We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic varieties. As a first application, we provide a general incidence estimate that is tight in its dependence on the size, degree and dimension of the varieties involved.

中文翻译：

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变体上的多项式方法

我们建立了使多项式方法适应任意变体的敏锐估计。其中包括划分定理、对固定集合上消失的多项式的估计以及实代数簇的连通分量数的界限。作为第一个应用，我们提供了一个普遍的发生率估计，它与所涉及品种的大小、程度和维度密切相关。