AI 中文总结
本文证明函数域上 Waring 问题在 n>2d 时的预期渐近性，通过聚合小弧估计和相交理论获得最优范围及显式下界。
AI 中文摘要
我们证明了在 $\mathbb F_q[T]$ 上 Waring 问题的预期渐近性，并带有幂次节省误差项，只要 $n>2d$，特征大于 $(d-1)^2$，且 $q$ 满足一个显式下界。该范围对于目标多项式均匀的预期渐近性而言，一般情况下是最优的。我们的主要新输入是一个聚合小弧估计：我们根据其相关奇异轨迹的余维数对泛函进行计数，并使用相交理论来界定所得参数空间的次数。特别地，若 $n\ge (2+\varepsilon)d$，则对 $q$ 所需的下界是关于 $d$ 的次数为 $2+4/\varepsilon$ 的多项式。
英文摘要
We prove the expected asymptotic in Waring's problem over $\mathbb F_q[T]$, with a power-saving error, whenever $n>2d$, the characteristic is greater than $(d-1)^2$, and $q$ satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if $n\ge (2+\varepsilon)d$, the required lower bound on $q$ is polynomial in $d$ of degree $2+4/\varepsilon$.
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