关于实多维有理K-矩问题

The multidimensional truncated Moment Problem: The Moment Cone

Let $\mathsf{A}=\{a_1,\dots,a_m\}$, $m\in\mathbb{N}$, be measurable functions on a measurable space $(\mathcal{X},\mathfrak{A})$. If $μ$ is a positive measure on $(\mathcal{X},\mathfrak{A})$ such that $\int a_i dμ<\infty$ for all $i$, then the sequence $(\int a_1 dμ,\dots,\int a_m dμ)$ is called a moment sequence. By Richter's Theorem each moment sequence has a $k$-atomic representing measure with $k\leq m$. The set $\mathcal{S}_\mathsf{A}$ of all moment sequences is the moment cone. The aim of this paper is to analyze the various structures of the moment cone. The main results concern the fac…

计算多维残差

Computing Multidimensional Residues

给定n个多项式在n个变量与有限数量的复根，对于任何他们的根有一个当地的剩余运营商分配一个复数的任何多项式。这是系数的代数函数，但通常不是有理函数。另一方面，整体剩余，这是所有根上的局部剩余之和，合理地依赖于系数。本文论述了符号算法评估的有理函数。在变形到初始形式是平坦的假设下，对于变量上的某些权重的选择，我们将整体剩余表示为关于初始形式的单个剩余积分。当输入方程是一个Groebner的基础，这导致一个有效的级数展开算法的全球残留物，并消失定理相对于相应的锥在Groebner风扇。多项式的整体剩余等于其（Groebner基）范式的最高系数，相反，整个范式用整体剩余表示。这产生了一种方法，用于评估零维完全相交的轨迹。应用包括计算实根，计算多项式映射的次数，以及多元对称函数的求值。所有的算法都说明了一个显式系统中的三个变量。

The multidimensional truncated Moment Problem: Shape and Gaussian Mixture Reconstruction from Derivatives of Moments

In this paper we introduce the theory of derivatives of moments and (moment) functionals to represent moment functionals by Gaussian mixtures, characteristic functions of polytopes, and simple functions of polytopes. We study, among other measures, Gaussian mixtures, their reconstruction from moments and especially the number of Gaussians needed to represent moment functionals. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians but not less. Hence,…

Rational singularities, quiver moment maps, and representations of surface groups

We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the representation zeta function of the special linear group over the integers and over the p-adic integers.

全负箭图矩映射的有理奇点

Rational singularities for moment maps of totally negative quivers

本文证明全负箭图矩映射的零纤维具有有理奇点，通过推广Budur的射流空间维数界方法，并将该性质传递到2-Calabi-Yau范畴的模空间，进而推广Wyss的有限域射流计数渐近结果并给出p进体积解释。