In this paper, we uncover a new uncertainty principle that governs the complexity of Boolean functions. This principle manifests as a fundamental trade-off between two central measures of complexity: a combinatorial complexity of its supported set, captured by its Vapnik-Chervonenkis dimension ($\mathrm{VC}(f)$), and its algebraic structure, captured by its polynomial degree over various fields. We establish two primary inequalities that formalize this trade-off: $\mathrm{VC}(f)+\mathrm{deg}(f)\ge n,$ and $\mathrm{VC}(f)+\mathrm{deg}_{\mathbb{F}_2}(f)\ge n$. In particular, these results recover the classical uncertainty principle on the discrete hypercube, as well as the Sziklai--Weiner's bound in the case of $\mathbb{F}_2$.

翻译：本文揭示了一个支配布尔函数复杂性的新不确定性原理。该原理表现为两种核心复杂性度量之间的基本权衡：一是由其支撑集的组合复杂性所捕获的Vapnik-Chervonenkis维数（$\mathrm{VC}(f)$），二是其在不同域上的多项式次数所捕获的代数结构。我们建立了两个核心不等式来形式化这种权衡关系：$\mathrm{VC}(f)+\mathrm{deg}(f)\ge n$ 与 $\mathrm{VC}(f)+\mathrm{deg}_{\mathbb{F}_2}(f)\ge n$。特别地，这些结果不仅恢复了离散超立方体上的经典不确定性原理，也涵盖了$\mathbb{F}_2$域情形下的Sziklai--Weiner界。