齐次型不等式
Homogeneous Forms Inequalities
AI总结：
本文研究齐次型不等式，显式确定实半代数区域体积并联系 Sato-Bernstein 多项式最大根，证明其重数等于 zeta-分布最小极点阶，解决 D-模经典问题；同时建立随机扭曲下 Oppenheim 猜想的有效度量版本并解决 Sarnak 猜想。
摘要：
给定一组由齐次型确定的不等式，建立了以下相互关联的结果：（1）由这些不等式确定的实半代数区域的体积被显式确定；证明它与所谓的 Sato-Bernstein 多项式（该多项式关联于从给定齐次型集合导出的一个多元多项式）的最大根有关；（2）与该结果相关，证明了一个多元多项式的 Sato-Bernstein 多项式的最大根的重数，与附着于该多项式的复亚纯 zeta-分布的最小极点的阶一致。这解决了 D-模理论中的一个经典问题；（3）在齐次型被随机幺模矩阵扭曲的情形下，建立了 Oppenheim 猜想的一个度量、一致且有效的版本。这回答了 Athreya 和 Margulis（2018）提出的一个问题。一个相关的度量估计也得到解决，该估计计数扭曲不等式组在整数格点中的解的数量。（4）在齐次型集合固定的确定性情形下，证明了所考虑的不等式组的整数解计数函数满足一个上界。该估计中的误差项被证明具有幂次节省，只要由几何层析成像产生的一个平坦性定量测度足够大。这解决了 Sarnak（1997）提出的一个猜想。
英文摘要：
Given a set of inequalities determined by homogeneous forms, the following intertwined results are established: (1) the volume of the real semi-algebraic domain determined by these inequalities is explicitly determined; it is shown to be related to the largest root of the so-called Sato-Bernstein polynomial associated to a multivarite polynomial derived from the given set of homogeneous forms; (2) in relation with this result, the multiplicity of the largest root of the Sato-Bernstein polynomial of a multivariate polynomial is shown to coincide with the order of the smallest pole of the complex meromorphic zeta-distribution attached to it. This settles a classical problem in the theory of D-modules; (3) in the case that the homogeneous forms are twisted by random unimodular matrices, a metric, uniform and effective version of the Oppenheim conjecture is established. This answers a problem raised by Athreya and Margulis (2018). So does a related metric estimate counting the number of solutions in integer lattice points to the set of twisted inequalities. (4) in the deterministic case where the set of homogeneous forms is fixed, an upper bound is proved to hold for the function counting the number of integer solutions to the system of inequalities under consideration. The error term in this estimate is shown to admit a power saving provided that a quantitative measure of flatness emerging from geometric tomography is large enough. This settles a conjecture stated by Sarnak (1997).
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