Title: Roots of real-valued zero mean maps: Compositions of linear functionals and equivariant maps Show Chinese title

Title: 实值零均值映射的根：线性泛函和等变映射的复合

Abstract: We develop a novel topological framework that yields results constraining the distribution of zeros of certain zero mean real-valued maps, namely those obtained from composing a fixed equivariant map with linear functionals. We use this framework to establish upper bounds for the topology of set systems in the domain where (multivariate) trigonometric polynomials do not change their sign, generalizing and, in certain regimes, strengthening results in the literature. Our results more generally contain restrictions on the distribution of zeros of Chebyshev spaces as special cases. Lastly, we apply this framework to derive existence results for efficient cubature rules for compositions of affine functionals and equivariant maps. Show Chinese abstract

Abstract: 我们开发了一个新的拓扑框架，该框架产生结果，限制了某些零均值实值映射的零点分布，即那些通过将固定等变映射与线性泛函复合得到的映射。 我们使用这个框架来建立在域中（多变量）三角多项式不改变符号的集合系统的拓扑上界，推广并加强了文献中的某些结果。 我们的结果更普遍地包含切比雪夫空间的零点分布限制作为特殊情况。 最后，我们将这个框架应用于推导仿射泛函和等变映射复合的高效求积规则的存在性结果。