关于函数域中的覆盖半径
On Covering Radii in Function Fields
AI总结：
本文研究了正特征设定下数的几何学中的覆盖半径问题，找到了凸体覆盖半径的闭式，证明了Woods和Minkowski猜想的正特征类似物，并得出正特征下Gruber-Mordell谱在所有维度均平凡的结论。
摘要：
在本文中，我们将讨论正特征设定下的数的几何学主题，例如覆盖半径。我们找到了关于凸体的覆盖半径的闭式，这将导出该设定下 Woods 猜想的正特征类似物的证明。然后，我们将证明关于乘性覆盖半径的 Minkowski 猜想的正特征类似物。为此，我们将证明 Solan 结果的正特征类似物，即每条对角轨道都与良格集相交。这意味着正特征下的 Gruber-Mordell 谱在所有维度中都是平凡的。
英文摘要：
In this paper, we shall discuss topics in geometry of numbers in the positive characteristic setting, such as covering radii. We find a closed form for covering radii with respect to convex bodies, which will lead to a proof of the positive characteristic analogue of Woods' conjecture in this setting. Then, we will prove a positive characteristic analogue of Minkowski's conjecture about the multiplicative covering radius. To do this, we shall prove a positive characteristic analogue of Solan's result that every diagonal orbit intersects the set of well rounded lattices. This implies that the Gruber-Mordell spectrum in positive characteristic is trivial in every dimension.
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