly (i.e., with multiplicities, including complex solutions and solutions on the infinite hyperplane), then the equality holds. This paradigm can be generalized for the transcendental case as described above. It turns out some counting problems admit similar bounds depending only on the degrees and dimensions, whereas other can be treated only locally, i.e., admit bounds for the number of solutions in some domains of limited size. There is only one class of counting problems, which admits global bounds, but besides the degree and dimension, the answer depends on the height of the corresponding Pfaffian system. The unifying feature for these results is the core fact that lies at the heart of their proofs. This fact can be regarded as a variety of distant generalizations of the Rolle theorem known from the undergraduate calculus, claiming that between any two roots of a univariate differentiable function on a segment must lie a root of its derivative. We discuss generalizations of the Rolle theorem for vector-valued and complex analytic functions (none of them straightforward) and for germs of holomorphic maps.
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