改进的亚线性投影Schwartz-Zippel界及（亚）二次维度增长界在任意余维数中的应用
Improved, sublinear projective Schwartz-Zippel and (sub)quadratic dimension growth bounds in arbitrary codimension
AI总结：
本文改进了投影Schwartz-Zippel界，通过亚线性方法在低维情况下获得亚二次维度增长界，并将二次界扩展到任意余维数。
摘要：
我们致力于解决Cluckers和Glazer在[CG25]中提出的问题，即缩小最坏情况下的维度增长上界和下界之间的差距。为此，我们引入了一个亚线性的Schwartz-Zippel界的改进版本。我们证明了多个案例，包括线性次结构的配置。这导致在某些低维情况下得到亚二次维度增长界，优于Binyamini、Cluckers和Kato在[BCK25]中得到的二次依赖关系。我们引入了一个自然的投影论证，利用拉回并用此方法解决Cluckers和Glazer提出的第二个问题，将[BCK25]中的二次维度增长界扩展到任意余维数。
英文摘要：
We introduce a sublinear sharpened version of a classical projective variant of the Schwartz-Zippel bounds. Here, the sublinearity is in the degree, whereas the classical Schwartz-Zippel bounds are linear in the degree. We prove such sublinear bounds in several cases, including for unions of linear varieties. These bounds lead to a subquadratic dependence on the degree for projective dimension growth bounds in low dimensions. This improves on the quadratic dependence obtained by Binyamini, Cluckers and Kato in [BCK25] and thus brings the dimension growth upper bounds and lower bounds for the worst case closer together, a question asked by Cluckers and Glazer in [CG25]. We also address another question from [CG25], to treat the case of general codimension. To this end, we introduce a natural projection argument with pull-backs. We also give our main results for any global field instead of just $\mathbb{Q}$.
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