设 f:=(f^1,...,f^n) 为一个稀疏随机多项式系统。这意味着每个 f^i 具有固定的支撑集（可能非零系数的列表），且每个系数服从任意方差的高斯概率分布。我们将 f 在区域 U 内根的期望数量表示为 U 上某个混合体积形式的积分。当 U = (C^*)^n 时，可恢复经典的混合体积。本文的主要结果是对 f 在区域 U 上的条件数大于 1/epsilon 的概率的一个界。该界依赖于混合体积形式在 U 上的积分，以及 U 作为环流形子集的某个内在不变量。还考虑了实系数多项式，并给出了实根期望数量和条件数的界。讨论了稀疏随机多项式系统的零点、Kähler 几何与力学（动量映射）之间的联系。
英文摘要：
Let f:=(f^1,\...,f^n) be a sparse random polynomial system. This means that each f^i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance.
We express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form. When U = (C^*)^n, the classical mixed volume is recovered.
The main result in this paper is a bound on the probability that the condition number of f on the region U is larger than 1/epsilon. This bound depends on the integral of the mixed volume form over U, and on a certain intrinsic invariant of U as a subset of a toric manifold.
Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given.
The connection between zeros of sparse random polynomial systems, Kahler geometry, and mechanics (momentum maps) is discussed.
Reconstructing sparse exponential polynomials from samples: Stirling numbers and Hermite interpolation
Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung & FORWISS, University of Passau（帕绍大学数字图像处理重点数学研究所与FORWISS）；University of Passau（帕绍大学）
Sparse polynomial approximation for optimal control problems constrained by elliptic PDEs with lognormal random coefficients
Oden Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Department of Mechanical Engineering, and Department of Geological Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校机械工程系和地质科学系）
A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions
Computer Science Research Institute, Sandia National Laboratories（桑迪亚国家实验室计算机科学研究所）；Scientific Computing and Imaging (SCI) Institute and Mathematics Department, University of Utah（犹他大学科学计算与成像研究所及数学系）；Institute of Computational Mathematics and the Chinese Academy of Sciences（中国科学院计算数学研究所）
Department of Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium（荷语鲁汶大学电气工程系（ESAT））；Department of Mathematics, University of British Columbia, Vancouver, BC, Canada（不列颠哥伦比亚大学数学系）
