Complex Variables Analogues of Hilbert's Seventeenth Problem

建议：DMS-9970024首席研究员：John P.D‘Angelo摘要：D’Angelo的项目将继续他对希尔伯特第十七问题(由于Catlin和D‘Angelo)的复变量版本的研究，并将其应用于复杂的几何、分析和代数。他建议得到全纯丛的Catlin-D‘Angelo等距嵌入定理的推论，并将结果推广到某些退化度量。这项工作直接涉及到不同维度的球之间的适当全纯映射的分类问题。希尔伯特第十七个问题的复变量版本已被证明，其结果如下。假设在复欧氏空间上给出一个多项式，它不会在闭单位球上消失。这个多项式必须是球之间有理适当映射的分母(减少到最低项)，但目标维度需要选择得足够大。因此，限制目标尺寸限制了球之间适当映射的复杂性。法兰的工作进一步证明了这一点。D‘Angelo试图证明一个一般定理，该定理为不同维度的单位球之间的适当有理映射度提供了一个锐界。这将为关于不同维度的CR流形之间的CR映射的更一般的猜想提供证据。在1900年的国际数学家大会上，David Hilbert提出了一系列研究问题，这些问题对本世纪数学的发展产生了巨大的影响。Artin在1925年前后解决了Hilbert最初的第17个问题，证明了多个实变量中的非负值的多项式一定是有理函数的平方和。这一结果对于多个复变数多项式的自然类比是不成立的。鉴于复变量理论在物理、工程和纯数学中的重要性，考虑复多项式的希尔伯特问题的其他形式是很自然的。近年来，D‘Angelo发现，这种类似物在研究被称为不同复杂维度的球之间的“适当映射”的对象中发挥着至关重要的作用。D‘Angelo和Catlin后来在复杂的环境中建立了一个版本的Artin定理。它在适当映射中的应用使人们能够理解单位球在更高维度中的作用，从而为研究静电学、扩散和流体流动提供了一个多维工具。最近，Catlin和D‘Angelo证明了一个主要结果，该结果推广了他们早期的定理，把它们放在一个抽象的框架中，即所谓的全纯向量丛的框架中。这个证明给出了Bergman核函数(分析的一部分)在代数问题中的意外应用，以及对希尔伯特问题的几何重新解释。这一结果可能对现代数学物理(如弦理论和超对称)中全纯向量丛起重要作用的部分具有相当大的意义。D‘Angelo是1999年伯格曼奖的获得者，这项工作在引文中被提到。D‘Angelo目前的提案建议对这些想法进行各种扩展和应用。

Proposal: DMS-9970024Principal Investigator: John P. D'AngeloAbstract: D'Angelo's project will continue his investigation of a complex variables version of Hilbert's seventeenth problem (due to Catlin and D'Angelo) and give applications of it to complex geometry, analysis, and algebra. He proposes to derive corollaries of the Catlin-D'Angelo isometric imbedding theorem for holomorphic bundles, and to extend the result to certain degenerate metrics. This work bears directly on the problem of classifying proper holomorphic mappings between balls in different dimensions. The complex variables version of Hilbert's seventeenth problem already proved has the following consequence. Suppose one is given a polynomial on complex Euclidean space that does not vanish on the closed unit ball. This polynomial must then be the denominator of a rational proper mapping (reduced to lowest terms) between balls, but the target dimension needs to be chosen sufficiently large. Thus restricting the target dimension constrains the complexity of a proper map between balls. Work of Faran gives further evidence of this. D'Angelo seeks to prove a general theorem providing a sharp bound for the degree of a proper rational mapping between unit balls in different dimensions. This will provide evidence for a more general conjecture about CR mappings between CR manifolds of different dimensions.In his famous address to the International Congress of Mathematicians in 1900, David Hilbert posed a series of research problems that have greatly influenced the development of mathematics in this century. Artin solved Hilbert's original Seventeenth Problem around 1925 by proving that a polynomial in several real variables whose values are nonnegative must be a sum of squares of rational functions. The natural analogue of this result for polynomials of several complex variables does not hold. Given the importance of complex variable theory in physics, engineering, and pure mathematics, it is natural to consider other facsimiles of Hilbert's problem for complex polynomials. In recent years, D'Angelo discovered that such analogues play a crucial role in the study of objects known as "proper mappings" between balls in different complex dimensions. D'Angelo and Catlin later established a version of Artin's theorem in the complex setting. Its application to proper mappings enables one to understand the role of the unit sphere in higher dimensions and thus furnishes a multidimensional tool for the study of electrostatics, diffusion, and fluid flow. More recently, Catlin and D'Angelo have proved a major result that generalizes their earlier theorems by casting them in an abstract framework, the framework of so-called holomorphic vector bundles. The proof gives an unexpected application of the Bergman kernel function (a part of analysis) to a problem in algebra, as well as a geometric reinterpretation of Hilbert's problem. The result may have considerable significance for the parts of modern mathematical physics (such as string theory and supersymmetry) where holomorphic vector bundles play a large role. D'Angelo is the 1999 winner of the Bergman Prize, and this work is mentioned in the citation. D'Angelo's current proposal suggests various extensions of and applications for these ideas.